13.1.2.1 Pricing in a One Period Binary Model

Let us consider two games related to the same tossing of a coin. The first game is such that the player receives € when heads occur and € when tails occurs, where . The participation to this game can be compared to buying a stock. We denote with the random variable with and .

The second game is such that the player receives 1 € when heads occur and 0 € when tails occurs. The participation to this game can be compared to buying a derivative. Indeed, the second game can be considered as a derivative because the payoff in the second game is random variable for , where and . Random variable has the distribution and . The third asset is a bond with value . The price of bond is 1 and the price of stock is denoted with . We want to find the price of the derivative.

The derivative can be replicated with the bond and the stock: Consider the portfolio with bonds and stocks. We choose

The replicating portfolio is . Indeed, we have that , because

By the law of one price, to exclude the possibility of arbitrage, the price of the derivative has to be equal to the price of the portfolio:⁴

The price of the portfolio is

Thus, the price of the derivative is

13.1

The price of the derivative is in general not equal to 0.5. If , then the price of the derivative is . If , then the price of the derivative satisfies .⁵

We have given the price of the derivative in (13.1) in terms of the price of the stock. This is an example of relative pricing: a price of an asset is expressed in terms of the prices of another asset.

13.1.2.2 Arbitrage

Arbitrage is both a term of everyday language and a technical term used in mathematical finance.

Arbitrage is used in everyday language to denote a financial operation where one obtains a profit with probability one by a simultaneous selling and buying of assets. We give two examples of this type of arbitrage.

1. 1. The stock of Daimler is listed both in Frankfurt and Stuttgart stock exchanges. If the stock can be bought in Frankfurt with the price of 10 € and sold in Stuttgart with the price of 11 €, we obtain a risk free profit of 1 € (minus the transaction costs).
2. 2. Suppose the price of a stock is 10 € and a call option with strike price € with the expiration time in 1 week can be bought with the price of 1 €. Then, we can sell the stock short and buy the call option. The profit of the operation will be € (buying the call costs 1 €, selling the stock short gives 10 €, and exercising the option costs 8 €).

   In general, we have a lower bound for the price of a call option, where is the price of the stock at the time of buying the option, and is the strike price. See (14.9) for a more precise lower bound.

In mathematical finance, an arbitrage is a financial operation whose payoff is always nonnegative and sometimes positive, that is, the probability of a nonnegative payoff is one and the probability of a positive payoff is greater than zero. More formally, arbitrage portfolio is such that its value at time satisfies but its value at a later time satisfies and . A reasonable system of prices should be such that arbitrage is excluded, so that there does not exist an arbitrage portfolio.

The absence of arbitrage implies the law of one price.

13.1.2.3 The Law of One Price and the Monotonicity Theorem

The law of one price states that if two financial assets have the same payoffs then they have the same price: If two portfolios satisfy

then their prices are equal at a previous time :

The absence of arbitrage implies that the law of one price holds. Indeed, consider the case where the law of one price does not hold. Then we have two assets with different prices at time , say , and the prices of the assets are the same with probability 1 at a later time : . Then we can by the cheaper asset at time and sell the more expensive asset at time to obtain the amount . This amount can be put into a bank account. At time the two assets have the same price, and thus we have locked the profit of time . We have shown that there exists an arbitrage opportunity. Thus we have shown that the absence of arbitrage implies that the law of one price holds.

The monotonicity theorem states that if two financial assets satisfy

then their prices satisfy

at time . Furthermore, if , then their prices satisfy at time . This formulation of the monotonicity theorem is similar to the formulation of Blyth (2014, p. 48).

The law of one price implies the linearity of the pricing function. Let

be a portfolio, and let be the prices of the basis assets at time . Then the price of the portfolio at time is

13.2

13.1.2.4 Pricing using The Law of One Price

The law of one price can be used to price linear assets by replication.⁶ Furthermore, the law of one price can be used to price all assets in complete markets. By a market we mean a collections of tradable assets together with assumptions about the probability distributions of the asset values. A complete market is such that any possible payoff can be obtained by a portfolio of assets. That is, assume that the market has tradable assets . Assume that an arbitrary payoff can be obtained, so that . The law of one price implies that price of this payoff is

where we applied the linearity in (13.2).

Futures are linear derivatives, and thus the law of one price can be used to price futures; see Section 14.1. Futures can be priced by the law of one price because futures can be defined as a portfolio of the underlying asset and a bond: the payoff of a futures contract is a linear combination of the payoffs of the underlying asset and a bond.

The payoff of an option is not a linear function of the payoff of the underlying. Thus options cannot be priced as easily as futures. The law of one price can be used to price options in the Black–Scholes model, because the Black–Scholes model is a complete model for the markets, so that all derivatives can be replicated (linearly).

The law of one price can be used to derive the put-call parity, which says that the prices of two options satisfy an equation. The law of one price can also be used to give bounds to option prices without assuming the Black–Scholes model, or any other market model. See Section 14.1.2 for the derivation of the put-call parity.

13.1.3.1 Pricing in a One Period Ternary Model

Let us have two games related to the same tossing of a dice. The first game is such that the player receives € when the dice shows 1 or 2, € when the dice shows 3 or 4, and € when the dice shows 5 or 6, where . The participation to this game is an analogy to buying a stock and we denote with the random variable with .

The second game is such that the player receives 0 € when the dice shows 1, 2, 3, or 4 and 1 € when the dice shows 5 or 6. The participation to this game is an analogy to buying a derivative and we denote with the random variable , where is defined by when and when . Now and . The third asset is a bond with value . The price of the bond is 1 and the price of the stock is denoted with . We want to find the price of the derivative.

The derivative cannot be replicated with the bond and the stock: Consider the portfolio with bonds and stocks. The portfolio is . We have when and satisfy

We can typically not find such and because, in general, two parameters cannot satisfy three equations simultaneously. To obtain an approximate replication we could choose and so that is minimized. We have that

Since the probabilities are all equal to , we get the least squares solution for :

where

The solution is

We set the price of the derivative to be equal to the price of the approximately replicating portfolio:

If , then .

13.1.3.2 Statistical Arbitrage and the Law of Approximate Price

Statistical arbitrage is a financial operation where a profit is obtained with a high probability. The principle of excluding statistical arbitrage is a pricing principle, which can be used when the principle of excluding arbitrage does not apply. However, the concept of statistical arbitrage can be defined in many ways. Let us compare the principle of excluding arbitrage to the idea of excluding statistical arbitrage.

1. 1. Excluding arbitrage. The value of a derivative is at time . Let us have another asset whose value is at time . Assume that the values are equal with probability 1: . Then it should hold that the value of the derivative and the other asset are equal at all previous times: for all previous times . Otherwise, there would be an arbitrage opportunity: sell the more expensive instrument and buy the cheaper instrument to obtain a risk free profit at time .
2. 2. Excluding statistical arbitrage. The value of a derivative is at time . Let us have an other asset whose value is at time . If the random variables and are “close,” then the prices and should be close at all previous times . The closeness of random variables can be defined in many ways. For example, we can say that two random variables and are close when is small. A derivative can be priced with statistical arbitrage if we can construct an asset, which replicates the payoff of the derivative with high probability.

Pricing with statistical arbitrage requires that we define the best approximation to a random payoff . As an example, we can consider a call option written at time , with the strike price and with the expiration time . The payout of the option at the expiration time is , where is the price of the underlying instrument at time . The best constant approximation of random variable in the sense of mean squared error is its expectation:⁷

where the minimization is taken with respect to all real numbers. Thus, expectation can give a first approximation to the price of . We can use the underlying asset to provide a better approximation. The best approximation of with a function of is the conditional expectation:

where the minimization is taken with respect to functions , and function takes values . Thus, the conditional expectation could be a candidate for the fair price. However, the conditional expectation is typically not a tradable asset, and we will make a further restriction to find such function , which is tradable, which leads to linear approximations.

13.2.1.1 Filtered Probability Spaces

The underlying probability space is accompanied with a filtration of sigma-algebras .⁸ The price process of stocks is adapted with respect to the filtration: is measurable with respect to .⁹ The price process of the bond is predictable with respect to the filtration: is measurable with respect to , , and is measurable with respect to . Thus, the value of is known at time , which makes locally riskless. The prices are assumed to be nonnegative. We assume that¹⁰

Thus, and elements of are constants (with probability 1).

13.2.1.2 Trading Strategies

A trading strategy is

The values and express the quantity of the bond and the th asset held between and . The trading strategy is predictable: and are measurable with respect to . This means that and are determined at time , using the information available at time .

13.2.1.3 Examples

Let us give examples of the locally riskless bond . We can take , where is a constant, or , where . In addition, we can take , and

for , where is predictable. We can also take and , where is predictable.

Consider the two period binary model as an example of adaptability and predictability. Now and . The initial stock price is . The next price is and the final price is , where and are random variables. Random variable satisfies and for , where is a fixed constant. Random variable has the same distribution as , and is independent of . Choose

where and refer to the upwards and downwards movements of the stock. Set describes all possible trajectories of the process. Now,

Let . It follows that

In order for the stock prices to be adapted to the filtration we need

We have that

It follows that in order for the stock prices to be adapted to the filtration we need

Let the initial bond price be . Let the next bond price be , where is a constant. Let the final bond price be , where . Now bond prices are predictable with respect to the filtration. Bond price at time depends only on the stock price at time . Thus, the bond price at time is a random variable which is known at time .

13.2.2.1 The Wealth and Value Processes

We use the following notation for the inner product:

At time 0 the initial wealth is and after that,

Indeed, the portfolio vector is chosen at time and hold during the period .

We assume that for all and choose the bond as a numéraire. The discounted price process is defined by

We denote

13.3

The value process is defined as and

13.2.2.2 The Wealth Process under Self-financing

We assume in most cases that the trading strategy is self-financing. The local quadratic hedging without self-financing in Section 16.2.3 is a case where self-financing is not assumed.

Let us describe trading under the condition of self-financing. At time 0 the initial wealth is . The wealth is allocated among the available assets: the quantities are chosen so that

The prices change from to , and the wealth changes accordingly from to . After that, wealth is allocated among the available assets. We obtain

We continue in this way to obtain

The final wealth is

At time we need not do the reallocation, because it is the last time instance.

We have described a process of trading, which is self-financing. We say that a trading strategy is self-financing if

13.4

When the trading strategy is self-financing, then no external funds are received, and no funds are reserved for consumption.

Under the assumption (13.4) of self-financing, the change of wealth can be written as

for . Thus, the wealth at time can be written as¹¹

13.5

where .

13.2.2.3 The Value Process Under Self-Financing

Let us assume that the rebalancing is made respecting the condition of self-financing: is chosen so that the wealth is allocated among the assets. Equation sets a linear constraint on the vector . It is convenient to write the wealth process so that the quantity of bonds is eliminated, and this can be done using the value process.

The self-financing condition in (13.4) implies that the discounted price process in (13.3) satisfies

13.6

Similarly as for the wealth process, it holds that

13.7

Under the condition (13.6) of self-financing, an increment of the value process can be written as

where the last equality follows because the first element of is 1 for all . Thus, the value at time can be written as

13.8

Note that the value process is written in terms of the quantity of stocks. The quantity of bonds is obtained from the equations

13.9

which follow from self-financing equations (13.7).

The gains process is defined as

For a self-financing strategy

13.10

The gains process is a discrete stochastic integral. The gains process is a transformation of by means of .¹²

The value process can be used to derive some expressions for the wealth. For example, when , then¹³

13.2.3.1 The Existence of a Risk Neutral Measure Implies No-Arbitrage

We think that it is instructive to prove the result first for the case , and after that for the general case .

A Proof in the One Period Model

Assume that there exists an equivalent martingale measure . The martingale measure satisfies

for . The value process is

Take a portfolio such that . Then,

Thus, we cannot have . Thus, we cannot have , and we cannot have , and cannot be an arbitrage opportunity.

A Proof in the Multiperiod Model

A proof can be found in Shiryaev (1999, p. 417). We assume that there exists a martingale measure , which is equivalent to and such that is a -dimensional martingale with respect to , where . We noted in (13.10) that the value process satisfies

where

Note that since is a martingale with respect to , then sequence is a martingale transformation with respect to , when a martingale transformation is defined by (13.11).

Let be a strategy with , and , so that , and . Let us assume that for , where is a constant. Then is a martingale, and . Since , then , which implies , and .

The case of unbounded is handled in Shiryaev (1999, p. 98, Chapter II §1c)¹⁵ and in Föllmer and Schied (2002, Theorem 5.15, p. 229).

13.2.3.2 A Construction of an Equivalent Martingale Measure

We have taken the proof from Shiryaev (1999, p. 413), which follows the ideas of Rogers (1994). The construction of equivalent martingale measures is based on the Esscher conditional transformations. The Esscher transforms were used also in Gerber and Shiu (1994) to construct an equivalent martingale measure. Note that the most proofs found in literature are not constructive, but apply a separation theorem in finite-dimensional Euclidean spaces, for example.¹⁶ It is instructive to first consider the case of the one period model with one risky asset, second consider the case of the multiperiod model with one risky asset, and third consider the general case.

A Martingale Measure in the One Period Model with One Risky Asset

Let us consider the one period model () with one risky asset (). We assume for simplicity that . Let

The absence of arbitrage implies that¹⁷

We need to construct measure so that

Let

for , where

We can assume that for each such that .¹⁸ In addition, and . We define the probability measure

Now . We have that , and thus is strictly convex on . Let

We have to prove that

13.12

If (13.12) holds, then we define

Now is the required measure because and

Let us prove (13.12). Let us assume that (13.12) does not hold and derive a contradiction. Let be a sequence such that

13.13

Then or . Otherwise, we can choose a convergent subsequence, the minimum is attained at a finite point, and (13.12) holds. Let

We have that

Thus there exists such that

Thus,

as . For sufficiently large we have

which contradicts (13.13).

A Martingale Measure in the Multiperiod Model with One Risky Asset

Let us consider the multiperiod model () with one risky asset (). We assume for simplicity that . Let

where . The absence of arbitrage implies that¹⁹

13.14

-almost surely, for . We need to construct measure so that

Then is a martingale difference with respect to , and is a martingale with respect to . Let

for , We can assume that is finite.²⁰ For a fixed function is strictly convex, as follows from (13.14). There exists a unique finite

such that is attained at , which can be shown similarly as (13.12). We can show that is -measurable.²¹ Let and

for . Now , are -measurable, and they form a martingale:

-almost surely. We define the probability measure

Now , , and for .

A Martingale Measure in the Multiperiod Model with Several Risky Assets

Let and . We assume for simplicity that . Let

where . Now are vectors of length . The portfolio vector is a -dimensional -measurable vector. The components are bounded, so that for and . The absence of arbitrage implies that

-almost surely, for . We need to construct measure so that

Then is a martingale difference with respect to , and is a martingale with respect to . Let

for . There exists a unique finite such that is attained at , and is -measurable.²² Let and

for . We define the probability measure

and is the required equivalent martingale measure.

13.2.3.3 Estimation of an Equivalent Martingale Measure

We estimate the Esscher martingale measures using S&P 500 daily data, described in Section 2.4.1. We consider both a one period model and a two period model.

A Martingale Measure for S&P 500: One Period

We consider the one period model where the period consists of 20 days. Let

be the price increment. Our S&P 500 data provides a sample of identically distributed observations of : we use data , where and is the gross return over the period of 20 days. We use nonoverlapping increments. The risk free rate is .

The density of the Esscher martingale measure with respect to underlying physical measure of can be estimated as

where , is the sample average of , and is the minimizer of over . The underlying physical density of with respect to the Lebesgue measure can be estimated using the kernel estimator . The kernel density estimator is defined in (3.43). The density of the martingale measure with respect to the Lebesgue measure can be estimated as

Figure 13.1(a) shows the estimate of the density of the martingale measure with respect to the physical measure (red). The blue curve shows the density of the risk neutral log-normal density with respect to the estimated physical measure. We see that the measures put more probability mass on the negative increments than the physical measure. Fitting of a log-normal distribution is discussed in the connection of Figure 3.11.²³ Panel (b) shows the kernel estimate of the density of the physical measure as a red curve, and the estimate of the density of the Esscher martingale measure with respect to the Lebesgue measure as a red dashed curve. We apply the standard normal kernel and the normal reference rule to choose the smoothing parameter. The blue curves show the corresponding densities in the log-normal model.

Figure 13.2 shows density ratios. Panel (a) shows the ratio

and panel (b) shows the ratios

where is an estimate of the log-normal physical measure, is an estimate of the log-normal risk neutral measure, and .

A Martingale Measure for S&P 500: Two Periods

Let us estimate the Esscher martingale measure for the two period model with two periods of 10 days. Let

be the price increments. Our S&P 500 data provides a sample of identically distributed observations of . The observations are

where

where . We use nonoverlapping increments. Let

where is the sample average of , and is the minimizer of over . Let

where is the minimizer of over , and is a regression estimate evaluated at , when the response variable is and the explanatory variable is . We apply a kernel regression estimate of (6.20) and (6.21) to define

where , , are the observation of ,

are the kernel weights, is the Gaussian kernel function and is the smoothing parameter, chosen by the normal reference rule.

The density of the martingale measure with respect to the underlying physical measure of can be estimated as

13.15

We can also assume the independence of the increments and estimate by

13.16

Figure 13.3 shows estimates of the density of the Esscher measure with respect to the physical measure. In panel (a) we show estimate (13.15), which does not assume independence, and in panel (b) we show estimate (13.16), which assumes independence.

13.2.3.4 Examples of Equivalent Martingale Measures

We calculate the class of equivalent martingale measures in the one period binary model, in the one period ternary model, and in the one period model with a finite amount of states, which generalizes the two previous models.

The One Period Binary Model

Let us have two assets: bond and stock . The value of the bond at time 1 is , where . The value of the stock at time 1 is with probability and with probability , where and . That is,

Let the price of the bond be and the price of the stock be . Let us consider probability measure which is defined by

where . If is a martingale measure, then it satisfies

This holds if

13.17

Thus, there exists an equivalent martingale measure if and only if

13.18

Thus, the market is arbitrage-free if and only if (13.18) holds. The martingale measure is unique. The calculation will be repeated in (14.18), where derivative pricing is discussed.²⁴ Note that the pricing in the binary model was already studied in (13.1).

The One Period Ternary Model

Let us have two assets: bond and stock . The value of the bond at time 1 is , where . The value of the stock at time 1 is with probability , with probability , and with probability , where , and . That is,

Let the price of the bond be and the price of the stock be . Let us consider probability measure , which is defined by

where and . If is a martingale measure, then it satisfies

This holds if , where

We can write

Thus, there exists an equivalent martingale measure if and only if

13.20

Thus, the market is arbitrage-free if and only if (13.20) holds. There are several martingale measures.²⁵

A Finite Amount of States

Let us consider the one period model with a finite amount of states. Now the probability space has a finite number of elements. We have basic securities and possible states. In the binary model . In the ternary model and .

The th risky asset takes values, corresponding to the different states. Let be the matrix whose elements are , where is the th state and is the th risky asset.

Let be the vector of the probabilities of the states. Let be the vector of the prices of the risky assets at time 0. Let be the vector of the risky assets.

Let be a vector of probabilities of the states. Vector is a martingale measure if

13.21

We can assume that and , because the redundant basic assets can be removed. A redundant asset would correspond to a column of which could be expressed as a linear combination of the other columns.

When , then there exists a unique equivalent martingale measure, and this is the solution to (13.21):

13.22

When , then the system (13.21) of linear equations with variables has many solutions.

13.2.4.1 The Definition of an European Continent Claim

We use the terms “contingent claim” and “derivative” interchangeably. However, these terms can have a different meaning.

1. 1. A European contingent claim is a nonnegative random variable defined on the probability space .
2. 2. A derivative of the underlying assets is a European contingent claim, which is measurable with respect to the -algebra generated by the price process , .

We assume that , and thus in our case the two definitions lead to the same concept. Time is called the maturity, or the expiration date.²⁶ The examples of European contingent claims include the following.

1. 1. A call and a put on the th asset are defined by
   
2. where is the strike price and .
3. 2. An Asian call and put option are defined by
   
4. where
   
5. , and is the cardinality of .
6. 3. A knock-out option on the th asset is defined by
   
7. where is the strike price, and is the barrier.

13.2.4.2 Arbitrage-Free Prices of European Continent Claims

A European contingent claim is attainable (replicable, redundant), if there exists a self-financing trading strategy whose terminal portfolio value is equal to :

The trading strategy is called a replicating strategy for . A contingent claim is attainable if and only if the discounted claim

has the form

for a self-financing trading strategy with value process . Now it is natural to take the initial value

to be the fair price of , because a different price would lead to an arbitrage opportunity. The corresponding arbitrage-free price of the contingent claim is

We need to define an arbitrage-free price also for those contingent claims which are not attainable. In fact, in typical market models a contingent claim cannot be replicated. Föllmer and Schied (2002, p. 238) formulate the following definition. An arbitrage-free price of a discounted claim is a real number , if there exists an adapted stochastic process such that

1. 1. ,
2. 2. for ,
3. 3. , and
4. 4. the enlarged market model with price process is arbitrage-free.

According to this definition, an arbitrage-free price of a contingent claim is such that trading with this price at time 0 does not allow an arbitrage opportunity. A corresponding arbitrage-free price of the continent claim is then

We can express the class of arbitrage-free prices with the help of equivalent martingale measures. Föllmer and Schied (2002, Theorem 5.30, p. 239) formulate the following theorem.

The price of contingent claim can now be written as

13.23

In the Black–Scholes model we use continuous compounding, where , and , so that for a call option ; see (14.47). In many cases we denote by the time of writing the option and then , so that .

13.2.4.3 Pricing Kernel

The pricing kernel (discount factor) , related to the martingale measure , is defined as the discounted density of with respect to the physical measure :

The price of is given in (13.23) as . Now the price of derivative can be written as

The One Period Binary Model

In the one period binary model, the martingale measure was defined as the measure

where the probability is defined in (13.17). The pricing kernel is function , defined by

Let be a derivative, where . The price of is

A Finite Amount of States

Let us continue to study the one period model with a finite amount of states. Let be the vector of the probabilities of the states. Let be the vector of the prices of the risky assets at time 0. Let be the vector of the risky assets. Let be the matrix whose elements are the values of the th asset at the th state.

Let be an equivalent martingale measure, which is a solution to the equation in (13.21). In the case we may obtain from (13.22) as . Let

for . Let be a derivative. The price of is

The Arrow–Debreu securities take value 1 in one state and value 0 in other states: for the th Arrow–Debreu security it holds that , and for . Then is the identity matrix. Then and .²⁷

13.2.5.1 The One Period Binary Model

In the one period binary model, we have two assets: bond and stock . The bond satisfies , where . The stock satisfies

where and . Let the price of the bond be and the price of the stock be . The space of attainable payoffs is

Let us consider contingent claim , where . To replicate the contingent claim, we need to choose and so that

This leads to equations

We have two equations and two free variables. The model is complete.

13.2.5.2 The One Period Ternary Model

Let us have two assets: bond and stock . The bond satisfies , where . The stock satisfies

where , and . Let the price of the bond be and the price of the stock be . The space of attainable payoffs is

Let us consider contingent claim , where . To replicate the contingent claim, we need to choose and so that

This leads to equations

We have three equations and two free variables. The model is not complete.

13.2.5.3 A Finite Amount of States

Let us continue to study the one period model with a finite amount of states. Let be the vector of the probabilities of the states. Let be the vector of the prices of the risky assets at time 0. Let be the vector of the risky assets. Let be the matrix whose elements are the values of the th asset at the th state.

We can assume that and , because the redundant basic assets can be removed. A redundant asset would correspond to a column of which could be expressed as a linear combination of the other columns. A redundant basic asset could be considered as a derivative.

A derivative security is random variable which takes possible values. Let those values be in vector . To replicate , we need to find vector so that . This leads to the matrix equation

When , then

When , then we do not always have a solution, because there are free variables and equations.

We can choose an approximate replication by minimizing the sum of squared replication errors. Let the replication error be

where is the Euclidean norm in . The minimizer is

which is the same formula as the formula for the least squares coefficients in the linear regression . Note that matrix has rank , when has rank , and thus is invertible.

The Arrow–Debreu securities provide an example of derivatives. An Arrow–Debreu security has price 1 in one state and 0 in the other states. There are as many Arrow–Debreu securities as there are states. When , the columns of give the portfolio weights for replicating the Arrow–Debreu securities.

13.2.6.1 European and Bermudan Options

An European contingent claim is obtained as a special case of an American contingent claim, when we choose for . The value of the American option is larger or equal to the value of the corresponding European option.

A Bermudan option can be exercised at times . Formally we can define a Bermudan contingent claim as a non-negative adapted process , , on the filtered space . A Bermudan option can be obtained as an American option with for .

On the other hand, an American option can be considered as a special case of a Bermudan option with . Also, from the point of view of the continuous time model with time space , an American option in a discrete time model could be considered as a Bermudan option whose possible exercise times are .

13.2.6.2 The Set of Arbitrage-Free Prices

Let be a discounted American claim and let be the payoff which is obtained for a fixed exercise strategy . Now can be considered as a discounted European contingent claim, whose set of arbitrage-free prices is given in Theorem 13.2 as

Föllmer and Schied (2002, Definition 6.31) give the following definition for an arbitrage-free price. A number is called an arbitrage-free price of a discounted American claim if

1. 1.

   There exists some and such that .

   (The price is not too high.)

2. 2.

   There does not exist such that for all .

   (The price is not too low.)

The set of arbitrage-free prices is characterized in Föllmer and Schied (2002, Theorem 6.33). It is assumed that for all and . The set of arbitrage-free prices is an interval with endpoints and , where

The interval can be a single point, open, or half open.

13.2.6.3 Exercise Strategies for the Buyer

Let us call an exercise strategy optimal if

13.24

where

Thus, an exercise strategy is defined to be optimal, if it maximizes the expectation among the class of the payoffs. Note that the definition can be generalized so that we maximize , where is a utility function, and is a probability measure, possibly different from the physical measure .

Föllmer and Schied (2002, Theorem 6.20) shows that the exercise strategy is optimal, when we define

where

13.25

It is assumed that for . Process is called a Snell envelope of with respect to the measure . Föllmer and Schied (2002, Proposition 6.22) notes that any optimal exercise strategy satisfies , so that is the minimal optimal exercise strategy.

Föllmer and Schied (2002, Theorem 6.23) shows that is also an optimal exercise strategy, when we define

where means . In addition, is the largest optimal exercise strategy in the sense that any optimal exercise strategy satisfies .

13.2.6.4 American Options in Complete Models

Let us assume that the market model is complete, so that there exists the unique equivalent martingale measure . We defined the optimal exercise time in (13.24) using the physical measure , and the construction of the optimal exercise time was made in (13.25) using the physical measure . Let us define the Snell envelope using the equivalent martingale measure . The value can be considered as the unique arbitrage-free price, because

holds -almost surely, when is optimal with respect to ; see Föllmer and Schied (2002, Corollary 6.24).