Chapter 2

Primer on Pricing Risky Securities

2.1 Stocks and Stock Price Models

2.1.1 Underlying Assets and Derivative Securities

Let us start with a basic classification of financial instruments. First, we differentiate between underlying assets (or securities) and derivative financial contracts. A financial security is a legal claim to some future benefit. Bonds, bank deposits, common stocks, and the like are all examples of financial securities. In contrast, a general financial contract links nominally two (or more) parties. Such a contract specifies conditions under which payments or payoffs are to be made between the parties. The main example is derivative contracts whose payoff depends on the value of another financial variable such as price of a stock, price of a bond, market index, or exchange rate called underlying assets. Examples include forward contracts, futures, swaps, and options.

In Chapter 1, we discussed various types of financial assets and investments, such as bank accounts, bonds, annuities, mortgages, and other types of loans. They all have one thing in common. It is the assumption that all future payments are guaranteed. However, in any investment portfolio, where a certain amount is invested today and future cash flows are returned over time, there is always the possibility that not all future payments will actually be received or the amount of future payments is not certain. For most investments, future cash flows are contingent upon the financial position of the company issuing the investment. On the other hand, there are many financial assets (or securities), such as stocks, foreign currencies, or commodities, whose future market values are unpredictable, because they depend on the choices and decisions made by a great number of agents acting under conditions of uncertainty. Such assets can be viewed as “risky” assets in comparison with “risk-free” assets with certain future cash flows. There are several types of risk involved in the purchase and sale of financial assets. These include economic risk, interest rate sensitivity, the possibility of company failure, company management problems, competition with other companies, and governmental rulings that may negatively affect the company. It is reasonable to assume that future prices of risky assets depend on random factors. That is, the asset values can be treated as random variables.

A corporation that needs funds for its development or expansion may issue stock to investors. Stock represents ownership of a corporation. Owners of common stock have voting rights and are entitled to the earnings of the company after all obligations are paid. If an investor purchases shares of stock in the company, then such an investor assumes a large amount of risk in return for the possibility of growth of the company and a corresponding increase in the value of the shares. It is important to realize that the corporation receives its money when the stock shares are issued. Any trading after that point takes place between the shareholders and the people wishing to purchase the stock, and does not directly represent a profit or loss to the company. Investors who purchase stock may receive dividends periodically (usually quarterly). Thus, the investor may profit by either an increase in the value of the stock or by receipt of dividends. The price that an investor is willing to pay for a share of common stock is based upon the investor's expectations regarding dividends and the future price of the stock.

In order to buy or sell shares of stocks, the investor uses an investment firm registered with the appropriate governmental agencies. The fee or commission charged for the transaction is an important consideration. The person actually making the transactions for the firm is called a broker. If an investor believes that a stock is going to decrease in value, then the investor may borrow shares from the broker (if such a transaction is possible), and then sell the stock. Such a financial operation is called short selling. If the stock decreases in value, then the investor may purchase an equivalent number of shares at the lower price and use those shares to repay the loan.

Stock market indices are used to compute an “average” price for groups of stocks. A stock market index attempts to mirror the performance of the group of stocks it represents through the use of one number, the index. Indices may represent the performance of all stocks in an exchange or a smaller group of stocks, such as an industrial or technological sector of the market. In addition, there are foreign and international indices. Examples are Dow Jones, Standard and Poor's, and NASDAQ indices.

The most well-known example of a derivatives is an option contract, which is defined just below.

Definition 2.1.

An option is a contract that gives its buyer the right, but not the obligation, to buy (for a call option) or to sell (for a put option) a specified asset at a specified price (called the exercise price or strike price) on or before a specified date (called the expiry date or maturity date).

An option is an example of a derivative financial contract, whose value is derived from the values of other underlying assets. Options can be written for numerous products, such as gold, wheat, tulip bulbs, foreign exchange, movie scripts, stocks, etc. The purchase price of an option is called the premium. An option is said to be in the money if exercising the option yields a profit, excluding the premium. It is out of the money if exercising the option is unprofitable. If the purchase price of the underlying asset is equal to the exercise price, then the option is said to be at the money.

In this chapter we deal with stock options (also called equity options), which give the holder the right to buy (or to sell) a stock for a specified price during a specified time period. Thus, the buyer of an option has the right, but not the obligation, to buy or sell the stock. A call option gives the right to buy one unit (share) of the underlying stock at the strike price. A put option gives the right to sell one unit of the underlying stock at the strike price. The seller (writer) of an option must sell or buy the asset at the strike price once the option is exercised. An American option gives its holder the right to buy or sell an asset for the strike price at any time before or at the expiration date. A European option gives the same right, except the option may only be exercised on the expiration date. The terms American and European refer to the type of option, not the geographical region where the options are bought or sold.

2.1.2 Basic Assumptions for Asset Price Models

In practice, financial mathematicians generally make the following assumptions when dealing with asset price models.

Not moving the market. Our actions do not affect the market prices. In other words, we can buy or sell any amount of assets without affecting their prices. Clearly, this is not true for a free market, since an increasing demand moves market prices up.

Liquidity. At any time we can buy or sell as much as we wish at the market price without being forced to wait until a counter-party can be found. A liquid asset can be sold rapidly, any time within market hours. Cash is the most liquid asset. A market is liquid when there are ready and willing buyers and sellers at all times.

Shorting. We can have a negative amount of an asset by selling assets we do not hold. In this case we say that a short position is taken or that the asset is shorted. A short position in bonds means that the investor borrows cash and the interest rate is determined by the bond prices. A short position in stock means that the investor borrows the stock, sells it, and uses the proceeds to make other investments. The opposite of going “short,” i.e., holding an asset, is sometimes called being “long” in the asset.

Fractional quantities. We are able to purchase fractional quantities of any asset. It is a reasonable assumption when the size of a typical financial transaction is sufficiently larger than the smallest unit one can hold.

No transactions costs. We can buy and sell assets without paying any additional fees. In the market, one of the typical ways to collect transaction costs is that buy and sell prices differ slightly. Such a difference in prices is called a bid–ask spread. The size of the bid–ask spread is closely related to liquidity. For a very liquid asset, the bid–ask spread will be quite small.

Stochastic prices. The future prices of financial assets are uncertain. Thus, we deal with stochastic asset price models. All factors that can affect the outcome of an economy are commonly called risk factors. We assume that all prices are equilibrium prices. We are not concerned with modelling the mechanism by which prices equilibrate.

Model assumptions. At the time we introduce asset price models, some assumptions on the models will be stated.

Basic Components of a Financial Model:

• The time horizon T > 0 is a date at which trading activity stops.
• Trading dates are calendar dates between the initial time t = 0 and time horizon t = T at which trading can be allowed to take place.
• A state of the world ω characterizes the “real-world” state, which is relevant to the economic environment we wish to model. Such outcomes ω are also called market scenarios or states of the world. The set of all feasible scenarios or states of the world, denoted by Ω, is called the state space.
• Tradable base assets are underlying assets available for trading. The price of any derivative instrument depends on the current price or history of prices of the corresponding underlying asset.
• Trading rules such as allowance of short selling, presence of taxes and transaction costs, should be specified.

Financial models can be categorized as either of two general types: continuous-time or discrete-time. For a continuous-time model, the allowable time moments and trading dates form a continuous interval [0, T] with specified initial and final calendar times 0 and T, respectively. For discrete-time models, assets are observed at only a finite set of calendar dates {0, 1, ..., T}. The discrete-time models can be further subdivided into single period models with only two relevant trading dates consisting of the current date t = 0 and the maturity date t = T and multi-period models, where trading takes place at several dates. Financial models can also be categorized with respect to the state space Ω, which can be finite or infinite. In this section, Ω is assumed to be finite.

Consider some asset (or financial security) denoted by A or S. The price of asset A at time t will be denoted by At (or A(t) in some cases). The price is assumed to be positive for all times t. The collection of asset prices indexed by time, {At}t∊T, is called the price process of asset A. Here, we have T = {0,T} for a single-period model, T = {0, 1, ..., T} for a multi-period discrete-time model, and T = [0, T] for a continuous-time model. At the present time t = 0, the current price A0 is known to all investors. In general, the future values of any asset are uncertain. From the mathematical point of view, the prices At, 0 ≤ t ≤ T, are positive random variables on the state space Ω. Therefore, the price process {At}t∊T is nothing but a sequence of random variables indexed by time. Such a sequence of random variables is called a random or stochastic process. The notation At(ω) is used to denote the price at time t given that the market follows scenario ω ∊ Ω. Fixing ω gives a particular realization of the asset price process for the given scenario. Such a realization At(ω) considered as a function of time t ∊ T is called a sample asset price path or simply a sample path.

2.2 Basic Price Models

2.2.1 A Single-Period Binomial Model

Consider an investor who is faced with an economy whose future state is not known for certain. Let us construct the simplest possible model that describes such an economy. First we suppose that there are only two dates: a current date labeled 0 and a future (maturity) date labeled T. Since the economy is uncertain, there should be at least two future states, each corresponding to one of two possible scenarios or outcomes. Let these two states in our model be denoted by ω− and ω+ . For example, these states can respectively represent “bad” news and “good” news. So our first financial model is a single-period model with the state space Ω = {ω−,ω+}.

Now we come back to the investor. The wealth V of the investment in underlying assets is a function of time: V = Vt, t ∊ {0, T}. It is reasonable to assume that the initial wealth V0 of the investment is known at time t = 0. However, the terminal wealth VT at the maturity date is uncertain and is a function of state ω ∊ Ω. There are two states; hence VT can take on two possible values, VT(ω−) and VT(ω+).

To further develop our model, we first need to quantify the chances of finding our economy in either of the possible states of Ω. Second, we need to specify what assets are available to form an investment portfolio. Assume that state ω+ occurs with probability p ∊ (0, 1). Thus, state ω− occurs with probability 1 − p. So the terminal wealth VT can be viewed as a random variable on the finite probability space (Ω, $ℱ$, ℙ) with Ω as a scenario set and ℙ as a probability distribution function defined on the collection of events F = {∅, {ω−}, {ω+}, Ω} as follows:

$\mathbb{P}\left(E\right)=\{\begin{array}{ll}0\hfill & \text{if}E=\phi ,\hfill \\ p\hfill & \text{if}E=\left\{{\omega }^{+}\right\},\hfill \\ 1-p\hfill & \text{if}E=\left\{{\omega }^{-}\right\},\hfill \\ 1\hfill & \text{if}E=\Omega =\left\{{\omega }^{-},{\omega }^{+}\right\}.\hfill \end{array}$

We note that ℙ(E) represents the real-world (physical) probability of event E. We will see shortly that the fair price of a derivative contract does not depend on the real-world probabilities {p, 1 − p}.

We consider the case with only two tradable base assets, namely, a risk-free bond B and a risky asset such as a stock S. Let St and Bt, t ∊ {0, T}, be the respective time-t prices of the stock and the bond. The initial prices S0 and B0 are positive constants. The terminal prices ST and BT are positive random variables on the probability space (Ω, F, ℙ). The bond is a risk-free asset iff the variable BT is certain, i.e., BT(ω+) = BT(ω−) ≡ BT. The stock is a risky asset iff the variable ST is uncertain, i.e., ST(ω+) ≠ ST(ω−). This is depicted in Figure 2.1, where for convenience we take S+ ≡ ST(ω+) > ST(ω−) ≡ S−.

A static portfolio in the bond and stock is a pair of real numbers (x, y) ∊ ℝ2 that represents the positions (fixed in time) of the investment in the stock and bond:

$x=\#\text{of}\text{shares}\text{or}\text{units}\text{in}\text{the}\text{stock}\text{and}y=\#\text{of}\text{shares}\text{or}\text{units}\text{in}\text{the}\text{bond}.$

If x > 0 (or y > 0) then the position in the stock (or bond) is said to be long. If x < 0 (or y < 0) then the position in the stock (or bond) is said to be short. Hence, the value of a portfolio (x, y), denoted by Π(x, y), is given by

${\prod }_T^{\left(x,y\right)}=x{S}_t+y{B}_t,t\in \left\{0,T\right\}.$

Note that in this economic model the investor takes on positions (x, y) in the two base assets at initial time t = 0 and holds these positions until the end of the period at time t = T. The initial value Π0 = Π0(x, y) is constant, while the terminal value ΠT is a generally nonconstant random variable on (Ω, F, ℙ). Note that the position x represents the number of units held in the risky asset. Hence, ΠT is constant iff x = 0, in which case the portfolio contains only positions in the bond.

Before proceeding further with the discussion of this two-state single period economy, we note the basic model assumptions as follows.

Divisibility. Positions in the base assets, x and y, may have noninteger value.

Liquidity. There are no bounds on x and y. That is, any asset can be bought or sold on demand at the market price in arbitrary quantities.

Short Selling. The positions x and y may be negative. In this case we say that a short position in the asset is taken or that the asset is shorted (otherwise, we say that an investor has a long position in the asset). A short position in the bond means that the investor borrows cash and the interest rate is determined by the bond prices. A short position in the stock means that the investor borrows shares of the stock, sells them, and uses the proceeds to make other investments.

Solvency. The portfolio value must be nonnegative at all times, Πt ≥ 0 for t ∊ {0, T}.A portfolio satisfying this condition is said to be admissible.

Clearly, there are many possible ways to form an investment portfolio. While the initial wealth of such portfolios is the same, the terminal values may have different distributions. Different investments can be compared by calculating their returns. Return on investment, is the ratio of the terminal value of the investment to the initial value of the investment. The total return on the portfolio (x, y) from time 0 to time T, denoted by RΠ, can be expressed as a weighted sum of returns on the base assets:

$\begin{array}{l}{R}_{\prod }=\frac{{\prod }_T^{\left(x,y\right)}}{{\prod }_0^{\left(x,y\right)}}=\frac{x{S}_T+y{B}_T}{x{S}_0+y{B}_0}=\frac{x{S}_0}{\underset{\equiv w_1}{\underbrace{x{S}_0+y{B}_0}}}\frac{S_T}{\underset{\equiv R_S}{\underbrace{S_0}}}+\frac{y{B}_0}{\underset{\equiv w_2}{\underbrace{x{S}_0+y{B}_0}}}\frac{B_T}{\underset{\equiv R_B}{\underbrace{B_0}}}\hfill \\ =w_1{R}_S+w_2{R}_B,\hfill \end{array}\text{ (2.1)}$

where $R_S:=\frac{S_T}{S_0}$ and $R_B:=\frac{B_T}{B_0}$ are the total returns on the stock and bond, respectively. Note that the return on the bond is nonrandom, i.e., RB = RB(ω±) is a constant. The weights w1 and w2, with w1 + w2 = 1, correspond to the respective fractions of initial wealth invested in the stock and bond. Since the sum of the weights is one, the rate of return on the portfolio (x, y) is a weighted sum of the respective rates of return on the stock and bond:

$\begin{array}{l}{r}_{\prod }=R_{\prod }-1=w_1{R}_S+w_2{R}_B-1=w_1\left(r_S+1\right)+w_2\left(r_B+1\right)-1\hfill \\ =w_1{r}_S+w_2{r}_B.\hfill \end{array}\text{ (2.2)}$

As noted above, it is convenient in what follows to denote the stock price at maturity in the two possible states as S± ≡ ST(ω±) and let S+ > S−. The return on the stock is the random variable whose value on the two states we can simply denote by rS± ≡ rS(ω±), where rS± = S±/S0 − 1 and S± = S0(1 + rS±). From (2.2), we see that the rate of return on the portfolio, rΠ, is a random variable with two possible values: rΠ(ω+)= w1rs+ + w2rB for the higher return outcome and rΠ(ω−) = w1rs− + w2rB for the lower return outcome. Hence, the return on any portfolio without short selling (with positive x, y) always falls in between the largest possible return and smallest possible return on the stock and bond:

$\min \left\{r_B,r_S^{-},r_S^{+}\right\}\le r_{\prod }\le \max \left\{r_B,r_S^{-},r_S^{+}\right\}.$

The terminal value of a portfolio in base assets is a function of the form ΠT : Ω → ℝ. Any such function is called a payoff function (or payoff for short). A nonnegative payoff is called a claim. A payoff is in fact a random variable defined on the probability space (Ω, F, ℙ). Every contract C in our economic model can be specified by the terminal payoff CT and by the initial market price C0 at which the contract is traded. Clearly, a sum (or linear combination) of several payoffs is again a payoff function. Therefore, we can consider a portfolio of contracts, which, in fact, is another contract. A typical financial problem is the evaluation of the fair initial price of a contract with a given terminal payoff. This can be done by constructing a portfolio whose terminal value is equivalent to the payoff of the contract. Such a portfolio is said to replicate the payoff. Being given a replicating portfolio, the fair initial price of the contract is simply equal to the initial cost of setting up the portfolio. An important question that arises, and which we now answer, is as follows.

How do we replicate the payoff of a specified target financial contract with a portfolio in the two base assets B and S? In other words, how do we form an investment portfolio (x, y) whose terminal value coincides with the payoff of the contract in every state?

The claims that we wish to replicate are generally contingent (derivative) claims with uncertain payoff dependent on the outcome. In the two-state economy, any payoff CT has two possible values, C+ ≡ CT(ω+) and C− ≡ CT(ω−). For a contingent payoff, C− ≠ C+ holds; otherwise, when C− = C+, the payoff is said to be deterministic. To replicate a claim CT with a portfolio in B and S, we must form a so-called replicating portfolio (x, y) such that ΠT(x, y)(ω) = CT(ω) for both outcomes ω = ω+ and ω = ω−. The replication is equivalent to solving a system of two linear equations:

$\begin{array}{l}x{S}^{+}+y{B}_T=C^{+},\hfill \\ x{S}^{-}+y{B}_T=C^{-}.\hfill \end{array}$

Since S+ ≠ = S− in the model, this system has a unique solution:

$x=\frac{C^{+}-C^{-}}{S^{+}-S^{-}},y=\frac{C^{-}{S}^{+}-C^{+}{S}^{-}}{B_T\left(S^{+}-S^{-}\right)}.\text{ (2.3)}$

This solution states that, given an arbitrary claim with payoffs C± in the two possible outcomes ω± , we can form a unique replicating portfolio (x, y) with x, y given by (2.3) where ΠT(x, y)(ω±) = C±. We can rewrite (2.3) in terms of the initial prices S0 and B0, the return on the bond, where BT = (1 + rB)B0, and the return on the stock in the two states, where S± = (1 + rS±)S0, as follows:

$x=\frac{C^{+}-C^{-}}{S_0\left(r_S^{+}-r_S^{-}\right)},y=\frac{C^{-}\left(1+r_S^{+}\right)-C^{+}\left(1+r_S^{-}\right)}{B_0\left(1+r_B\right)\left(r_S^{+}-r_S^{-}\right)}.\text{ (2.4)}$

As we now see, and as discussed later in Section 2.3 and more formally and mathematically in-depth in later chapters, replication is the key to fair pricing and valuation of derivative contracts. By replicating the exact payoff structure of a target contract, by means of a portfolio in tradable assets, we are arriving at the fair price of the contract, which is given by the initial cost of setting up the replicating portfolio: C0 = Π0(x, y). In particular, by substituting the above values for x, y, we can represent the initial value of the replicating portfolio, and hence the fair price C0 of the derivative contract, in the following equivalent ways:

$\begin{array}{ll}{C}_0\hfill & ={\displaystyle {\prod }_0^{\left(x,y\right)}=x{S}_0+y{B}_0=\frac{C^{+}-C^{-}}{S^{+}-S^{-}}{S}_0+\frac{C^{-}{S}^{+}-C^{+}{S}^{-}}{B_T\left(S^{+}-S^{-}\right)}{B}_0}\hfill \\ \hfill & =\frac{B_0}{B_T}\left[\left(\frac{\left(B_T/B_0\right)S_0-S^{-}}{S^{+}-S^{-}}\right)C^{+}+\left(\frac{S^{+}-\left(B_T/B_0\right)S_0}{S^{+}-S^{-}}\right)C^{-}\right]\hfill \\ \hfill & =\frac{1}{1+r_B}\left[\left(\frac{\left(1+r_B\right)S_0-S^{-}}{S^{+}-S^{-}}\right)C^{+}+\left(\frac{S^{+}-\left(1+r_B\right)S_0}{S^{+}-S^{-}}\right)C^{-}\right]\hfill \\ \hfill & =\frac{1}{1+r_B}\left[\left(\frac{\left(1+r_B\right)S_0-\left(1+r_S^{-}\right)S_0}{\left(r_S^{+}-r_S^{-}\right)S_0}\right)C^{+}+\left(\frac{\left(1+r_S^{+}\right)S_0-\left(1+r_B\right)S_0}{\left(r_S^{+}-r_S^{-}\right)S_0}\right)C^{-}\right]\hfill \\ \hfill & =\frac{1}{1+r_B}\left[\left(\frac{r_B-r_S^{-}}{r_S^{+}-r_S^{-}}\right)C^{+}+\left(\frac{r_S^{+}-r_B}{r_S^{+}-r_S^{-}}\right)C^{-}\right].\hfill \end{array}\text{ (2.5)}$

It is clear from (2.5) that the value of the replicating portfolio, and hence the initial fair price of the derivative contract, does not depend on the real-world probabilities of the outcomes ω± . Later we shall formally introduce the notions of arbitrage and risk-neutral probabilities, which will bring a more complete meaning to the result encapsulated in (2.5).

Example 2.1.

Suppose B0 = $100, BT = $110, S0 = $100, S+ = $120, S− = $90.

• (a) Determine a portfolio (x, y) whose value at time T is given by ΠT(ω+) = $930 and ΠT(ω−) = $780.
• (b) Find the initial value Π0 of the portfolio constructed in (a) and the rate of return, r, on the portfolio.
• (c) Assuming that $\mathbb{P}\left({\omega }^{+}\right)=\frac{2}{5}$, find the expected rate of return E[r] and the expected terminal value E[ΠT].

Solution.

• (a) Apply (2.3) with C+ = ΠT(ω+) = $930, C− = ΠT(ω−) = $780 to obtain:

  $x=\frac{930-780}{120-90}=5,y=\frac{780\cdot 120-930\cdot 90}{110\cdot \left(120-90\right)}=3.$

• (b) From (2.5) we have Π0 = xS0 + yB0 = 5 · 100 + 3 · 100 = $800. Equation (2.2) gives $r\left({\omega }^{\pm }\right)=\frac{{\displaystyle {\prod }_T\left({\omega }^{\pm }\right)-{\Pi }_0}}{{\Pi }_0}$, hence

  $\begin{array}{l}r\left({\omega }^{+}\right)=\left(930-800\right)/800=0.1625=16.25\%,\hfill \\ r\left({\omega }^{-}\right)=\left(780-800\right)/800=-0.025=-2.5\%.\hfill \end{array}$

• (c) The real-world expected return on the portfolio given that $\mathbb{P}\left({\omega }^{+}\right)=p=\frac{2}{5}$ and $\mathbb{P}\left({\omega }^{-}\right)=1-p=\frac{3}{5}$ is

  $\text{E}\left[r\right]=pr\left({\omega }^{+}\right)+\left(1-p\right)r\left({\omega }^{-}\right)=\frac{2}{5}\cdot 0.1625+\frac{3}{5}.\left(-0.025\right)=0.05=5\%.$

  The expected terminal value is

  $\text{E}\left[{\prod }_T\right]=p{\prod }_T\left({\omega }^{+}\right)+\left(1-p\right){\prod }_T\left({\omega }^{-}\right)=\frac{2}{5}\cdot 930+\frac{3}{5}\cdot 780=\$840.$

  This number is consistent with the fact that the expected return is

  $\text{E}\left[r\right]=\text{E}\left[{\prod }_T/{\prod }_0-1\right]=\text{E}\left[{\prod }_T\right]/{\prod }_0-1,\text{giving}\text{E}\left[{\prod }_T\right]={\prod }_0\left(1+\text{E}\left[r\right]\right).$

  As a check: 840 = 800 · (1 + 0.05).

In a discrete-time financial model one is generally interested in basic characteristics of the model, such as whether or not the following statements hold true:

• (1) There exists a replicating portfolio for every arbitrary derivative contract or claim.
• (2) Every replicating portfolio is unique for a given claim.
• (3) If there are different replicating portfolios for a given claim, then they all have the same initial cost.
• (4) The initial cost of a portfolio replicating a positive claim CT (i.e., CT ≥ 0 and CT(ω) > 0 for at least one ω ∊ Ω) is necessarily positive.

In the above simplest two-state model with two base assets we have already seen that statements 1 and 2 hold; hence statement 3 is irrelevant. Statement 4 can be guaranteed once we impose the extra condition that rs− < rB < rs+, which is equivalent to requiring that there is no arbitrage in the model. This is discussed in detail in Section 2.3.

2.2.2 A Discrete-Time Model with a Finite Number of States

Let us make our model more realistic by increasing the number of observation periods and adding more states. We assume that trading can take place at discretely monitored times t = 0, 1, 2, ..., T. That is, the time horizon T is an integer, and time is measured in periods. One observation period may correspond to one year, one week, one day, or even one second. The state space Ω is finite and contains M states of the world ω1,ω2, ..., ωM. Consider an asset such as a stock (A = S) or a bond (A = B) with the price At monitored at times t = 0, 1, 2, ..., T. Assume that At > 0 for all t. At the present time t = 0, the current price A0 is assumed to be known. The future prices At for t > 0 remain uncertain until information about the market state is revealed. So, mathematically, for every fixed t, price At is a positive random variable on a finite probability space.

2.2.2.1 Asset Returns

Consider an asset A (or a portfolio of assets) with the price process {At}t = 0, 1, ..., T. The total return and the rate of return on the asset A over a time interval [s, t] with 0 ≤ s<t, respectively denoted by $R_{\left[s,t\right]}^A$ and $r_{\left[s,t\right]}^A$, are random variables defined by

$R_{\left[s,t\right]}^A:=\frac{A_t}{A_s}\text{and}{r}_{\left[s,t\right]}^A:=\frac{A_t-A_s}{A_s},$

respectively. They are related by RA = rA + 1. Often, for simplicity, the term return is used for both notions. Returns on risky assets (or portfolios of risky assets) are uncertain and are therefore functions of ω ∊ Ω, e.g., $R_{\left[s,t\right]}^A\left(\omega \right)=\frac{A_t\left(\omega \right)}{A_s\left(\omega \right)}$. For returns on asset A over a single period [t − 1, t] where t = 1, 2, ..., T, we use notations RtA := R[t − 1, t]A and rtA := r[t − 1, t]A. For simplicity and when the context is clear, we will omit the superscript A and will simply denote the total return by R and the rate of return by r.

Now the asset prices can be written in terms of single period returns on the stock. For every t = 1, 2, ..., T, we have that

$r_t=R_t-1=\frac{A_t-A_{t-1}}{A_{t-1}}=\frac{A_t}{A_{t-1}}-1\Rightarrow A_t=R_t{A}_{t-1}\text{and}{A}_t=\left(1+r_t\right)A_{t-1}.$

By applying successively this rule to At − 1, At − 2, ..., A1, we obtain

$\begin{array}{ll}{A}_t=\left(1+r_t\right)A_{t-1}\hfill & =\left(1+r_{t-1}\right)\left(1+r_t\right)A_{t-2}=...\hfill \\ \hfill & =\left(1+r_1\right)\left(1+r_2\right)\cdots \left(1+r_t\right)A_0.\hfill \end{array}$

Equivalently, we have At = R1 R2 ··· Rt A0 for t = 0, 1, ..., T. Therefore, the dynamics of asset prices can also be described by asset returns and initial price A0. Note that the aggregate returns $r_{\left[s,t\right]}=\frac{A_t-A_s}{A_s}$ and $R_{\left[s,t\right]}=\frac{A_t}{A_s}$ on asset A from time s to time t with 0 ≤ s < t ≤ T respectively satisfy

$1+r_{\left[s,t\right]}=\left(1+r_{s+1}\right)\left(1+r_{s+2}\right)\cdots \left(1+r_t\right)\text{and}{R}_{\left[s,t\right]}=R_{s+1}{R}_{s+2}\cdots R_t.\text{ (2.6)}$

Example 2.2.

Consider a market that assumes three possible scenarios: Ω = {ω1, ω2, ω3}. Suppose that stock S takes on the following values over a two-period interval:

Find the returns r1, r2 and compare them with r[0, 2].

Solution. The one-period returns $r_1=\frac{S_1-S_0}{S_0}$ and $r_2=\frac{S_2-S_1}{S_1}$ and the two-period return $r_{\left[0,2\right]}=\frac{S_2-S_0}{S_0}$ take on the following values:

As is seen from the table above, the returns satisfy (2.6), which takes the following form for a two-period model: 1 + r[0, 2] = (1+ r1)(1 + r2).

The returns on a risky asset are random variables. If the scenario probabilities are given, then expected values of returns can be calculated. Suppose that the probabilities pk := ℙ(ωk) are known for every ωk ∊ Ω. Suppose that the probabilities pk are strictly positive and sum up to one:

$p_1,p_2,...,p_M>0,p_1+p_2+\cdots +p_M=1.$

For any given scenario ωk, k = 1, 2, ..., M, the return r[s, t](ωk) is known. The expected return for the period [s, t] with 0 ≤ s < t ≤ T can be calculated as follows:

$\text{E}\left[r_{\left[s,t\right]}\right]=r_{\left[s,t\right]}\left({\omega }^1\right)\cdot p_1+r_{\left[s,t\right]}\left({\omega }^2\right)\cdot p_2+\cdots +r_{\left[s,t\right]}\left({\omega }^M\right)\cdot p_M.\text{ (2.7)}$

If the one-period returns rs+1, rs+2, ..., rt are independent random variables, then the expected aggregate return can be expressed in terms of expected one-period returns as follows:

$1+\text{E}\left[r_{\left[s,t\right]}\right]=\left(1+\text{E}\left[r_{s+1}\right]\right)\left(1+\text{E}\left[r_{s+2}\right]\right)\cdots \left(1+\text{E}\left[r_t\right]\right).$

Suppose that the one-period returns rt, t ≥ 1, are independent and identically distributed (i.i.d.) random variables with the common expected value of a one-period return, rA := E[r1]. Then, we obtain

$1+\text{E}\left[r_{\left[s,t\right]}\right]={\left(1+\text{E}\left[r_1\right]\right)}^{t-s}={\left(1+r_A\right)}^{t-s}.$

Since At = (1+ r[0,t]) A0 and A0 is certain, the expected asset price at time t is

$\text{E}\left[A_t\right]=\left(1+\text{E}\left[r_{\left[0,t\right]}\right]\right)A_0={\left(1+r_A\right)}^t{A}_0.\text{ (2.8)}$

Note that this expression is very similar to the formula of the accumulated value under periodic compounding of interest. In practice it may be difficult to estimate probability distributions of returns. What can be easily computed from historical data is an average return over a certain period. As a result, one can estimate expected future cash flows.

The log-return on asset A over a time interval [s, t], denoted L[s, t]A, is given by

$L_{\left[s,t\right]}^A:=\text{ln}\left(\frac{A_t}{A_s}\right)=\text{ln}{R}_{\left[s,t\right]}^A=\text{ln}\left(\text{1+}{r}_{\left[s,t\right]}^A\right).$

A single-period log-return, denoted LtA, is

$L_t^A:=L_{\left[t-1,t\right]}^A,t=1,2,...,T.$

Since R[s, t]A = Rs+1A Rs+2A ··· RtA, we obtain that the log-returns are additive:

$L_{\left[s,t\right]}^A=L_{s+1}^A+L_{s+2}^A+\cdots +L_t^A.$

The bond prices Bt, t ≥ 0, are nonrandom (deterministic). In other words, they do not depend on the world state: Bt(ω) = Bt for all ω ∊ Ω and t ≥ 0. The returns on the bond $r_t=\frac{B_t-B_{t-1}}{B_{t-1}},t\ge 1$, are deterministic as well. We can express the bond prices Bt, t ≥ 1, in terms of the initial price B0 and one-period returns as

$B_t=B_0\left(1+r_1\right)\left(1+r_2\right)\cdots \left(1+r_t\right).$

Assuming that the one-period returns rt all have the same constant value rB, we arrive at a formula that is analogous to (2.8):

$B_t=B_0{\left(1+r_B\right)}^t.\text{ (2.9)}$

Equations (2.8) and (2.9) enlighten us on how to compare performances of a risky asset and risk-free asset—we can simply compare the expected one-period returns on assets of interest.

In summary, to construct a discrete-time financial model we need to know the initial price and the probability distribution of one-period returns for each base asset. For a model with a finite number of states, all returns are discrete random variables defined on a common probability space. Usually we can distinguish one underlying whose returns are certain, e.g., a risk-free bond. A more detailed analysis of single-period models and multi-period models will be respectively done in later chapters. In the next section, we present the binomial tree model, which is the simplest example of a multi-period model.

2.2.3 Introducing the Binomial Tree Model

In this subsection we introduce a discrete-time model with one risky stock S and one risk-free asset B such as a bond (or a bank account). Assume that there are T periods. The one-period return on the risk-free asset is denoted by r ≡ rB > 0. In fact, r is a risk-free interest rate compounded periodically. The bond prices are Bt = B0(1 + r)t , t = 1, 2, ..., T. Assume that the one-period returns Rt ≡ RtS, t =1, 2, ..., T, on the stock are i.i.d. random variables such that

$R_t=\{\begin{array}{l}u\text{with}\text{probaility}-p,\hfill \\ d\text{with}\text{probaility}-1-p,\hfill \end{array}\text{ (2.10)}$

where the factors d and u are such that 0 < d < u, and p ∊ (0, 1) is the probability that the stock price moves up. Notice that the average one-period return on the stock is given by

$E\left[R_t\right]=pu+\left(1-p\right)d.$

Since Rt = St/St−1, the stock price St at time t is expressed in terms of the price St−1 at the previous time moment as follows:

$S_t=R_t{S}_{t-1}=\{\begin{array}{l}{S}_{t-1}u\text{with}\text{probaility}-p,\hfill \\ S_{t-1}d\text{with}\text{probaility}-1-p.\hfill \end{array}\text{ (2.11)}$

In other words, at each time t the stock price St can move up by a factor u or down by a factor d (relative to St−1). The inequality d > 0 guarantees the positiveness of stock prices, i.e., St > 0 for all t ≥ 1, provided that S0 > 0. Equivalently, we may work with rates of return, rt = Rt − 1. Equations (2.10)–(2.11) take the form

$S_t=\left(1+r_t^S\right)S_{t-1},\text{where}{r}_t^S=\{\begin{array}{l}u-1\text{with}\text{probaility}-p,\hfill \\ d-1\text{with}\text{probaility}-1-p.\hfill \end{array}$

A useful way to represent (2.11) is to write

$S_t=u^{X_t}{d}^{1-X_t}{S}_{t-1},\text{where}{X}_t=\{\begin{array}{l}1\text{with}\text{probaility}-p,\hfill \\ 0\text{with}\text{probaility}-1-p.\hfill \end{array}\text{ (2.12)}$

Note that Xt is a Bernoulli random variable. For each t, the variable Xt is a function of the return Rt. Since the returns on the stock are independent, the variables Xt are independent as well. In other words, {Xt}t≥1 is a collection of i.i.d. Bernoulli random variables with probability p of success. Iterating Equation (2.12) gives

$\begin{array}{l}{S}_1=S_0{u}^{X_1}{d}^{1-X_1},\hfill \\ S_2=S_1{u}^{X_2}{d}^{1-X_2}=S_0{u}^{X_1+X_2}{d}^{2-\left(X_1+X_2\right)},\hfill \\ ⋮\hfill \\ S_n=S_0{u}^{X_1+X_2+\cdots +X_n}{d}^{n-\left(X_1+X_2+\cdots +X_n\right)}.\hfill \end{array}$

The last expression gives the stock price at time t = n in terms of its initial price and the sum X1 + X2 + ··· + Xn. The sum of n i.i.d. Bernoulli random variables, denoted Yn, can be interpreted as the number of successes in n independent trials with probability p of success on each trial. It has the binomial distribution; the probability mass function of Yn ~ Bin(n, p) is

$b\left(k;n,p\right)=\mathbb{P}\left(Y_n=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-k},k=0,1,2,...,n.$

Thus, at time t = n we have

$S_n=S_0{u}^{Y_n}{d}^{n-Y_n},\text{where}{Y}_n~Bin\left(n,p\right).$

Note that the stock price Sn can only take on a value in the set

$\left\{S_{n,k}:=S_0{u}^k{d}^{n-k}:k=0,1,...,n\right\}.$

By the equivalence of events {Sn = Sn, k} and {Yn = k}, the probability distribution of Sn is then

$\mathbb{P}\left(S_n=S_{n,k}\right)=\mathbb{P}\left(Y_n=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-k},k=0,1,...,n.\text{ (2.13)}$

Equation (2.13) tells us that the stock price Sn at time t = n can admit n + 1 different values, i.e., the values Sn, k, k = 0, 1, ..., n. There are (kn) different scenarios (n-step price paths) with exactly k upward and n − k downward price moves that produce the same (terminal) stock price Sn, k at time n. The set Ω of all the possible scenarios can be compactly represented by the n-step recombining binomial tree or binomial lattice (see Figure 2.2). Each n-step path or scenario leading to Sn, k has equal probability pk(1 −p)n−k of occurring. Every such path starts at S0, moves up a total of k times and down a total of n − k times. Since there are (kn) paths leading to the same value Sn, k, then summing this over values k = 0, 1, ..., n must give the total number of all possible paths, which is 2n. This, of course, corresponds to the binomial formula:

$2^n={\left(1+1\right)}^n={\displaystyle \sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)}={\displaystyle \sum _{k=0}^n\left\{\begin{array}{l}\#\text{of}\text{scenarios}\text{with}k\text{upward}\text{and}\hfill \\ n-k\text{downward}\text{price}\text{moves}\hfill \end{array}\right\}}.$

For the sake of comparison, let us consider three two-step binomial tree models under three different assumptions on the probability distributions of returns.

Example 2.3.

Let S0 = 100. Find the distributions of prices S1 and S2 assuming one of the following.

• (a) The returns R1 and R2 are i.i.d. and $R_t=\{\begin{array}{cc}1.1& \text{with}\text{prob}.1.2,\\ 0.9& \text{with}\text{prob}.1/2.\end{array}$

• (b) The returns R1 and R2 are independent, but not identically distributed:

  $R_1=\{\begin{array}{l}1.1\text{with}\text{prob}.1/2,\hfill \\ 0.9\text{with}\text{prob}.1/2;\hfill \end{array}{R}_2=\{\begin{array}{l}1.15\text{with}\text{prob}.1/3,\hfill \\ 0.9\text{with}\text{prob}.2/3.\hfill \end{array}$

• (c) The returns R1 and R2 are not independent and not identically distributed:

  $\begin{array}{ll}{R}_1\hfill & =\{\begin{array}{ll}1.1\hfill & \text{with}\text{prob}.1/2,\hfill \\ 0.9\hfill & \text{with}\text{prob}.1/2;\hfill \end{array}\hfill \\ R_2|\left\{R_1=1.1\right\}\hfill & =\{\begin{array}{ll}1.15\hfill & \text{with}\text{prob}.1/3,\hfill \\ 0.9\hfill & \text{with}\text{prob}.2/3;\hfill \end{array}\hfill \\ R_2|\left\{R_1=0.9\right\}\hfill & =\{\begin{array}{ll}1.1\hfill & \text{with}\text{prob}.1/4,\hfill \\ 0.85\hfill & \text{with}\text{prob}.3/4.\hfill \end{array}\hfill \end{array}$

Solution. Since the probability distribution of R1 is the same for all three cases, the distribution of S1 is the same as well:

$S_1=S_0{R}_1=\{\begin{array}{l}100\cdot 1.1\text{with}\text{prob}.1/2\hfill \\ 100\cdot 0.9\text{with}\text{prob}.1/2\hfill \end{array}=\{\begin{array}{l}110\text{with}\text{prob}.1/2\hfill \\ 90\text{with}\text{prob}.1/2.\hfill \end{array}$

The stock price at the end of period 2 is S2 = S1 R2. So we find the probability distribution of S2 for each of the three cases based on the values of R2 and S1.

• (a) In this case, we have a two-period binomial tree model discussed just above with parameters u =1.1, d = 0.9, and p =1/2. Applying (2.13) gives

  $\begin{array}{l}{S}_2=S_{2,2}=100\cdot {1.1}^2=121\text{with}\text{probability}\left(\begin{array}{c}2\\ 0\end{array}\right){0.5}^2=0.5,\hfill \\ S_2=S_{2,1}=100\cdot 1.1\cdot 0.9=99\text{with}\text{probability}\left(\begin{array}{c}2\\ 1\end{array}\right){0.5}^2=0.25,\hfill \\ S_2=S_{2,0}=100\cdot {0.9}^2=81\text{with}\text{probability}\left(\begin{array}{c}2\\ 2\end{array}\right){0.5}^2=\mathrm{0.25.}\hfill \end{array}$

• (b) Given that S1 = 110, the price S2 = 110 R2 takes one of two values: 110·1.15 = 126.5 or 110·0.9 = 99. If S1 = 90, then S2 = 90 R2 is either 90·1.15 = 103.5 or 90·0.9 = 81. So there are four distinct values for S2. We can compute the probabilities by using the independence of the returns R1 and R2, i.e., using that P(R1 = x, R2 = y)= ℙ(R1 = x) · ℙ(R2 = y) for all x and y (hence, S1 and R2 are independent random variables):

  $\begin{array}{l}\mathbb{P}\left(S_2=126.5\right)=\mathbb{P}\left(R_1=1.1,R_2=1.15\right)=\mathbb{P}\left(R_1=1.1\right)\cdot \mathbb{P}\left(R_2=1.15\right)=\frac{1}{2}\cdot \frac{1}{3}=\frac{1}{6},\hfill \\ \mathbb{P}\left(S_2=103.5\right)=\mathbb{P}\left(R_1=0.9,R_2=1.15\right)=\mathbb{P}\left(R_1=0.9\right)\cdot \mathbb{P}\left(R_2=1.15\right)=\frac{1}{2}\cdot \frac{1}{3}=\frac{1}{6},\hfill \\ \mathbb{P}\left(S_2=99\right)=\mathbb{P}\left(R_1=1.1,R_2=0.9\right)=\mathbb{P}\left(R_1=1.1\right)\cdot \mathbb{P}\left(R_2=0.9\right)=\frac{1}{2}\cdot \frac{2}{3}=\frac{1}{3},\hfill \\ \mathbb{P}\left(S_2=81\right)=\mathbb{P}\left(R_1=0.9,R_2=0.9\right)=\mathbb{P}\left(R_1=0.9\right)\cdot \mathbb{P}\left(R_2=0.9\right)=\frac{1}{2}\cdot \frac{2}{3}=\frac{1}{3}.\hfill \end{array}$

• (c) If R1 = 1.1 (and hence S1 = 110), then S2 = 110·R2 takes on the value 110·1.15 = 126.5 or 110 · 0.9 = 99. If R1 = 1.1 (and hence S1 = 90), then S2 = 90 · R2 is either 90 · 1.1 = 99 or 90 · 0.85 = 76.5. The distribution of S2 is given by conditioning on R1:

  $\begin{array}{ll}\mathbb{P}\left(S_2=126.5\right)\hfill & =\mathbb{P}\left(R_1=1.1,R_2=1.15\right)\hfill \\ \hfill & =\mathbb{P}\left(R_2=1.15|R_1=1.1\right)\cdot \mathbb{P}\left(R_1=1.1\right)=\frac{1}{3}\cdot \frac{1}{2}=\frac{1}{6},\hfill \\ \mathbb{P}\left(S_2=99\right)\hfill & =\mathbb{P}\left(R_1=0.9,R_2=1.1\right)+\mathbb{P}\left(R_1=1.1,R_2=0.9\right)\hfill \\ \hfill & =\mathbb{P}\left(R_2=1,1|R_1=0.9\right)\cdot \mathbb{P}\left(R_1=0.9\right)\hfill \\ \hfill & +\mathbb{P}\left(R_2=0.9|R_1=1.1\right)\cdot \mathbb{P}\left(R_1=1.1\right)=\frac{1}{4}\cdot \frac{1}{2}+\frac{2}{3}\cdot \frac{1}{2}=\frac{11}{24},\hfill \\ \mathbb{P}\left(S_2=76.5\right)\hfill & =\mathbb{P}\left(R_1=0.9,R_2=0.85\right)\hfill \\ \hfill & =\mathbb{P}\left(R_2=0.85|R_1=0.9\right)\cdot \mathbb{P}\left(R_1=0.9\right)=\frac{3}{4}\cdot \frac{1}{2}=\frac{3}{8}.\hfill \end{array}$

Example 2.4.

Suppose that the stock price follows a binomial tree with current price S0 = $100. Find the factors d and u if (a) S1 ∊{$120, $90}; (b) minω S2(ω) = $64 and maxω S2(ω) = $121. These are simple examples of a two-period binomial model calibration.

Solution.

• (a) In one period, the stock prices are S+ = S0u = 100u = 120 and S− = S0d = 100d = 90. Solve these equations for d and u to obtain d = 0.9 and u = 1.2.

• (b) In two periods:

  $\underset{\omega }{\min }S\left(2,\omega \right)=S_0\underset{0\le k\le 2}{\min }{u}^k{d}^{2-k}=S_0\min \left\{d^2,ud,u^2\right\}=S_0{d}^2=100d^2$

  and

  $\underset{\omega }{\max }S\left(2,\omega \right)=S_0\underset{0\le k\le 2}{\max }{u}^k{d}^{2-k}=S_0\max \left\{d^2,ud,u^2\right\}=S_0{u}^2=100u^2.$

  Hence, $u=\sqrt{1.21}=1.1$ and $d=\sqrt{0.64}=0.8$.

2.2.4 Self-Financing Investment Strategies in the Binomial Model

Here we give a brief introduction to the concept of a self-financing trading strategy within the binomial model. Recall that a static portfolio (x, y) in the two base assets is an investment consisting of fixed x positions in the stock (the risky asset) and fixed y positions in the bond or money market account (the risk-free asset). For such a portfolio there is no trading, or re-balancing, of the positions over all time periods. In contrast, in the binomial model trading is allowed to take place at the beginning of each period so that the positions held in the two base assets are generally not static over time. That is, within each period the positions are allowed to be re-balanced at the beginning of the period and are subsequently held fixed until the end of the period. The model allows for trading at the discrete times t = 0, 1, ..., T. Imagine that an investor begins by setting up a portfolio (x0, y0) at time t = 0 and holds it until the end of the first period. At time t = 1, trading is allowed and the investor re-balances the portfolio positions to (x1, y1) and this new portfolio is held to the end of the second period at time t = 2. At time t = 2, trading takes place and the investor again re-balances the positions to (x2, y2) and holds this portfolio until time t = 3, and so on until maturity time t = T. The sequence of portfolios (x0, y0), (x1, y1), ..., (xT−1, yT−1) forms what is called a trading or investment strategy (with maturity time T) in the binomial model. This is also referred to as a portfolio strategy in the two base assets.

At each time t, after re-balancing, the portfolio value is

${\displaystyle {\prod }_t=x_t}{S}_t+y_t{B}_t.$

During the period from time t to t + 1 the portfolio positions are held static and the new portfolio value, before re-balancing, at time t +1 is

${\displaystyle {\prod }_{t+1}=x_t}{S}_{t+1}+y_t{B}_{t+1}.$

Note that the change in value for the one period is due solely to the change in prices of the base assets:

${\displaystyle {\prod }_{t+1}-{\displaystyle {\prod }_t=x_t}}\left(S_{t+1}-S_t\right)+y_t\left(B_{t+1}-B_t\right).\text{ (2.14)}$

The first term on the right of this equation gives the change in portfolio value due to the change in the share price of the stock. Assuming the number of shares xt is positive (i.e., long the stock), then xt(St+1 − St) corresponds to a gain (or loss) if the share price increases (or decreases) whereas the opposite is true if xt is negative (short the stock). The second term gives the change in value due to the bond component. If yt > 0, then yt(Bt+1 − Bt) > 0 is a gain or earning based on the interest paid by the bond or a bank account during the period. If yt < 0, the position in the bond is negative and this gives a loss corresponding to borrowing money from a bank account. At time t + 1, the portfolio value after re-balancing must be

${\displaystyle {\prod }_{t+1}=x_{t+1}}{S}_{t+1}+y_{t+1}{B}_{t+1}.$

At this point we bring in the important notion of self-financing within the above strategy. So far, we allowed for the positions to be re-balanced at each time step without any restriction. Of course, any decision made by the investor about when to change the asset positions and what assets to sell or to buy is only based on the historical information about the market currently available. Generally, the investor may alter the positions at any time by selling some assets and investing the proceeds in others. However, we now enforce the condition that there cannot be any consumption or injection of funds within the strategy at any time past initial time t0 = 0. In other words, after initially setting up the portfolio, we only allow for self-financed strategies. Hence, at each time t, the portfolio value before re-balancing the positions must be the same as its value after re-balancing. The above two expressions for Πt+1 must therefore be equal and this gives rise to the so-called self-financing condition (s.f.c.):

$S_{t+1}\left(x_{t+1}-x_t\right)+B_{t+1}\left(y_{t+1}-y_t\right)=0\text{ (2.15)}$

for all 0 ≤ t ≤ T − 1. The first term corresponds to the cost (at time t + 1) of re-balancing in the stock with share price St+1 and position change δxt := xt+1 − xt. The second term gives the re-balancing cost (at time t + 1) for the bond holdings where the position change is δyt := yt+1 − yt. Note that (i) δxt < 0 iff δyt > 0 and (ii) δxt > 0 iff δyt < 0. In case (i) re-balancing involves selling |δxt| stock shares at price St+1 and investing the proceeds in buying δyt bonds at price Bt+1. In case (ii) re-balancing involves selling |δyt| bonds at price Bt+1 and investing this amount in buying δxt stock shares at price St+1.

Let us denote the one-period changes in the portfolio value and asset prices by

$\delta {\displaystyle {\prod }_t:={\displaystyle {\prod }_{t+1}-}{\prod }_t},\delta S_t:=S_{t+1}-S_t,\delta B_t:=B_{t+1}-B_t,$

respectively. Since St+1 = St + δSt and Bt+1 = Bt + δBt, the above s.f.c. takes the form

$\left(S_t+\delta S_t\right)\delta x_t+\left(B_t+\delta B_t\right)\delta y_t=0.\text{ (2.16)}$

In later chapters we shall see that this form is similar to the s.f.c. for continuous-time modelling of positions and prices. If we now compute the change in the re-balanced portfolio values from time t to t + 1, we see that the s.f.c. is equivalent to the statement that this change is due only to the changes in the asset prices:

$\begin{array}{l}\delta {\displaystyle {\prod }_t=x_{t+1}{S}_{t+1}}+y_{t+1}{B}_{t+1}-\left(x_t{S}_t+y_t{B}_t\right)\hfill \\ =\left(x_t+\delta x_t\right)\left(S_t+\delta S_t\right)+\left(y_t+\delta y_t\right)\left(B_t+\delta B_t\right)-\left(x_t{S}_t+y_t{B}_t\right)\hfill \\ =x_t\delta S_t+y_t\delta B_t+\left(S_t+\delta S_t\right)\delta x_t+\left(B_t+\delta B_t\right)\delta y_t\hfill \\ =x_t\delta S_t+y_t\delta B_t.\hfill \end{array}$

This recovers (2.14) above and follows by enforcing the s.f.c. in (2.16) in the last expression.

We can write the s.f.c. in terms of the one-period (t → t + 1) returns, where St+1 = (1 + rt+1S)St and Bt+1 = (1+ rB) Bt. Given the positions xt, yt at time t, then by (2.14)

${\displaystyle {\prod }_{t+1}=x_t}\left(1+r_{t+1}^S\right)S_t+y_t\left(1+r_B\right)B_t$

and the s.f.c. takes the form

$S_t\left(1+r_{t+1}^S\right)\left(x_{t+1}-x_t\right)+B_t\left(1+r_B\right)\left(y_{t+1}-y_t\right)=0.\text{ (2.17)}$

In particular, assume the rate of return rt+1S is one of two known values rS+ or rS−, i.e., two scenarios are possible for one period. Then, given knowledge of St, Bt, rB, xt, and yt, the above s.f.c. gives a linear relation between the position in the stock and that in the bond at time t + 1 for either (+ or −) stock return scenarios:

$S_t\left(1+r_S^{+}\right)\left(x_{t+1}^{+}-x_t\right)+B_t\left(1+r_B\right)\left(y_{t+1}^{+}-y_t\right)=0,\text{ (2.18)}$

where the time t + 1 portfolio (xt+1, yt+1) can be explicitly denoted by (xt+1+, yt+1+) in case rt+1S = rS+ and by (xt+1−, yt+1−) for rt+1S = rS−. Note that if we are given another independent linear relation between the positions at time t+1, then the portfolio (xt+1, yt+1) is uniquely given. The example below describes such a situation where we impose an added independent condition on the positions besides the s.f.c.

Example 2.5.

Consider a one-period binomial model with rB = 10%, rS− = −10%, rS+ = 20%, S0 = $100, and B0 = $10. Construct a self-financing strategy with an initial value of $1000 such that 50% of the wealth is always invested in risk-free bonds.

Solution. We solve this problem by applying the above formulae for a single period t = 0 to t + 1 = 1. Initially, Π0 = 1000 = x0S0 + y0B0, such that y0B0 = 0.5 · 1000 = 500. Hence, $y_0=\frac{500}{B_0}=50$ units of the bond and $x_0=\frac{500}{S_0}=5$ shares of stock, i.e., (x0, y0) = (5, 50). At time 1, S1 = S0(1 + rS) and Π1 = x1S1 + y1B1 = x1S0(1 + rS)+ y1B0(1 + rB). Since our strategy is constrained to have 50% equally invested in the stock and in the bond, then x1S0(1 + rS)= y1B0(1 + rB). Combining this relation with the above one-period s.f.c. in (2.18) (with t = 0) gives us two linear equations in the two unknowns x1, y1:

$\begin{array}{l}{S}_0\left(1+r_S\right)x_1+B_0\left(1+r_B\right)y_1={\prod }_1,\hfill \\ S_0\left(1+r_S\right)x_1-B_0\left(1+r_B\right)y_1=0,\hfill \end{array}$

where Π1 = x0S0(1 + rS) + y0B0(1 + rB). The solution is:

$x_1=\frac{{\prod }_1}{2S_0\left(1+r_S\right)},y_1=\frac{{\prod }_1}{2B_0\left(1+r_B\right)}.$

• (a) For rS = rS+ = 0.2, S0(1 + rS) = 120, B0(1 + rB) = 11, Π1 = 5 · 120 + 50 · 110 = 1150, giving the portfolio $\left(x_1,y_1\right)=\left(\frac{115}{24},\frac{575}{11}\right)\equiv \left(x_1^{+},y_1^{+}\right)$. In this case, $\delta x\equiv x_1-x_0=\frac{115}{24}-5=-\frac{5}{24}$ and $\delta y\equiv y_1-y_0=\frac{575}{11}-50=\frac{25}{11}$. Hence, at time t = 1, re-balancing involves selling $\frac{5}{24}$ stock shares at price S1 = $120 and investing the proceeds in buying $\frac{25}{11}$ bonds at price B1 = $110.
• (b) For rS = rS− = −0.1, S0(1 + rS) = 90, Π1 = 5·90 + 50·110 = 1000, giving the portfolio $\left(x_1,y_1\right)=\left(\frac{50}{9},\frac{500}{11}\right)\equiv \left(x_1^{-},y_1^{-}\right)$. Since $\delta x=\frac{50}{9}-5=\frac{5}{9}$ and $\delta y=\frac{500}{11}-50=-\frac{50}{11}$, we re-balance by buying $\frac{5}{9}$ stock shares at price S1 = $90 and finance this by selling off $\frac{50}{11}$ bonds at price B1 = $110.

Finally, note that in both cases we have the s.f.c. δx S1 + δy B1 = 0.     □

Let us end this subsection with the definition of an admissible strategy.

Definition 2.2.

An admissible strategy is a self-financing strategy with nonnegative values for all dates from time zero until maturity.

At any date the holder of an admissible strategy will have no potential liabilities which he or she would not able to honour. In particular, since the terminal value will be nonnegative in any state of the world, the liquidation of the terminal portfolio will not result in a loss.

2.2.5 Log-Normal Pricing Model

The binomial tree model has apparent disadvantages as a discrete-time and discrete-price model. We shall remove these restrictions by passing to the continuous-time limit from the binomial tree model. As a result, we will obtain a continuous model of the stock price S(T). Although S(T) is uncertain at time 0, we assume that the mathematical expectation μ and variance σ2 of the log-return on the stock over the time interval [0, T] given by

$\mu :=\frac{1}{T}\text{E}\left[\text{In}\frac{S\left(T\right)}{S\left(0\right)}\right]\text{and}{\sigma }^2:=\frac{1}{T}\text{Var}\left[\text{In}\frac{S\left(T\right)}{S\left(0\right)}\right]$

can be estimated from historical observations.

For each N ≥ 1, consider an N-period binomial tree model with trading dates

$0,{\delta }_N,2{\delta }_N,...,N{\delta }_N=T$

spaced uniformly in [0, T] with the time step ${\delta }_N:=\frac{T}{N}$. Let $\left\{S_t^{\left(N\right)};t=0,1,...,N\right\}$ denote the stock price process in the N-period recombining binomial tree model; let the initial price be S0 ≡ S(0). Recall that the stock prices are given by a product of the initial stock price and single-period returns on the stock S,

$S_t^{\left(N\right)}=S_0{\displaystyle \prod _{k=1}^t{R}_k^{\left(N\right)}}.$

The returns Rk(N), k = 1, 2, ..., N, are i.i.d. random variables having the common probability distribution

$R_k^{\left(N\right)}=\{\begin{array}{l}{u}_N\text{with}\text{probability}{p}_N,\hfill \\ d_N\text{with}\text{probability}\text{1}-p_N.\hfill \end{array}$

The next step is to parametrize the binomial tree model. The upward and downward factors, uN and dN, can be obtained by matching the first two moments of the stock price returns. In the binomial model, the aggregate log-returns on the stock, $L_{\left[0,N\right]}^{\left(N\right)}:=\ln \frac{S_N^{\left(N\right)}}{S_0}$, have the following first two moments:

$\text{E}\left[L_{\left[0,N\right]}^{\left(N\right)}\right]=N\text{E}\left[L_1^{\left(N\right)}\right]=N\left(\text{In}\left(u_N\right)p_N+\text{In}\left(d_N\right)\left(1-p_N\right)\right),\text{ (2.19)}$

$\text{var}\left[L_{\left[0,N\right]}^{\left(N\right)}\right]=N\text{var}\left[L_1^{\left(N\right)}\right]=N\left(\text{In}{\left(u_N\right)}^2{p}_N+\text{In}{\left(d_N\right)}^2\left(1-p_N\right)\right).\text{ (2.20)}$

Equating the respective moments of the log-returns ln $\frac{S\left(T\right)}{S\left(0\right)}$ and $L_{\left[0,N\right]}^{\left(N\right)}$ Lgives us two equations:

$\text{In}\left(u_N\right)p_N+\text{In}\left(d_N\right)\left(1-p_N\right)=\mu {\delta }_N\text{ (2.21)}$

$\text{In}{\left(u_N\right)}^2{p}_N+\text{In}{\left(d_N\right)}^2\left(1-p_N\right)={\sigma }^2{\delta }_N.\text{ (2.22)}$

Note that there are three unknowns, i.e., uN and dN and the probability pN. Hence, we need a third equation. A convenient choice is the symmetry condition

$u_N.d_N=1.\text{ (2.23)}$

The following solution is the most commonly used in binomial models:

$u_N=e^{\sigma \sqrt{{\delta }_N}},\text{ (2.24)}$

$d_N=e^{-\sigma \sqrt{{\delta }_N}},\text{ (2.25)}$

$p_N=\frac{1}{2}+\frac{1}{2}\frac{\mu }{\sigma }\sqrt{{\delta }_N}.\text{ (2.26)}$

Under this parametrization, the log-returns on the stock have the following properties:

$\text{E}\left[L_{\left[0,N\right]}^{\left(N\right)}\right]=N\mu {\delta }_N=\mu T,\text{ (2.27)}$

$\text{var}\left[L_{\left[0,N\right]}^{\left(N\right)}\right]=N{\sigma }^2{\delta }_N+N{\mu }^2{\delta }_N^2={\sigma }^{2}T+{\left(\mu T\right)}^2/N.\text{ (2.28)}$

That is, the solution (2.24)–(2.26) satisfies (2.19)–(2.20) up to $\mathcal{O}\left(\delta t\right)$. In other words, the binomial tree models we constructed here conserve the expected log-return on the stock and asymptotically (as N → ∞) conserve the variance of the log-return.

The binomial price SN(N) Sat the end of the Nth period is a discrete random variable taking on a value in the set {SN,k = S0uNkdNN − k; k = 0, 1, ..., N. As the number of periods N increases to ∞, the length δN of one period approaches zero. Moreover, the density of the points SN, k increases and their range expands. Let us find the limiting distribution of the binomial prices. At the maturity date T, we define the limiting asset price:

$S\left(T\right):=\underset{N\to \infty }{\text{lim}}{S}_N^{\left(N\right)}=\underset{N\to \infty }{\text{lim}}S\left(0\right)\exp \left(L_{\left[0,N\right]}^{\left(N\right)}\right).$

To understand the distribution of the limiting price S(T), we take a closer look at the distribution of the log-return $L_{\left[0,N\right]}^{\left(N\right)}=L_1^{\left(N\right)}+L_2^{\left(N\right)}+...+L_N^{\left(N\right)}$. The one-step log-returns Lk(N), k = 1, 2, ..., N, are i.i.d. random variables. For each value of N, we introduce a sequence of i.i.d. Bernoulli random variables Xk(N), k = 1, 2, ..., N, having the following two-point probability distribution:

$X_k^{\left(N\right)}=\{\begin{array}{l}1\text{with}\text{probability}{p}_N,\hfill \\ 0\text{with}\text{probability}1-p_N.\hfill \end{array}$

We have that Xk(N) = 1 and 1 − Xk(N) = 0 with probability pN , and Xk(N) = 0 and 1 − Xk(N) = 1 with probability 1 − pN. Therefore, we can express the log-return Lin Lk(N) in terms of Xk(N) as follows:

$L_k^{\left(N\right)}=\text{In}\left(u_N\right)X_k^{\left(N\right)}+\text{In}\left(d_N\right)\left(1-X_k^{\left(N\right)}\right)=\text{In}\left(\frac{u_N}{d_N}\right)X_k^{\left(N\right)}+\text{In}\left(d_N\right).$

As a result, the aggregate log-return is given by

$L_{\left[0,N\right]}^{\left(N\right)}={\displaystyle \sum _{k=1}^N{L}_k^{\left(N\right)}=}\text{In}\left(\frac{u_N}{d_N}\right){\displaystyle \sum _{k=1}^N{X}_k^{\left(N\right)}+}N\text{In}\left(d_N\right).$

Denote $Y_N:={\displaystyle {\sum }_{k=1}^N{X}_k^{\left(N\right)}}$. A sum YN of N i.i.d. Bernoulli variables has the binomial probability distribution: YN ~ Bin(N, pN). By the de Moivre–Laplace Theorem 2.1 (provided below), for large N the distribution of YN is approximately normal with mean NpN and variance NpN(1 − pN). Therefore, for large N, the distribution of the log-return L[0, N](N) = ln(uN/dN)YN + N ln(dN) is also approximately normal with mean μT and variance σ2T (see formulae (2.27) and (2.28)). In the limiting case, we obtain that the probability distribution of limN→ ∞ L[0, N](N) is Norm(μT, σ2T). Thus, the limiting stock price S(T) = limN→ ∞ S(0) exp(L[0, N](N)) has the log-normal probability distribution and admits the following representation:

$S\left(T\right)=S\left(0\right){\text{e}}^{\mu \text{T+}\sigma \sqrt{TZ}},\text{where}Z~Norm\left(0,1\right).\text{ (2.29)}$

The parameter μ is called the drift parameter; σ is the volatility parameter.

Here and below, Norm(a, b2) denotes the normal probability distribution with mean a and variance b2. The cumulative distribution function (CDF) of the standard normal distribution Norm(0, 1), denoted N (or Φ in some other texts), is

$\mathcal{N}\left(z\right):=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^z{\text{e}}^{\text{-}{x}^2}/2dx}.\text{ (2.30)}$

If Z is a standard normal variate, then for any real a and b, the random variable a + bZ has the Norm(a, b2) probability distribution. Hence, the CDF F of Norm(a, b2) is

$F\left(z\right)=\mathbb{P}\left(a+bZ\le z\right)=\mathbb{P}\left(Z\le \frac{z-a}{b}\right)=\mathcal{N}\left(\frac{z-a}{b}\right)=\frac{1}{\sqrt{2\pi b}}{{\displaystyle {\int }_{-\infty }^z\text{e}}}^{-\frac{{\left(x-a\right)}^2}{2b^2}}dx.$

A rigorous justification of (2.29) is based on the following theorem, which we are giving without a proof.

Theorem 2.1

(Moivre–Laplace). Consider a sequence {pn}n≥1 in (0, 1) that converges to p ∊ (0, 1) as n → ∞. Let {Yn}n≥1 be a sequence of independent binomial random variables with Yn ~ Bin(n, pn). Then the sequence of rescaled (normalized) random variables

$Y_n^{*}:=\frac{Y_n-\text{E}\left[Y_n\right]}{\sqrt{\text{Var}\left(Y_n\right)}}=\frac{Y_n-n{p}_n}{\sqrt{n{p}_n\left(1-p_n\right)}}$

converges weakly (in distribution) to a standard normal variable.

Recall that a sequence of random variables {Yn}n≥1 converges in distribution to another random variable Y, denoted $Y_n\underrightarrow{d}Y$, as n → ∞, if the CDF's of Yn converge to the CDF of Y, as n → ∞, i.e., for almost all x ∊ ℝ, we have $F_{X_n}\left(x\right)\to F_X\left(x\right)$, as n → ∞.

Corollary 2.2

Suppose that a sequence {Yn}n≥1 converges weakly to a standard normal random variable: $Y_n\underrightarrow{d}Norm\left(0,1\right)$, as n → ∞. Consider two converging sequences of real numbers: an → a and bn → b ≠ 0, as n → ∞. Then

$a_n+b_n{Y}_n\underrightarrow{d}Norm\left(a,b^2\right),asn\to \infty .$

We have a sequence of binomial random variables YN ~ Bin(N, pN), where the probability $P_N=\frac{1}{2}+\frac{1}{2}\frac{\mu \sqrt{T}}{\sigma \sqrt{N}}\to \frac{1}{2}$, as N → ∞. Therefore, by Theorem 2.1,

$\frac{Y_N-\text{E}\left[Y_N\right]}{\sqrt{\text{Var}\left(Y_N\right)}}=\frac{Y_N-N{p}_N}{\sqrt{N{p}_N\left(1-p_N\right)}}\underrightarrow{d}Norm\left(0,1\right),\text{as}N\to \infty .$

On the other hand, for the log-returns LN ≡ L[0, N](N), we have the identity

$\frac{L_N-\text{E}\left[L_N\right]}{\sqrt{\text{Var}\left(L_N\right)}}=\frac{Y_N-\text{E}\left[Y_N\right]}{\sqrt{\text{Var}\left(Y_N\right)}}\text{ (2.31)}$

the proof of which is left as an exercise for the reader. Thus, $L_N^{*}:=\frac{L_N-E\left[L_N\right]}{\sqrt{\text{Var}\left(L_N\right)}}\to Norm\left(0,1\right)$, as N → ∞. Since we can express LN in terms of LN*,

$L_N=\text{E}\left[L_N\right]+L_N^{*}\sqrt{\text{Var}\left(L_N\right)}=\mu T+L_N^{*}\sqrt{{\sigma }^{2}T+{\left(\mu T\right)}^2/N},$

and $\sqrt{{\sigma }^{2}T+{\left(\mu T\right)}^2/N}\to \sigma \sqrt{T}$, as N → ∞, then, by the corollary, we have

$L_N\underrightarrow{d}Norm\left(\mu T,{\sigma }^{2}T\right),\text{as}N\to \infty .$

To summarize, in the binomial tree model, the stock price SN(N) is a discrete random variable, which is a function of a binomial random variable. In the log-normal model, the stock price S(T) is a continuous random variable having the log-normal probability distribution. As seen in Figure 2.3, the shape of the probability distribution of binomial prices is close to that of log-normal prices.

Example 2.6.

(The log-normal distribution). Find the cumulative distribution function (CDF) and the probability density function (PDF) of the log-normal price

$S\left(T\right)=S_0{\text{e}}^{uT+\sigma \sqrt{TZ}}\text{with}T>0,$

where Z ~ Norm(0, 1).

Solution. First, obtain the CDF F of S(T):

$\begin{array}{ll}F\left(s\right)\hfill & =\mathbb{P}\left(S\left(T\right)\le s\right)=\mathbb{P}\left(S_0{\text{e}}^{\mu T+\sigma \sqrt{T}Z}\le s\right)\hfill \\ \hfill & =\mathbb{P}\left(\mu T+\sigma \sqrt{T}Z\le \ln \left(s/S_0\right)\right)=\mathbb{P}\left(Z\le \frac{\ln \left(s/S_0\right)-\mu T}{\sigma \sqrt{T}}\right)\hfill \\ \hfill & =\mathcal{N}\left(\frac{\ln \left(s/S_0\right)-\mu T}{\sigma \sqrt{T}}\right)\text{for}s>0,\hfill \end{array}$

where $\mathcal{N}$ is the standard normal CDF defined by (2.30): $\mathcal{N}\left(z\right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{-\infty }^z{\text{e}}^{-x^2/2}\text{d}x}$. The standard normal PDF is given by ${\mathcal{N}}^{\prime }\left(z\right)=\frac{1}{\sqrt{2\pi }}{\text{e}}^{-z^2/2}$. Differentiating the CDF F(s) w.r.t. s gives us the PDF f of the log-normal price:

$\begin{array}{c}f\left(s\right)=F^{\prime }\left(s\right)={\mathcal{N}}^{\prime }\left(\frac{\ln \left(s/S_0\right)-uT}{\sigma \sqrt{T}}\right)\cdot \frac{\text{d}}{\text{ds}}\left(\frac{\ln \left(s/S_0\right)-uT}{\sigma \sqrt{T}}\right)\\ =\frac{1}{s\sigma \sqrt{2\pi T}}{\text{e}}^{-{\left(\ln \left(s/S_0\right)-uT\right)}^2/\left(2{\sigma }^{2}T\right)},s>0.\end{array}$

2.3 Arbitrage and Risk-Neutral Pricing

An arbitrage opportunity arises when someone can buy an asset at a low price to immediately sell it for a higher price. For example, such a combination of matching deals can be done by taking advantage of an asset price difference between two or more markets. Both buying in one market and selling on the other must occur simultaneously to avoid exposure to any type of market risk. In practice, such simultaneous transactions are only possible with assets and financial products which can be traded electronically. The prices should not change before both transactions are complete; the cost of transport, storage, transaction, or insurance should not eliminate the arbitrage opportunity. In other words, arbitrage is a risk-free opportunity of gaining money.

A trader who engages in arbitrage is called an arbitrageur. Arbitrage opportunities are often hard to come by, due to transaction costs, the costs involved with finding an arbitrage opportunity, and the number of people who are also looking for such opportunities. Arbitrage profits are generally short-lived, as the buying and selling of assets will change the price of those assets in such a way as to eliminate that arbitrage opportunity. This is particularly the case in an efficient market.

Arbitrageurs can often be found in currency markets. Such financial markets have the advantage of being quite liquid, so we do not take the risk of acquiring an asset that may take some time to sell. The transaction costs are minimal for large currency transactions. Since foreign currency markets are an ideal environment for arbitrageurs, arbitrage opportunities tend to be very limited, as any discrepancies in exchange rates tend to be corrected quite quickly by investors trying to exploit those differences.

While arbitrage opportunities may exist in financial markets, in what follows we assume that the financial models we deal with do not allow for arbitrage. All asset prices are equilibrium prices and all arbitrage opportunities are eliminated. We are going to develop a non-arbitrage pricing theory. In a market model that admits arbitrage, wealth can be created from nothing. Thus, it is reasonable to assume that the financial model of consideration does not admit arbitrage opportunities. Let us start with a basic definition of arbitrage without reference to any model.

Definition 2.3.

An arbitrage opportunity is a trading strategy that costs nothing to begin with (i.e., zero initial capital is used to set up) and has no chance of incurring any loss, but has a nonzero chance of making a gain.

This is reminiscent of a free lottery ticket. One of the fundamental properties of a good mathematical model for a financial market is that it does not allow for arbitrage.

2.3.1 The Law of One Price

As is mentioned in Section 2.2, replication is a key to pricing derivatives. Indeed, the following theorem states that if the future values of any two assets (or two portfolios of base assets) at some time are equal to each other in all possible market scenarios, then the present prices of these assets must be the same as well. Therefore, being given a derivative security which can be replicated by a portfolio or trading strategy in base assets (it means that the future values of the derivative and the portfolio are equal in all market states), the initial price of the derivative has to be equal to the initial value of the replicating portfolio in the absence of arbitrage.

Theorem 2.3

(Law of One Price). Assume that the market is arbitrage free. Let there be two assets X and Y whose respective initial prices are X0 and Y0. Suppose at some time T > 0 the prices of X and Y are equal in all states of the world: XT(ω)= YT(ω), for all states ω ∊ Ω. Then X0 = Y0.

Proof. We shall show that if X0 ≠ Y0, then there exists an arbitrage. Without loss of generality, we suppose that X0 >Y0 (if X0 < Y0 then we may just relabel X and Y). Let us construct an arbitrage portfolio in these two assets. Starting with $0, we first borrow and sell one unit of X and realize $X0. We then buy one unit of Y, costing us $Y0. Both transactions give us a positive amount $(X0 − Y0), which we can keep in cash or invest in a risk-free asset. So at time zero we have a portfolio of one unit of Y , negative one unit of X, and the cash amount of $(X0 − Y0). The initial value of this portfolio is zero:

${\displaystyle {\prod }_0=-X_0+Y_0}+\left(X_0-Y_0\right)=0.$

Note that this portfolio requires no initial investment.

At time T, we sell the unit of Y to obtain $YT. We buy and return the unit of X. This costs $XT. Since XT = YT, the net cost of these two trades is zero. However, we still have the positive cash amount X0 − Y0 (and possible interest earned), and hence we have exhibited an arbitrage opportunity:

${\displaystyle {\prod }_T=-X_T+Y_T}+X_0-Y_0=X_0-Y_0>0.$

Therefore, to eliminate the arbitrage, we must have X0 = Y0.

In this proof we have assumed that there are no transaction costs in carrying out the trades required, short sells are allowed, and the assets involved can be bought and sold at any time at will.

2.3.2 A First Look at Arbitrage in the Single-Period Binomial Model

In the single-period binomial model of a financial market considered in the previous section, an arbitrage strategy simply reduces to an arbitrage portfolio. Consider a portfolio (x, y) in stock S and bond B with initial value Π0 = Π0(x, y) = xS0 + yB0 and terminal values ΠT(ω±) = ΠT(x, y)(ω±) = xST(ω±) + yBT ≡ xS± + yBT. The above definition of arbitrage hence implies that (x, y) will be an arbitrage portfolio when the following conditions are met:

• (a) Π0 = 0,
• (b) ΠT(ω±) ≥ 0, and ΠT(ω+) > 0 or ΠT(ω−) > 0.

Note that condition (b) can be stated using probabilities:

• (b) ℙ(ΠT ≥ 0) = 1 and ℙ(ΠT > 0) > 0.

Hence, the single-period binomial model admits an arbitrage iff there exists a portfolio (x, y) satisfying conditions (a) and (b). As Theorem 2.4 shows, there is no such arbitrage portfolio (x, y) when the return on the bond falls strictly in between the higher and lower returns on the stock.

Theorem 2.4

(Arbitrage: Single-period binomial model). The single-period binomial model admits no arbitrage iff rS− < rB < rS+.

Proof. First, note that condition rS− <rB < rS+ is equivalent to S− < (1 + rB)S0 < S+ . So we can formulate our proof using either returns or prices.

• Consider any zero-cost, nontrivial portfolio (x, y), i.e.,Π0(x, y) = xS0 + yB0 = 0. This implies that the portfolio (x, y) has either form (x, −xS0/B0) with x > 0 or (−yB0/S0, y) with y > 0. The first portfolio type corresponds to buying the stock and borrowing money while the second is a portfolio short in stock and positively invested in a bond.
• For a portfolio of the form (x, −xS0/B0), we have: ΠT(x, y)(ω±) = x[S± − (1 + rB)S0]. In the worst case scenario ΠT(x, y)(ω−) = x[S− − (1 + rB)S0]. Hence S− ≥ (1 + rB)S0 implies ΠT(x, y)(ω±) ≥ 0 and ΠT(x, y)(ω+) = x[S+ − (1 + rB)S0] > 0, since S− < S+. Therefore, such a portfolio is an arbitrage unless S− < (1 + rB)S0.
• Similarly, for the second type we have ΠT(x, y)(ω±) = y(B0/S0)[(1 + rB)S0 − S±]. In the best case scenario ΠT(x, y)(ω+) = y(B0/S0)[(1 + rB)S0 − S+]. Hence S+ ≤ (1 + rB)S0 implies ΠT(x, y)(ω±) ≥ 0 and ΠT(x, y)(ω−)= y(B0/S0)[(1 + rB)S0 − S−] > 0, since S− < S+ . The portfolio is an arbitrage unless S+ > (1 + rB)S0.

Hence, by combining the two cases, we have shown that there is no arbitrage iff

$S^{-}<\left(1+r_B\right)S_0<S^{+}.$

Note that the result of Theorem 2.4 can be formulated in terms of asset values: there is no arbitrage iff $\frac{S^{-}}{S_0}<\frac{B_T}{B_0}<\frac{S^{+}}{S_0}$. If market prices do not allow for a profitable arbitrage, then the prices are said to constitute an arbitrage-free market. Later we shall see that the assumption that there is no arbitrage is used in quantitative finance to calculate unique (no-arbitrage) prices for derivatives that can be replicated.

In conclusion, let us demonstrate that if the initial price C0 of a claim is equal to the initial cost Π0 of the replicating portfolio in (2.5), then there is no arbitrage. In other words, if the actual initial price is less than or greater than the price in (2.5), then there is an arbitrage portfolio in the base assets B and S and the claim C. One way to argue this statement is to apply the Law of One Price. Indeed, since the payoff CT is identical to that of the replicating portfolio, ΠT , the present values C0 and Π0 have to be the same, or else an arbitrage exists. On the other hand, we can always form an arbitrage portfolio when C0 ≠ Π0, as demonstrated in the following example.

Example 2.7.

Consider a single-period binomial model with rB = 10%, rS− = −10%, rS+ = 20%, S0 = $100, and B0 = $10. Contract C has the following payoff: C− = 0 and C+ = $50.

• (a) Find portfolio (x, y) that replicates the payoff (C−, C+) and then calculate its initial cost Π0 = Π0(x, y).
• (b) Suppose that C0 > Π0(x, y), where (x, y) is the portfolio replicating the contract. Construct an arbitrage portfolio.

Solution. Applying (2.4) gives us the replicating portfolio:

$x=\frac{50-0}{100\cdot \left(0.2-\left(-0.1\right)\right)}=\frac{5}{3}\text{and}y=\frac{0\cdot \left(1+0.2\right)-50\cdot \left(1-0.1\right)}{10\cdot \left(1+0.1\right)\cdot \left(0.2-\left(-0.1\right)\right)}=\frac{150}{11}.$

The initial value of the portfolio is

${\displaystyle {\prod }_0^{\left(x,y\right)}=x{S}_0}+y{B}_0=\frac{5}{3}\cdot 100-\frac{150}{11}\cdot 10=\frac{1000}{33}\cong \$\mathrm{30.303.}$

Suppose that the actual price of the contract C0 is larger than Π0. Write and sell the contract for C0. Use the proceeds to buy $x=\frac{5}{3}$ shares of stock. If $C_0<x{S}_0=\frac{500}{3}$, then borrow $\frac{50}{3}-\frac{C_0}{10}$ bonds; otherwise, we invest the balance in bonds. So the portfolio (x, y, z) contains $x=\frac{5}{3}$ shares of stock S, $y=\frac{C_0}{10}-\frac{50}{3}$ bonds B, and z = −1 contracts C. The initial cost is zero. At the end of the period, we sell stock and pay CT to the holder of the contract. The balance is positive for every market scenario whenever $C_0>\frac{1000}{33}$:

${\displaystyle {\prod }_T^{\left(x,y,z\right)}=\underset{=0}{\underbrace{\frac{5}{3}{S}_T-\frac{150}{11}{B}_T-C_T}}}+\left(\frac{150}{11}-\frac{50}{3}+\frac{C_0}{10}\right)B_T=\frac{11}{10}\left(C_0-\frac{1000}{33}\right).$

2.3.3 Arbitrage in the Binomial Tree Model

In a multi-period model, there is more flexibility for an investment portfolio. The investor may alter the positions in the portfolio at any time by selling some assets and investing the proceeds in others. Therefore, the definition of an arbitrage investment strategy is a bit different in the comparison with the definition of an arbitrage portfolio for the single-period case.

Definition 2.4.

An admissible strategy such that Π0 = 0 and ℙ(Πt > 0) > 0 for some t =1, 2,... is called an arbitrage (strategy).

The cost of setting up an arbitrage strategy is zero. The self-financing condition means that there are no injections of funds at intermediate dates. The admissibility guarantees that the holder will not face a potential loss. At the maturity date, there is no loss since the terminal value is nonnegative. Moreover, there exists at least one scenario where liquidating the portfolio will result in a positive gain. In summary, an arbitrage strategy is a possibility of having a potential gain at no cost and without potential losses. Since the wealth of an admissible strategy is always nonnegative, the definitions of an arbitrage portfolio and an arbitrage strategy are the same for the single-period binomial model.

Theorem 2.5.

The binomial tree model admits no arbitrage iff d < 1 + r < u.

Proof. First, consider the case of a one-period binomial tree (i.e., T = 1). As was justified in Theorem 2.4, the rate r of interest on a risk-free investment has to satisfy d < 1 + r < u, or else an arbitrage possibility would arise. Indeed, for the one-step returns on the stock we have 1 + rS− = d and 1 + rS+ = u; the one-step return on the bond is r. There is no arbitrage iff rS− < rB < rS+, which is equivalent to d < 1 + r < u.

Now let us consider a multi-period binomial model. Suppose that 1 + r ≤ d or u ≤ 1 + r holds. To construct an arbitrage portfolio, we proceed as follows. At time 0, construct a portfolio which is long in stock if 1 + r ≤ d or short in stock if u ≤ 1 + r with zero initial value. At time 1, close the position in stock and invest the proceeds in risk-free bonds. As a result of these manipulations, we obtain a positive amount of cash invested in bonds.

Let d < 1 + r < u and suppose that there is an arbitrage strategy, i.e., there is a self-financing strategy with zero initial value such that Πt ≥ 0 for all t ≥ 0 with probability 1 and ΠT > 0 with nonzero probability at maturity time T. Find the smallest time t > 0 for which Πt(ω) > 0 at some state ω. Since each state in the model is a path in the binomial tree, we can find a one-step subtree with two branches, so that Πt−1 = 0 at its root, Πt ≥ 0 at each node growing out of this root with Πt > 0 in at least one of these nodes. Note that the path ω is passing through the root and the node where Πt > 0. In the one-step case this is impossible if d < 1 + r < u, leading to a contradiction.

2.3.4 Risk-Neutral Probabilities

Although the future prices of stock are unknown with certainty, it is natural to compare the expected return on stock and the risk-free rate of return. The expected stock prices under the real-world probability function ℙ are given by

$\text{E}\left[S_t\right]=S_0{\left(1+\text{E}\left[r_S\right]\right)}^t,t=0,1,2,...\text{ (2.32)}$

where rS denotes a one-period rate of return on the stock. Since rS = u − 1 with probability p and rS = d − 1 with probability 1 − p, we obtain

$\text{E}\left[r_S\right]=p\left(u-1\right)+\left(1-p\right)\left(d-1\right)=pu+\left(1-p\right)d-1.$

Suppose that the amount S0 is invested in a risk-free bank account. It will grow to S0(1 + r)t after t steps, where r is the compound risk-free interest rate. Clearly, to compare the expected return on the stock, E[St/S0], and the risk-free return, (1 + r)t, we only need to compare the average one-step rate E[rS] on the stock and the risk-free rate r. There exist three main types of investors.

• A typical risk-averse investor requires that E[rS] > r, arguing that she or he should be rewarded with a higher expected return as a compensation for risk.
• A risk-seeker investor may be attracted by the reverse situation when E[rS] < r, if a risky return is very high with small nonzero probability and low with large probability.
• A border situation of a market in which E[rS] = r is referred to as risk-neutral.

We now introduce a new probability function $\tilde{\mathbb{P}}$ with the probabilities of one-period upward and downward moves of the stock price $\tilde{\mathbb{P}}\left(\text{up}\right)=\tilde{p}$ and $\tilde{\mathbb{P}}\left(\text{down}\right)=1-\tilde{p}$, respectively, such that the risk-neutrality condition

$\tilde{\text{E}}\left[r_S\right]=\tilde{p}u+\left(1-\tilde{p}\right)d-1=r\text{ (2.33)}$

is satisfied. This implies that

$\tilde{p}=\frac{1+r-d}{u-d}\text{and}1-\tilde{p}=\frac{u-r-1}{u-d}.\text{ (2.34)}$

We shall call $\tilde{p}$ and $1-\tilde{p}$ the risk-neutral probabilities of the stock price upward and downward moves, respectively. The corresponding probability function is called the risk-neutral probability function (or measure) and is denoted by $\tilde{\mathbb{P}}$. Here $\tilde{\text{E}}$ denotes the mathematical expectation with respect to the probability function $\tilde{\mathbb{P}}$; it is called the risk-neutral expectation.

Theorem 2.6.

The binomial tree model admits no arbitrage iff there exists the risk-neutral probability $\tilde{p}\in \left(0,1\right)$.

Proof. It is clear from (2.34) that $0<\tilde{p}<1$ iff d < 1 + r < u. The latter is a necessary and sufficient condition of the absence of arbitrage.

To explain why $\tilde{p}$ is called a risk-neutral probability, we are going to compare the real-world expected return E[rS] and the risk-neutral expected return $\tilde{\text{E}}\left[r_S\right]=r$. Let us define the risk of the investment in the stock to be the standard deviation of the one-step return rS:

${\sigma }_S=\sqrt{\text{Var}\left(r_S\right)}=\sqrt{\text{E}\left[r_S^2\right]-{\left(\text{E}\left[r_S\right]\right)}^2}.$

This parameter is often called the volatility of stock price return. It follows that

${\sigma }_S^2=\text{Var}\left(r_S\right)={\left(u-d\right)}^{2}p\left(1-p\right).\text{ (2.35)}$

Let us compare the expected returns E[rS] and $\tilde{\text{E}}\left[r_S\right]$:

$\text{E}\left[r_S\right]-\tilde{\text{E}}\left[r_S\right]=up+d\left(1-p\right)-u\tilde{p}-d\left(1-\tilde{p}\right)=\left(p-\tilde{p}\right)\left(u-d\right).\text{ (2.36)}$

Let us assume that E[rS] ≥ r, that is, the expected return is not less than the risk-free return. Combining (2.35) with (2.36) gives

$\text{E}\left[r_S\right]-r=\frac{p-\tilde{p}}{\sqrt{p^{\left(1-p\right)}}}\sigma s.$

We say that one asset is riskier than another when it has a higher volatility of return. If the volatility is zero (i.e., we deal with a risk-free asset), then the expected return is just r; if the volatility is nonzero, then we have a higher expected return. This result fits well with reality—if you want a higher expected return you must take on more risk. However, when $p=\tilde{p}$, i.e., we deal with a risk-neutral market, the expected return is always r no matter what value the volatility σ has.

In reality, the risk-neutral probability $\tilde{p}$ has no relation to the real-world probability p. However, the risk-neutral probability function is of great practical importance to us with respect to computing no-arbitrage prices of derivative contracts. Let us consider a single-period binomial model. The fair price C0 of a derivative contract with payoffs C± = CT(ω±) in the two possible outcomes ω± is given by (2.5). By using the notation rB = r, rS+ = u − 1, and rS− = d − 1, we can rewrite (2.5) as follows:

$C_0=\frac{1}{1+r}\left[\frac{1+r-d}{u-d}{C}^{+}+\frac{u-r-1}{u-d}{C}^{-}\right]\cdot$

Substituting (2.34) in the above equation gives us a simple valuation formula:

$C_0=\frac{1}{1+r}\left[\tilde{p}{C}^{+}+\left(1-\tilde{p}\right)C^{-}\right]=\frac{1}{1+r}\tilde{\text{E}}\left[C_T\right]=\frac{B_0}{B_T}\tilde{\text{E}}\left[C_T\right].\text{ (2.37)}$

In other words, the no-arbitrage price of claim C is given by a risk-neutral expectation of the discounted future payoff function. The discounting factor is $\frac{B_0}{B_T}=\frac{1}{1+r_B}$. The interesting fact is that this formula works for more sophisticated models and general payoff functions.

2.3.5 Martingale Property

Equation (2.37) can be rewritten as follows:

$\tilde{\text{E}}\left[\frac{C_T}{B_T}\right]=\frac{C_0}{B_0}.\text{ (2.38)}$

In this case, the process ${\left\{\frac{C_t}{B_t}\right\}}_{t\in \left\{0,T\right\}}$ is said to be a martingale under the risk-neutral probability function $\tilde{\mathbb{P}}$. Now we proceed to the multi-period case.

It follows from (2.32) that the expectation of St with respect to the risk-neutral probability function $\tilde{\mathbb{P}}$ is

$\tilde{\text{E}}\left[S_t\right]=S_0{\left(1+r\right)}^t,\text{ (2.39)}$

since $\tilde{\text{E}}\left[r_S\right]=r$. In other words, the expected return on the stock under $\tilde{\mathbb{P}}$ is equal to the risk-free return over the same time interval from 0 to t.

Equation (2.39) can be extended to any time step in the binomial tree model. Suppose that t time steps have passed and the stock price has changed from S0 to St. Let us find the risk-neutral expectation of the price St+1 given the price St. Formally, we need to find the conditional expectation of St+1 given St. We can write St+1 = St Rt+1S. Since in the binomial tree model all single-period returns are i.i.d., the distribution of the return Rt+1S does not depend on the time t. The risk-neutral expectation of Rt+1S isequal to 1 + r. The expectation of St+1 given St is

$\tilde{\text{E}}\left[S_{t+1}|S_t\right]=\tilde{\text{E}}\left[S_t{R}_{t+1}^S|S_t\right]=S_t\tilde{\text{E}}\left[R_{t+1}^S|S_t\right]=S_t\tilde{\text{E}}\left[R_{t+1}^S\right]=S_t\left(1+r\right).\text{ (2.40)}$

The above derivation is based on the following two facts. First, St is given and hence can be taken out of the expectation. Second, since stock returns are mutually independent, Rt+1S is independent of St and hence the last expectation becomes an unconditional one. Now we introduce discounted stock prices:

$\overline{S_t}=\frac{S_t}{B_t}=\frac{S_t}{B_0{\left(1+r\right)}^t},t\ge 0.$

Dividing both sides of (2.40) by the bond price Bt+1 and using Bt+1 = Bt (1 + r) give

$\tilde{\text{E}}\left[\frac{S_{t+1}}{B_{t+1}}|S_t\right]=\frac{S_t\left(1+r\right)}{B_{t+1}}=\frac{S_t\left(1+r\right)}{B_t\left(1+r\right)}=\frac{S_t}{B_t}.$

Thus, (2.40) takes the form

$\tilde{\text{E}}\left[{\overline{S}}_{t+1}|{\overline{S}}_t\right]={\overline{S}}_t,t\ge 0\text{ (2.41)}$

We say that the discounted stock price process ${\left\{{\overline{S}}_t\right\}}_{t\ge 0}$ is a martingale under the risk-neutral probability measure $\tilde{\mathbb{P}}$.

2.3.6 Risk-Neutral Log-Normal Model

In Subsection 2.2.5, we derived the log-normal price model as the limiting case of a sequence of binomial tree models as the number of periods N approaches infinity. Let r be the risk-free interest rate under continuous compounding. The equivalent one-period interest rate rN in the binomial tree model with N periods is rN = erδN − 1, where ${\delta }_N=\frac{T}{N}$. As was demonstrated in Subsection 2.2.5, the log-return L[0, N](N) = ln (SN(N)/S0) is approximately normal, as N → ∞. Let us find the parameters of the limiting normal distribution under the risk-neutral probability measure. In the N-period model, the risk-neutral probability of the upward movement of the stock price is

${\tilde{p}}_N=\frac{r_N+1-d_N}{u_N-d_N}=\frac{e^{{\delta }_{Nr}}-e^{-\sqrt{{\delta }_{N\sigma }}}}{e^{\sigma \sqrt{{\delta }_N}}-e^{-\sigma \sqrt{{\delta }_N}}}.$

Introduce the risk-neutral probability function ${\tilde{\mathbb{P}}}_N$ with the probability ${\tilde{p}}_N$ of an upward movement. Under this probability function, the normalized log-return, $L_N^{*}:=\frac{L_N-E\left[L_N\right]}{\sqrt{\text{Var}\left(L_N\right)}}$, where $L_N\equiv L_{\left[0,N\right]}^{\left(N\right)}$, is expressed in terms of a binomial random variable with probability ${\tilde{p}}_N$ of success, $Y_N~Bin\left(N,{\tilde{p}}_N\right)$, as given in (2.31). It is not difficult to compute the expectation and variance of the log-returns under the risk-neutral probability ${\tilde{\mathbb{P}}}_N$:

${\text{E}}_{{\tilde{\mathbb{P}}}_N}\left[L_N\right]=\left(2{\tilde{p}}_N-1\right)\sigma \sqrt{NT},{\text{Var}}_{{\tilde{\mathbb{P}}}_N}\left(L_N\right)={\tilde{p}}_N\left(1-{\tilde{p}}_N\right)4{\sigma }^{2}T.$

To find the distribution of limN → ∞ LN, we need to know the limiting values of the expectation and variance.

Proposition 2.7.

As N → ∞, we have the following limiting behaviour:

${\tilde{p}}_N\to \frac{1}{2},{\text{E}}_{{\tilde{\mathbb{P}}}_N}\left[L_N\right]\to rT-\frac{1}{2}{\sigma }^{2}T,{\text{Var}}_{{\tilde{\mathbb{P}}}_N}\left(L_N\right)\to {\sigma }^{2}T.$

Proof. The proof is left as an exercise for the reader.     □

By the de Moivre–Laplace theorem, the probability distribution of log-returns converges weakly to the normal distribution as N → ∞. Under the risk-neutral probability, the asymptotic distribution of LN, as N → ∞, is $Norm\left(\left(r-\frac{1}{2}{\sigma }^2\right)T,{\sigma }^{2}T\right)$. Therefore, in the limiting case, the risk-neutral probability distribution of the stock price S(T) = limN → ∞ SN(N) is the log-normal distribution:

$S\left(T\right)=S\left(0\right){\text{e}}^{\left(r-{\sigma }^2/2\right)T+\sigma \sqrt{TZ}},\text{ (2.42)}$

where Z ~ Norm(0, 1). The interesting fact is that the limiting distribution does not depend on the real-world expected return μ on the stock. In the risk-neutral binomial tree model, the expected return on the stock is the same as that of the risk-free bond. It is not difficult to check that in the limiting case we observe the same behaviour for the risk-neutral log-normal price model:

$\tilde{\text{E}}\left[S\left(T\right)\right]=S\left(0\right){\text{e}}^{rT}.$

2.4 Value at Risk

In the beginning of this chapter, we defined a risky asset as that with uncertain future cash flows. Examples of such assets include stocks, derivative contracts, defaultable bonds, and similar contingent claims subject to default risk. To distinguish risky and risk-free assets, we need to take a look at the distribution of their returns. The return of a risky asset is uncertain. Hence, from the mathematical point of view, it may be viewed as a random variable with nonzero variance. The return on a risk-free asset is certain, so its variance is zero. Therefore, the risk associated with an asset (with return R) can be measured by computing the standard deviation of the return on the asset: $\sigma =\sqrt{\text{Var}\left(R\right)}$. Clearly, a risk-averse investor prefers an asset with lower σ. However, the value of σ may not tell us how large the loss may be. The variance and expectation of the return on a risky asset alone define the shape of the profit and loss distribution only when the asset return has a normal distribution (or Student's t-distribution).

One can use other market risk metrics to measure the uncertainty in the portfolio return. For example, one may be interested in the probability of loss L on a specific financial asset (or a portfolio of assets) over some period of time being less than a given amount. That is, one may wish to evaluate the probability ℙ(L < A) for a given loss L and amount A. Let us reverse the question and find an amount A so that the probability of a loss not exceeding this amount is equal to a given probability, say 95% (although we may consider another confidence level such as 90% or 99%). That is, find A such that ℙ(L ≤ A) = 95%. This value is referred to as Value at Risk and denoted by VaR.

The VaR is a measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, probability level, and time horizon, VaR is defined as a threshold level such that the probability that the loss on the portfolio over the given time horizon exceeds this level is equal to the given probability.

VaR has two basic parameters: the significance level denoted α ∊ (0, 1) (or confidence level denoted 1 − α) and the risk horizon denoted h, which is the period of time over which we measure the potential loss. Traditionally, h is measured in trading days. Common parameters for VaR are 1% and 5% significance levels and 1-day and 10-day risk horizons, although other combinations are also in use. When VaR is computed, it is assumed that the current position in the portfolio of interest will remain unaltered over the chosen time period. Let Πt denote the value of the portfolio at the time t. The value of the same portfolio at the future time t + h, discounted to time t, is Z(t, t + h)Πt + h, where Z(t, t + h) is the price of a unit zero-coupon bond that matures at the time t + h. For example, for the case with continuously compounded interest, we have that Z(t, t + h) = e−jh (where j is the daily interest rate). So we use a zero-coupon bond as a discounting factor. The discounted profit-and-loss (P&L) over a risk horizon of h days is

$G_h:=Z\left(t,t+h\right){\displaystyle {\prod }_{t+h}-}{\prod }_t.$

In other words, Gh is the present value of the gain from an investment; hence, −Gh is the present value of the loss. Since the future value Πt + h is uncertain, the discounted P&L is a random variable. To calculate the VaR of the portfolio, we need to know the distribution of this random variable.

Given the significance level α and risk horizon h, the 100α% h-day VaR is defined as the present value of the largest possible loss amount that would be exceeded with only a probability α over an h-day time period:

${\text{VaR}}_{\alpha ,h}=-\inf \left\{x\in ℝ:\mathbb{P}\left(G_h>x\right)\le 1-\alpha \right\}=-\inf \left\{x\in ℝ:F_{G_h}\left(x\right)\ge \alpha \right\},\text{ (2.43)}$

where $F_{G_h}\left(x\right)=\mathbb{P}\left(G_h\le x\right)$ is the CDF of Gh. For a continuous price model with a continuous and strictly monotonic distribution function $F_{G_h}$, it is possible to solve the equation $F_{G_h}\left(x\right)=\alpha$ for x and then the VaR is defined as the α-quantile of the discounted h-day P&L distribution:

${\text{VaR}}_{\alpha ,h}=-g_{\alpha ,h},\text{where}\mathbb{P}\left(G_h\le g_{\alpha ,h}\right)=\alpha .$

Note that for a portfolio loss in h days, the present value change gα, h is negative and hence the VaR value is positive, i.e., VaR measures losses and is reported as a positive amount that corresponds to the loss.

When VaR is estimated from a P&L distribution, it is given in value terms. However, one may prefer to analyze the return distribution rather than the P&L distribution. In this case, the VaR is expressed as a percentage of the current value of the portfolio. The discounted h-day return on the portfolio is

$\frac{Z\left(t,t+h\right){\prod }_{t+h}-{\prod }_t}{{\prod }_t}.$

To calculate the VaR, we first find rα, h, the α-quantile of the return distribution:

$\mathbb{P}\left(\frac{Z\left(t,t+h\right){\prod }_{t+h}-{\prod }_t}{{\prod }_t}<r_{\alpha ,h}\right)=\alpha ,$

and then the VaR is given by

${\text{vaR}}_{\alpha ,h}=\{\begin{array}{l}-r_{\alpha ,h}\text{as}\text{a}\text{percentage}\text{of}\text{the}\text{portfolio}\text{value}{\prod }_t,\hfill \\ -r_{\alpha ,h}{\prod }_t\text{as}\text{a}\text{quantity}\text{in}\text{value}\text{terms}.\hfill \end{array}$

Example 2.8.

Assume that the discounted P&L is a normal random variable with mean μ and variance σ2. Calculate 1% VaR.

Solution. We have that the discounted P&L is G = μ + σZ for some Z ~ Norm(0, 1). We need to find g so that

$\alpha =0.01=\mathbb{P}\left(G<g\right)=\mathbb{P}\left(\frac{G-\mu }{\sigma }<\frac{g-\mu }{\sigma }\right)=\mathbb{P}\left(Z<\frac{g-\mu }{\sigma }\right)=\mathcal{N}\left(\frac{g-\mu }{\sigma }\right).$

Let us find the 0.01-quantile x for the standard normal distribution. We use the table of the standard normal CDF to find x such that $\mathcal{N}\left(x\right)=0.01$. By symmetry, $\mathcal{N}\left(-x\right)=1-0.01=0.99$. From the table we have $\mathcal{N}\left(z=2.33\right)=0.9901$ as the closest value. Hence, x = −z = −2.33, and the VaR value is given by −g = −(μ + σx)= −μ + 2.33σ. Therefore, among investments whose gains are normally distributed, the VaR criterion would select the one having the largest value of −μ + 2.33σ.

The VaR with significance level α gives us a value that has only a 100%α chance of being exceeded by the loss from an investment. However, this value does not tell us what the actual loss may be. It has been suggested that the conditional expected loss given that it exceeds the VaR is a better metric of the risk. This conditional expected loss is called the conditional value at risk or CVaR. The CVaR criterion is to choose the investment having the smallest CVaR.

Example 2.9.

Assume that the discounted P&L is a normal random variable with mean μ and variance σ2. Calculate 1% CVaR.

Solution. We have that the discounted P&L is G = μ + σZ with Z ~ Norm(0, 1). The CVaR is given by

$\begin{array}{ll}\text{CVaR}\hfill & =\text{E}\left[-G|-G>\text{VaR}\right]=\text{E}\left[-G|-G>-\mu +2.33\sigma \right]\hfill \\ \hfill & =\text{E}\left[\sigma \left(\frac{-G+\mu }{\sigma }\right)-\mu |\frac{-G+\mu }{\sigma }>2.33\right]\hfill \\ \hfill & =\sigma \text{E}\left[\frac{-G+\mu }{\sigma }|\frac{-G+\mu }{\sigma }>2.33\right]-\mu =\sigma \text{E}\left[Z|Z>2.33\right]-\mu .\hfill \end{array}$

For a standard normal random variable Z we have

$\begin{array}{ll}\text{E}\left[Z|Z>a\right]\hfill & =|\frac{\text{E}\left[Z{\mathbb{I}}_{\left\{Z>a\right\}}\right]}{\mathbb{P}\left(Z>a\right)}=\frac{1}{\mathbb{P}\left(Z>a\right)}{\displaystyle {\int }_a^{\infty }zn\left(z\right)\text{d}z}\hfill \\ \hfill & =\frac{1}{\mathbb{P}\left(Z>a\right)}{\displaystyle {\int }_a^{\infty }\frac{1}{\sqrt{2\pi }}{\text{e}}^{-z^2/2}\text{d}\left(z^2/2\right)=\frac{1}{\sqrt{2\pi }\mathbb{P}\left(Z>a\right)}{\text{e}}^{-a^2/2}}\hfill \end{array}$

for any real a. Hence, we obtain that

$\text{CVaR}=\sigma \frac{1}{\sqrt{2\pi }0.01}{\text{e}}^{-{2.33}^2/2}-\mu \cong -\mu +2.64\sigma ,$

since ℙ(Z > 2.33) ≅ 0.01.

2.5 Dividend Paying Stock

Consider a stock (with the price process {S(t)}t≥0) that pays dividends. Every moment a dividend payment is made, the price of one share instantaneously drops down by the amount of the dividend payment or otherwise an arbitrage opportunity would arise. Indeed, one can buy a share of stock right before a dividend payment is made, receive the payment, and then immediately sell the share. There is an arbitrage profit if the stock price is not adjusted when the dividend payment is made. Here, we assume that there is no delay between the ex-dividend date and the date when shareholders receive dividend payments. Note that in the U.S., the Internal Revenue Service (IRS) defines the ex-dividend date as “the first date following the declaration of a dividend on which the buyer of a stock is not entitled to receive the next dividend payment.”

Suppose that a dividend payment div(t*) is made at time t*. This payment can be given as a monetary amount or as a percentage of the spot price, i.e., div(t*) = d*S(t*) with the dividend percentage 0 ≤ d* ≤ 1. The price of one share immediately after the dividend payment must be S(t*) − div(t*) = S(t*) − d*S(t*) = (1−d*)S(t*) or otherwise an arbitrage opportunity exists. We illustrate this idea with a single-period model. Let S be the stock price at the beginning of period. At the end of period, the (pre-dividend) price is Su with probability p or Sd with probability 1 − p. After the dividend is paid, the price goes down by the dividend amount to become Su(1 − d*) or Sd(1 − d*). This situation is illustrated in Figure 2.5.

Suppose that the dividend on a single share is used to purchase

$\frac{{\text{d}}_{*}S\left(t_{*}\right)}{\left(1-{\text{d}}_{*}\right)S\left(t_{*}\right)}=\frac{{\text{d}}_{\text{*}}}{1-{\text{d}}_{\text{*}}}$

additional shares at time t*. So, each share in our portfolio will grow to $1+\frac{{\text{d}}_{*}}{1-{\text{d}}_{*}}=\frac{1}{1-{\text{d}}_{*}}$ shares right after time t*. The market value of one share is

${\Pi }_t=\{\begin{array}{ll}S\left(t\right)\hfill & \text{if}t<t_{*},\hfill \\ \frac{1}{1-{\text{d}}_{*}}S\left(t\right)\hfill & \text{if}t\ge t_{*}.\hfill \end{array}$

So, we may use the same asset price model without making dividend adjustments, if the dividends are assumed to be reinvested in the stock.

Let the dividends be paid at times tk = kΔt, k = 1, 2, ..., m, distributed evenly with the step size $\Delta t=\frac{T}{m}$. Let dm be the dividend percentage. Starting with one share at time 0, the market value of our portfolio at time T after the mth dividend payment is ${\Pi }_T=\frac{1}{{\left(1-d_m\right)}^m}S\left(T\right)$. Let m → ∞, so that dm/Δt → q with q > 0, then

$\frac{1}{{\left(1-{\text{d}}_m\right)}^m}\to {\text{e}}^{qT}.$

The stock is said to pay dividends continuously at a rate of q. If the dividends are reinvested in the stock, then an investment in one share held at time 0 will increase to become ${\text{e}}^{qT}$ shares at time T. Therefore, we need to start with ${\text{e}}^{-qT}$ shares at time 0 to obtain 1 share at time T.

2.6 Exercises

1. Exercise 2.1. At time 0, the value of a risk-free bond is B0 = 100, and the stock price is S0 = 100. Suppose that the annual risk-free interest rate is r = 5%, and the one-year return on the stock is

   $r_S=\{\begin{array}{ll}10\%\hfill & \text{with}\text{probability}60\%\hfill \\ -5\%\hfill & \text{with}\text{probability}40\%\hfill \end{array}$

   • (a) Find positions x and y so that the wealth Πt = xSt + yBt of the portfolio (x, y) at time t = T = 1 is

     ${\Pi }_T=\{\begin{array}{ll}\$1000\hfill & \text{if}\text{the}\text{stock}\text{price}\text{goes}\text{up}\hfill \\ \$1500\hfill & \text{if}\text{the}\text{stock}\text{price}\text{goes}\text{down}\hfill \end{array}$

   • (b) What is the expected return of the portfolio over the first year?
2. Exercise 2.2. A variant of the one-price theorem. Assume that there are no arbitrage portfolios. Suppose there are two assets X and Y with initial prices X0 and Y0. At some time T > 0, let XT(ω) ≥ YT(ω) hold for all states of the world and XT(ω′) > YT(ω′) hold for at least one state ω′. Prove that X0 > Y0. [Hint: Suppose the converse is true and then construct an arbitrage portfolio.]
3. Exercise 2.3. Consider two investments with respective wealth functions Vt and Wt, with t ∊ [0, T]. Suppose that V0 > W0 and ℙ(VT ≤ WT) = 1. Find an arbitrage opportunity.

4. Exercise 2.4. In the setting of Exercise 2.1

   • (a) find the risk-neutral probabilities $\left\{\tilde{p},1-\tilde{p}\right\}$ for this binomial model;

   • (b) verify that under the risk-neutral probabilities we have

     $\tilde{\text{E}}\left[\frac{S_1}{B_1}\right]=\frac{S_0}{B_0}.$

   In the latter case, the discounted stock price process St/Bt is said to be a martingale.

5. Exercise 2.5. Consider a single-period market model with a risk-free asset B, such that B0 = 10 and B1 = 11, and a risky asset, the price of which can follow three possible scenarios:

   Scenario	S0	S1
   ω1	50	70
   ω2	50	55
   ω3	50	40

   • (a) Show that the model admits no arbitrage opportunities.

   • (b) Find risk-neutral probabilities ${\tilde{p}}_i=\tilde{\mathbb{P}}\left({\omega }^i\right)$ of the scenarios, such that

     $\tilde{\text{E}}\left[S_1/B_1\right]=S_0/B_0$

     holds. Find the general solution that will depend on a variable parameter. Find the range for that parameter so that ${\tilde{p}}_1,{\tilde{p}}_2,{\tilde{p}}_3$ define a probability function.

6. Exercise 2.6. Consider a single-period market model with three states of the world and two base assets: a risky stock S and risk-free money market account A. Suppose that the possible stock prices at time t = T are as follows:

   $S_T=\{\begin{array}{ll}{S}^u\hfill & \text{with}\text{probability}{p}_1>0,\hfill \\ S^m\hfill & \text{with}\text{probability}{p}_2>0,\hfill \\ S^d\hfill & \text{with}\text{probability}{p}_3=1-p_1-p_2>0,\hfill \end{array}$

   where 0 < Sd < Sm < Su. Let the initial investment in a risk-free bond be equal to the current stock price, i.e., B0 = S0, Prove that at time t = T we have that Sd < BT < Su or else an arbitrage possibility would arise. In the latter case, construct an arbitrage portfolio.

7. Exercise 2.7. Consider a market with a risk-free bond B, for which B0 = 50, B1 = 55, and B2 = 60, and a risky stock with the spot price S0 = 50. Suppose that the stock price at times t = 1 and t = 2 can follow four possible scenarios:

   Scenario	S1	S2
   ω1	60	70
   ω2	60	55
   ω3	45	45
   ω4	45	40

   • (a) Find an arbitrage investment strategy if there are no restrictions on short selling.
   • (b) Is there an arbitrage opportunity if no short selling of the risky asset is allowed?
8. Exercise 2.8. Given the bond and stock prices in Exercise 2.7, is there an arbitrage strategy if short selling of stock is allowed, but transaction costs of 5% of the transaction volume apply whenever stock is traded (purchased or sold)?
9. Exercise 2.9. Given the bond and stock prices in Exercise 2.7, except that now assume S2(ω2) = S2(ω3) = 50, and the probabilities $\mathbb{P}\left({\omega }^1\right)=\frac{17}{33},\mathbb{P}\left({\omega }^2\right)=\frac{5}{33},\mathbb{P}\left({\omega }^3\right)=\frac{10}{33}$, and $\mathbb{P}\left({\omega }^4\right)=\frac{1}{33}$, show that the discounted stock price process St/Bt, t = 0, 1, 2, is a martingale under this measure ℙ, i.e., show that $\text{E}\left[\frac{S_{t+1}}{B_{t+1}}|S_t\right]=\frac{S_t}{B_t}$ for t = 0, 1.
10. Exercise 2.10. Consider a binomial model with rB = 10%, rS− = −10%, rS+ = 20%, S0 = 100, and B0 = 10. On the (x, y)-plane draw a domain representing the set of all portfolios (x, y) that lead to a self-financing one-step strategy with x1 = x stock shares and y1 = y bonds.
11. Exercise 2.11. Consider a one-period binomial tree model with rB = 5%, rS− = −10%, rS+ = 15%, S0 = $100, and B0 = $100. Construct a self-financing strategy with an initial value of $10,000 such that, at all times, the stock investment is twice that invested in risk-free bonds.
12. Exercise 2.12. Prove Proposition 2.7.
13. Exercise 2.13. Let Z ~ Norm(0, 1). Find the mathematical expectation of X = eaZ + b with a, b ∊ ℝ. Use the result obtained to find the variance Var(X) = E[X2] − E[X]2.
14. Exercise 2.14. Consider the log-normal price model $S\left(T\right)=S\left(0\right){\text{e}}^{\mu T+\sigma \sqrt{T}Z}$, where Z ~ Norm(0, 1), with drift parameter μ = 0.02 and volatility parameter σ = 0.2. If S0 = 100, find (a) E[S(5)], (b) ℙ(S(5) > 100), (c) ℙ(S(5) < 110).

15. Exercise 2.15. Consider the log-normal price model with the risk-neutral dynamic

    $S\left(T\right)=S\left(0\right){\text{e}}^{\left(r-{\sigma }^2/2\right)T+\sigma \sqrt{T}Z},$

    where Z ~ Norm(0, 1). Show that $\tilde{\text{E}}\left[{\text{e}}^{-rT}S\left(T\right)\right]=S\left(0\right)$.

16. Exercise 2.16. To calibrate an N-period binomial tree model, one needs to solve the simultaneous equations (2.21)–(2.23).

    • (a) Find the exact solution to (2.21)–(2.23).
    • (b) Let the condition uN dN = 1 in (2.23) be replaced by $p_N=\frac{1}{2}$. Find uN and dN satisfying (2.21)–(2.22) and this new condition.