14.1.1.1 Stock Futures

A futures contract on a stock is made at time . The contract specifies that the buyer of the contract has to buy the stock at a later time with price . We assume that the stock does not pay dividends during the time period from to . Let us denote with the price of the stock at time . The value of the futures contract at the expiration time is

because the buyer of the futures contract gives away and receives .

The price of a zero-coupon bond is taken as , where , is the annualized risk-free rate, and is the time between and in fractions of a year. This is the method of continuous compounding.¹

What is the fair value of the futures contract for ? We may replicate the futures contract buy buying the stock and borrowing the amount . At time the value of this portfolio is

with probability one. At time the value of this portfolio is with probability one. Thus,

14.1

for , by the law of one price, to exclude arbitrage.

When an investor enters a futures contract in a futures exchange, this does not imply any cash flows, although the exchange requires from the investor a liquid collateral in order to secure a possible future payment. The fair forward price is called such value of that makes the value of the futures contract zero. To get we need that

14.2

The future prices that are quoted in a futures exchange are the values (which are determined by the supply and demand). Numbers are called futures prices or forward prices.

A buyer of a stock future or a stock index future saves the carrying costs but looses the possible stock dividends. When the annualized dividend rate is known to be , then the fair future price is

14.1.1.2 Currency Forwards

A currency forward is made at time . We denote with the exchange rate USD/euro at time . Let be an exchange rate. The contract stipulates that the buyer will buy US dollars with euros at time , using the USD/euro exchange rate . The value of the contract for the buyer at time is

14.3

Indeed, the buyer uses euros to buy dollars. Then the buyer exchanges the dollars to euros to obtain euros. The profit of the buyer is .

What is the price of the currency forward for ? We can replicate the currency forward using zero-coupon bonds. Let be the USD price of the US zero-coupon bond and let be the euro price of an European zero-coupon bond. The zero-coupon bonds are such that USD and . Let us consider the portfolio with units of US zero-coupon bonds and with units of European zero-coupon bonds. The value of the portfolio at time is

We choose

The value of the portfolio a time is

14.4

Since (14.3) and (14.4) are equal, the portfolio replicates the currency forward, and the price of the currency forward is equal to at time .

The price is in euros

To make we need to choose the exchange rate

14.1.1.3 Forward Zero-Coupon Bonds

A time two parties make an agreement to exchange at a future time a zero-coupon bond whose maturity is at , with a cash payment . At time only the agreement is made, and at time the cash payment is made and the zero-coupon bond is received. This is called a forward zero-coupon bond. Forward zero-coupon bonds are considered in Section 18.2.1.

Let be the price of the forward zero-coupon bond. Let and be the prices of zero-coupon bonds with maturities and . Then,

Consider a portfolio with one unit of and short of units of . The portfolio has value

for . Thus, and we have replicated the forward zero-coupon bond. Thus, at time the prices are equal:

To make , we need to choose the cash payment as

14.5

14.1.1.4 Forward Rate Agreements

A forward rate agreement allows to change a floating Libor rate against a fixed rate. The contract is made at time . The reset time of the Libor is . At this time the Libor rate is settled to be , where is the maturity of the Libor. The agreement stipulates that the buyer of the contract pays the payment with the fixed rate , and receives the payment with the Libor rate . Thus, the payoff for the buyer at time is

14.6

where is the notional, and is the time between and . Forward rate agreements are considered in Section 18.2.2.

Let and be zero-coupon bonds. The forward rate agreement can be replicated by the trading strategy

where is the strategy of buying and reinvesting the payoff at time with the prevailing -period Libor rate. Thus, the payoff of at time is equal to . Thus, the trading strategy gives the payoff (14.6). The present value of the forward rate agreement is equal to

14.7

To make , we need to choose

14.1.1.5 Backwardation and Contango

Backwardation refers to the price relation where the spot price is higher than the forward price and contango refers to the case where the spot price is lower than the forward price:

At the expiration, the spot price and the futures price should be equal (to prevent arbitrage). Thus, the contango relationship means that distant delivery months trade at a greater price than near-term delivery months (the term structure is upward-sloping). Depending on the type of the underlying, either the contango or the backwardation is typical.

In the case of stock futures, or stock index futures, the theoretically fair price of a futures contract is greater than the spot price, according to (14.2). Thus, contango is typical for stock futures. However, futures prices can differ from the theoretically fair price, and thus both contango and backwardation are possible. In the case of stock futures we can change the terminology in the following way:

According to this terminology we do not compare futures prices to the spot prices but to the theoretically fair prices.²

When the underlying is a commodity, then there are borrowing costs for the seller of the futures contract, similarly as in the case of stock futures. In addition, there are costs for storing the commodity, for the seller of the futures contract. Thus, contango is typical for futures on commodities, and contango is typically more profound than for stock futures.

The buyer of a treasury bond future saves carrying costs (determined by the short-term rate) but looses the interest received by the bond owners (determined by the long-term rate). If the short-term rate is significantly less than the long-term rate, then the distant delivery months may trade at a lesser price than the near-term delivery months: this is backwardation.

14.1.2.1 A Derivation of the Put–Call Parity

Consider portfolio obtained by buying the call and writing the put:

At the expiration, we have . Indeed,

Let be the forward contract to buy stock at time with price . This forward contract was valued in (14.1), where we showed that

The replication was obtained by buying the stock and borrowing the amount . At the expiration, we have . Since with probability one, we have

for all times before , to exclude arbitrage. This is the statement (14.8) that we wanted to prove.

14.1.2.2 Bounds for the Call Price

We have bounds for the call price :

14.9

The upper bound is obvious, since the right to buy a stock must be less valuable than the stock itself. To prove the lower bound, we can note that is obvious, since the right to buy a stock involves no obligations.⁴ The put–call parity and the fact that gives

We have proved the lower bound.⁵

The lower bound in (14.9) implies that

14.10

for . This follows because when . The inequality (14.10) leads to the definition of the time value of the option; see (2.6). The lower bound says that the value of the option is larger than the intrinsic value . The difference of the price of the option and its intrinsic value is called the time value of the European option. For an European put option the lower bound fails unless . As a consequence of the put–call parity, the time-value of a put option whose intrinsic value is large (the option is in the money), is usually negative.

14.2.1.1 A Description of the One-Period Binary Model

In the single period model, there are time points and . The market consist of stock , bond , and contingent claim .

The bond takes value 1 at and value at :

where is the risk-free rate.

The initial value of the stock is . The stock can take only two values at time 1. At the value is with probability and the value is with probability :

where . We assume that

14.12

Note that if , then it would not be rational to invest in the bond, and if , then it would not be rational to invest in the stock. The “rationality” can be formalized as the absence of arbitrage.

The contingent claim takes two possible values and at :

For example, in the case of a call option we have for some strike price . Then and .

We want to find a fair value for the derivative at . Also, we want to find the optimal hedging coefficient , which is used to hedge the position of the writer of the option.

14.2.1.2 Pricing and Hedging in the One-Period Binary Model

We replicate the contingent claim with a portfolio

The portfolio consists of units of the bond and units of the stock. The portfolio takes values

To obtain we need to choose and so that

The first equation leads to

14.13

Inserting this value of to the second equation gives

14.14

The law of one price implies that the arbitrage-free price of the contingent claim equals the value of the replicating portfolio at time : , and thus

14.15

where and are defined in (14.13) and (14.14).

The number of bonds can be written as . Since , we can write

14.16

We can interpret the replicating portfolio from the point of view of the writer of the option as the portfolio where the writer receives the option premium , invests in the bank account, borrows the amount from the bank, and invests in the stock.

14.2.1.3 The Equivalent Martingale Measure

Price can be written as the expectation with respect to the equivalent martingale measure. We have

14.17

where

14.18

and means the expectation with respect to the probability measure with

Probability measure is called a risk-neutral measure because

and it is called a martingale measure because

Note that condition (14.12) guarantees that , so that is equivalent to .

14.2.1.4 Further Pricing and Hedging Formulas

It is of interest that the price can be written as

14.19

where

Indeed, similarly to (14.13), we get

which can be combined with (14.13) to get

Combining this formula for with the formula shows (14.19). In (14.19), we have written the derivative price as a discounted expectation of the derivative price, with an additional correction term.

The hedging coefficient can be written as

14.20

To derive (14.20), note that

Thus, (14.20) is equal to (14.14). We can also write

14.21

where is the gross return. This way of writing the hedging coefficient appears in (16.10), where quadratic hedging is considered.

14.2.2.1 A Description of the Multiperiod Binary Model

The market consists of stock , bond , and contingent claim . We define the discrete time processes

14.22

Note that in Section 13.2 we denoted the time steps of the discrete time markets by . We have changed the notation, because we construct a time series that approximates the geometric Brownian motion on . The approximation is done by dividing interval to periods of equal lengths and letting .

The bond takes value at step , where is the annualized interest rate, and is the time between two periods in fractions of a year.

At step , the stock can take values

14.23

where

The stock price is a random variable with

14.24

where ,

The stochastic process of stock prices can be described by a recombining binary tree, as in Figure 14.1, where . At step , the stock takes value (in the figure ). If the value of the stock at time is , then at step the stock can take values and , so that

The probabilities of the up and down movements are and :

The derivative can take at the expiration values , . The random variable takes the value , when . For example, when the contingent claim is the call option with , where is the strike price, then . We have

14.25

We want to find the arbitrage-free price of the derivative at time .

14.2.2.2 Pricing in the Multiperiod Binary Model

The evolution of the stock price in a multiperiod binary model has been described using a recombining binary tree. The price in a multiperiod binary model can be found by backward induction. We know the price at the expiration, when . We can use the single period model to calculate the price at step , and go backwards step by step to obtain the price at step . The price of the derivative is calculated using the backwards induction with the following steps.

1. 1. At the expiration step , the prices of the derivative are given by , .
2. 2. Let the current step be and the current node . Then the two possible prices for the derivative are and . We can use the single period model to calculate the price at step . We get the price from (14.17) as
   14.26
3. where
   
4. and the second equality follows from and .

We have described a recursive algorithm for the computation of the price, but we can also obtain the following explicit expression for the price.

The arbitrage-free price of the derivative is obtained not only at step , but the price is obtained at all steps . In fact, the price of the derivative, under the condition that the stock price at step is , is given by the formula

14.28

14.2.2.3 The Equivalent Martingale Measure

Let us define probability measure by

where . Measure is obtained from the physical measure , defined in (14.24), by replacing the probability of an up-movement by probability . The price in (14.27) can be written as the expectation

14.29

where is the random variable taking value , when takes value . The measure is called the equivalent martingale measure, or the risk-neutral measure.

The price of the derivative given in (14.28), under the condition that the stock price at step is , can be written as

14.30

Accordingly, we can define random variables

Defining

we obtain a more elegant formula

14.31

14.2.2.4 Hedging in the Multiperiod Binary Model

The following theorem gives the hedging coefficients of the replicating portfolio. The replication means that the wealth process of the trading strategy obtains the value of the derivative with probability one, or equivalently, the value process of the trading strategy obtains the discounted value of the derivative with probability one. The value process of a self-financing trading strategy is given in (13.8).

There are other ways to write the hedging coefficient. We have from the formula (14.21) of the one-period model that

where , , and . Here and mean conditional variance and covariance, conditional on , and under probability measure , which is the equivalent martingale measure.

We can also write the hedging coefficient as

14.33

Let us prove (14.33). We have from (14.31) that

Thus,

We have shown (14.33).

An additional formula for the hedging coefficient is

14.34

In fact, . Thus,

14.2.3.1 Choice of the Parameters

Let

We consider asymptotics when . We choose the up and down factors as

14.35

where . We choose the probabilities of the up and down movements as

where . With these choices the logarithmic return converges in distribution to the normal distribution , where . Note that when we choose and as in (14.35), then the probability of the up movement in the risk-neutral distribution is

14.36

14.2.3.2 Asymptotic Normality in the Multiperiod Binary Model

We show that the distribution of the stock price in the multiperiod binary model converges in distribution to a log-normal distribution, as the number of steps increases.

In the -period binary model the stock price , , can be written as

14.37

where are such i.i.d. random variables that when is a result of an up-movement and when is a result of a down-movement, so that

It holds that

14.38

as , where

We show a slightly more general result: For

14.39

as , where is such that , as . In particular, we can choose . Now (14.38) follows as a special case when we choose and .

We denote below . Let us denote . We can write

where

with . Now it holds that . We have that

1. 1. ,
2. 2. ,
3. 3. ,

as . Because , the claim (14.39) follows from items 1–3.

To prove item 1, we note that

Thus,

By the choice of , it holds as , and thus

as . Thus, item 1 follows by the central limit theorem.

To prove item 2, we use the fact that , which implies that as . Thus, the weak law of the large numbers implies item 2.

To prove item 3, we note that

Item 2 implies that and since , we have that , so that .

14.2.3.3 Convergence of the Price

The arbitrage-free prices (14.27) in the multiperiod binomial model are called the Cox–Ross–Rubinstein prices. We show that the Cox–Ross–Rubinstein put and call prices converge to the Black–Scholes put and call prices, as . This is done in two steps. First, we show that the put and call prices in the multiperiod binary model converge to the expected values of the option pay-offs, when the expectation is taken with respect to a log-normal distribution. Second, we calculate closed-form expressions for the expected values.

Bounded Continuous Payoff Functions

A fundamental theorem about weak convergence states that if

1. 1. ,
2. 2. is bounded and continuous,

then

as ; see Billingsley (2005, Theorem 25.8, p. 335).

1. 1. First, we apply the weak convergence in (14.38) to obtain
   14.40
2. where the stock price is a random variable taking values with probabilities
   
3. and .
4. 2. Second, we have noted in (14.29) that a Cox–Ross–Rubinstein price satisfies
   14.41
5. where the expectation is with respect to the risk-neutral measure , and is a random variable taking values , defined by (14.25).

Let be such function that Then the option payoff can be written as

If function is continuous and bounded, then (14.40) and (14.41) imply that

14.42

where the distribution of is defined by

14.43

and we used the fact that , because .

Unbounded Continuous Payoff Functions

The fundamental theorem about weak convergence applies for any sequence converging weakly. However, in our case we are interested in the special case of the convergence of a binomial distribution toward a log-normal distribution. In this special case Föllmer and Schied (2002, Proposition 5.39, p. 265) notes that the convergence of expectations can be proved also when we relax the condition of the boundedness. Let be measurable, almost everywhere continuous, and

14.44

Then,

as , where satisfies (14.40) and is distributed as (14.43).

Put Prices

The payoff function of a put option is

where is the strike price. Function , , is bounded and continuous. Thus, the Cox–Ross–Rubinstein price of an European put option converges to the expectation:

14.45

The right hand side is equal to the Black–Scholes put price, as shown in (14.54).

Call Prices

The payoff function of a call option is

where is the strike price. Function , , is continuous but not bounded. Thus, the convergence of the Cox–Ross–Rubinstein price to the expected value cannot be inferred similarly as in the case of put options. However, we can apply the put-call parity in (14.8) to conclude that the Cox–Ross–Rubinstein prices have to satisfy

where is the Cox–Ross–Rubinstein call price, and the put price. Since we have shown (14.45), it holds that

14.46

The right hand side is equal to the Black–Scholes call price, as shown in (14.52).

Calculation of the Expectations

We have proved in (14.45) and (14.46) that the arbitrage-free put and call prices in the multiperiod binary model approach the expected values of the option payoffs, when the expectations are with respect to the equivalent martingale measure (the risk neutral log-normal model). Thus, we want to calculate the expected values of the put and call payoffs. The expected values are the Black–Scholes prices, which we denote

14.47

and

14.48

where the expectations are taken with respect to the distribution of defined by

14.49

where and

Note that we have discussed the log-normal distribution in (3.50).

The density of the standard normal distribution is , where . Then,

where

By writing we have

Thus,

14.50

where is the distribution function of the standard normal distribution. Since ,

14.51

where

because and . This leads to the call price

14.52

Similarly,

14.53

which leads to the put price

14.54

We have calculated explicit expressions for the call and put prices at time , when the stock price is . The Black–Scholes call and put prices at time , when the stock price is , are given by

14.55

and

14.56

where the expectations are taken with respect to the distribution of defined by

where and . The corresponding explicit expressions are given in (14.58) and (14.59), and the Black–Scholes prices are discussed in Section 14.3.1.

14.2.3.4 Convergence of the Hedging Coefficient

We show that the hedging coefficients of puts and calls in the multiperiod binary model converge to the Black–Scholes hedging coefficients. The Black–Scholes hedging coefficients are called deltas, and they are obtained by differentiating the Black–Scholes prices with respect to the stock price.

Let the initial time and the initial stock price be

Let be either the Black–Scholes call price or the put price at time , when the stock price at time is , and the expiration is at time . These prices are written as expectations in (14.55) and (14.56), and in a more explicit form in (14.58) and (14.59).

Let us consider step of the multiperiod binary model. Let be one of the possible stock prices at step , where . These prices were defined in (14.23). Let be such that

as . In particular, we can choose , where is the largest integer . Choose

Then

as .

The hedging coefficient of the multiperiod binary model was written in (14.32) as

where and are the two possible prices of the derivative at step , when the value of the stock at step is .⁷ Now it holds that

14.57

where is the derivative of the Black–Scholes price with respect to stock price.

To show (14.57), note that from (14.46) and (14.45), and the continuity of the functions we have that

where means that . Thus,

Also, , , and , as .

14.2.3.5 Rate of Convergence

Figure 14.2 illustrates the convergence of (a) the ratio of the Cox–Ross–Rubinstein call price to the Black–Scholes call price, and (b) the ratio of the Cox–Ross–Rubinstein call hedging coefficient to the Black–Scholes call hedging coefficient. The ratios are plotted as a function of the number of the steps in the multiperiod binary model, where . The moneyness is 0.9 (green), 0.95 (black), and 1 (red). The annualized volatility is , the interest rate is , and the time to maturity is 1 month: . We see that the convergence of the at-the-money options is faster than the convergence of the out-of-the-money options.

14.2.3.6 Asian and Knock-Out-Options

We have proved the weak convergence at one point: Let and . Then

where is the price of the stock at step in the multiperiod binomial model, and is the price of the stock at time in the geometric Brownian motion model. If the option payoff is , then the price in the multiperiod binomial model converges to the price of the option whose payoff is in the geometric Brownian motion model. Asian options and knock-out options depend on values of stocks in more than one point.

Asian Options

In the case of Asian options, the option payoffs can be written as

where are steps, and are fixed time points. For example, in the case of an Asian call option

The corresponding prices converge for a suitable , when

Knock-Out Options

The payoffs of knock-out options depend on the trajectory of the prices through

Let us divide the time interval into subintervals. Denote the boundaries of the intervals by

Points fill the interval asymptotically. We can define a continuous time process , , by linearly interpolating : define

when . Geometric Brownian motion is defined in (5.62). We can show that process converges weakly to the geometric Brownian motion

where is the standard Brownian motion. The weak convergence

as , happens in the metric space of the continuous functions on . The prices of the options whose payoffs are

converge for suitable .

14.2.4.1 American Put Options in the One-Period Binary Model

In the one-period binary model, the American put option can be exercised at time or at time . Let us denote with the value of the American put option at time and let us denote with the value of the European put option at time . An arbitrage argument shows that

The value of an European put option can be obtained from (14.17) as

where

with

14.2.4.2 American Put Options in the Multiperiod Binary Model

The price of an American put option is determined in the -step binomial model by recursion. Remember that in the -step binomial model the possible prices at step , , are

with

The recursive steps are the following.

1. 1. At time the prices of the American put option are given by
   

   .

2. 2. At time , , when the stock has price , we know from the previous steps of the algorithm that the two possible prices for the derivative at time are and . We can use the single period model to calculate the price at time . We get the price from the one-step binary model as
   
3. where
   
4. and
   
5. with .

14.3.1.1 Computation of Black–Scholes Prices

For the application of the Black–Scholes formula the time is taken as the time in fractions of year. For example, when the time to expiration is 20 trading days, then . Alternatively, when time to expiration is 20 calendar days, then .

The risk-free rate is expressed as the annualized rate.

The only unknown parameter has to be estimated. Let be an equally spaced sample of stock prices and let us denote for . We assume that , are i.i.d. , so that the stock prices satisfy (3.50). We can estimate with the sample variance

where . Then an estimator of is

14.60

For example, if we sample stock prices daily, then and .⁸ If we sample stock prices monthly, then and The normalized sample standard deviation in (14.60) is called the annualized sample standard deviation.

14.3.1.2 Characteristics of Black–Scholes Prices

We study the qualitative behavior of the Black–Scholes prices as a function of five parameters , , , , and .

The prices of calls and puts increase as increases. We have

and

which are the bounds derived from the put–call parity in (14.9). The prices of calls and puts increase as the time to maturity increases. The price of a call increases as increases and the price of a call decreases as increases, but for puts the relations reverse. The price of a call increases as the interest rate increases but the price of a put decreases as the interest rate increases.

Figure 14.3 shows Black–Scholes prices for calls and puts as a function of the call moneyness . The call prices increase and the put prices decrease as a function of moneyness. Panel (a) shows the cases of annualized volatility (black), (red), and (green). The time to maturity is 20 trading days and the interest rate is . Panel (b) shows the cases of interest rates (black), (red), and (green). The time to maturity is 20 trading days and the annualized volatility is . We see from panel (a) that increasing volatility increases the prices, both for call and puts. The effect of increasing time to maturity is similar. We see from panel (b) that for calls increasing interest rates increases prices, but for puts increasing interest rates decreases prices.

Figure 14.4 shows that the call and put prices are not symmetric. The ratios of call prices to put prices are shown as a function of the call moneyness : We show the functions

where is the current stock price. The strike prices for the calls take values , and the corresponding strike prices for the puts are . Panel (a) shows the cases of annualized volatility (black), (red), and (green). The interest rate is . Panel (b) shows the cases of interest rates (black), (red), and (green). The annualized volatility is . The time to maturity is 20 trading days in both panels. We see that the call prices are higher than the put prices for out-of-the-money options. This is related to the asymmetry of the log-normal distribution (see Figure 3.11). For at-the-money options the difference between the call and the put prices is not large. Increasing the volatility makes the ratios of the call and put prices closer to one, and decreasing the interest rate makes the ratios of the call and put prices closer to one.

14.3.1.3 Black–Scholes Prices and Volatility

To apply Black–Scholes prices we need to estimate the volatility. We study how the Black–Scholes prices change when the volatility estimate changes. We apply the data of S&P 500 daily prices, described in Section 2.4.1.

Figure 14.5 shows time series of Black–Scholes call prices. The volatility is equal to the sequential annualized sample standard deviation. Panel (a) shows call prices with moneyness and panel (b) shows call prices with moneyness . The time to expiration is either 20 trading days (black curves) or 30 trading days (red curves). The risk-free rates are deduced from 1-month Treasury bill rates. We can see that the time series of Black–Scholes prices follow closely to the time series of sequential standard deviations in Figure 7.5.

Figure 14.6 studies Black–Scholes prices when the volatility is equal to the sequentially estimated GARCH() volatility. Panel (a) shows time series of call prices with moneyness (black) and (green). The time to expiration is 20 trading days. Panel (b) shows kernel density estimates of the distributions of the prices. The horizontal and the vertical lines show the Black–Scholes prices when the volatility is the annualized sample standard deviation computed from the complete sample. The risk-free rate is zero.

14.3.1.4 The Greeks

The greeks are defined by differentiating the option price with respect to the price of the underlying, the time to the expiration, the interest rate, or the volatility. In this section, we denote the Black–Scholes call and put prices by

where is the current time, is the current price of the underlying, is the strike price, is the time of the expiration, is the volatility, and is the interest rate. Sometimes we leave out some of the arguments and denote, for example, .

The Delta

The delta is the derivative of the price function with respect to the underlying. The delta is the hedging coefficient in the Black–Scholes hedging, as discussed in Section 14.4. The call and put delta are

14.61

We have that

In fact, if increases, then the price of the call increases and the price of the put decreases: and . The absolute value of the change in the value of a call or a put cannot exceed the absolute value of the change in the underlying: and .

The call delta is equal to

14.62

and the put delta equal to

14.63

Let us calculate the call delta. The delta of a call is given in (14.62) because

and finally⁹

Figure 14.7 shows the call delta as a function of moneyness . In panel (a) the annualized volatility is 10% (black) and 20% (red) with time to expiration 1 month. In panel (b) the time to expiration is 1 month (black) and 3 months (red) with volatility 10%. The solid lines show the case of interest rate and the dashed lines show the case of interest rates . The more interesting part is the moneyness . In this region the deltas are in both panels. Panel (a) shows that when moneyness is , then a larger volatility leads to a larger delta. Panel (b) shows that increasing time to maturity has a similar qualitative effect as increasing volatility.

Figure 14.8 shows a time series of Black–Scholes deltas using the data of S&P 500 daily prices, described in Section 2.4.1. The daily prices are used to create a time series with the sampling frequency of 20 and 30 trading days. Panel (a) shows call prices with moneyness and panel (b) shows call prices when moneyness . The time to expiration is either 20 trading days (black curves) or 30 trading days (red curves). The volatility is equal to the sequential sample standard deviation. The risk-free rates are deduced from 1-month Treasury bill rates. The corresponding time series of Black–Scholes prices is given in Figure 14.5.

The Gamma

The gamma is the second derivative of the price function with respect to the underlying:

The price functions are convex with respect to and thus

The call gamma and the put gamma are given by

Figure 14.9 shows the call gamma as a function of moneyness . In panel (a) the annualized volatility is 10% (black) and 20% (red) with time to expiration 1 month. In panel (b) the time to expiration is 1 month (black) and 3 months (red) with volatility 10%. The solid lines show the case of interest rate and the dashed lines show the case of interest rates .

Theta

The theta is the derivative of the price function with respect to time:

As increases the value of the option decreases (everything else being equal) and thus

The call theta is equal to

and the put theta is equal to

Figure 14.10 shows the call theta as a function of moneyness . In panel (a) the annualized volatility is 10% (black) and 20% (red) with time to expiration 1 month. In panel (b) the time to expiration is 1 month (black) and 3 months (red) with volatility 10%. The solid lines show the case of interest rate and the dashed lines show the case of interest rates .

Vega

The vega is the derivative of the price function with respect to volatility:

The call vega and the put vega are equal to

Figure 14.11 shows the call vega as a function of moneyness . In panel (a) the annualized volatility is 10% (black) and 20% (red) with time to expiration 1 month. In panel (b) the time to expiration is 1 month (black) and 3 months (red) with volatility 10%. The solid lines show the case of interest rate and the dashed lines show the case of interest rates .

Rho

The rho is the derivative of the price function with respect to interest rate:

The call rho is equal to

and the put rho is equal to

Figure 14.12 shows the call rho as a function of moneyness . In panel (a) the annualized volatility is 10% (black) and 20% (red) with time to expiration 1 month. In panel (b) the time to expiration is 1 month (black) and 3 months (red) with volatility 10%. The solid lines show the case of interest rate and the dashed lines show the case of interest rates .

14.3.2.1 Quoting Option Prices

It is helpful to quote option prices using implied volatilities. Option prices and implied volatilities are in a bijective correspondence, but it is easier to compare the prices of options with different maturities and strike prices using the implied volatilities than using the market prices. This is analogous to expressing the bond prices with annualized rates instead of quoted prices.¹⁰ The implied volatilities can be used to quote prices even when we do not think that the Black–Scholes prices are fair prices, similarly as the bond rates can be defined using various conventions.

14.3.2.2 The Volatility Surface

If the Black–Scholes model describes the true distribution of the asset prices, and if the market prices coincide with the Black–Scholes prices, then the implied volatilities of options with different strike prices and with different maturities are all equal. However, in practice the implied volatilities are different for the options with different strike prices and with different maturities.

The volatility surface gives for each strike price and for each maturity the corresponding implied volatility. Let be the market prices of call (put) options with strike prices and expiration dates , where , . The options are otherwise similar. The volatility surface is the function

where is the time to the expiration.

The volatility surface is typically not a constant function. Instead, for a fixed maturity , function

is typically u-shaped (smile) or skew (one sided smile). Instead of the strike one may take as the argument the moneyness (or ), or the delta of the options.

Options written on equity indices yield often skews. This might be due to the fact that a crash in stock markets leads to increased volatility, whereas a rise in stock markets is not usually involved with an increased volatility. Options on various interest rates yield more monotonous one sided smiles than the equity indices.

Currency markets yield often symmetric smiles. Currency markets are more symmetric than stock markets because a big movement in either direction results in an increased volatility (it is always a crash for one of the currencies).

The smile tends to flatten out with maturity. This might be due to the better Gaussian approximation when the maturity is longer.

14.3.2.3 Pricing of Options Using Implied Volatilities

Out-of-the-money options can be priced in the following way. (1) Find the implied volatility of at-the-money options. (2) Adjust the implied volatility (using experience) to get a new volatility. (3) Calculate the price of the out-of-the-money option using the new volatility.

14.3.2.4 VIX Index

The VIX index of CBOE (Chicago Board Options Exchange) uses prices of options to derive the volatility that is expected by the markets. Section 6.3.1 contains a discussion of the VIX index and Figure 6.5 shows a time series of the VIX index. Let us derive the formula of the VIX index.

It is assumed that the stock price follows a geometric Brownian motion, as defined in (5.62). That is,

where is the standard Brownian motion, , and . Under the equivalent martingale measure , where is the yearly risk-free interest rate. Thus, for the equivalent martingale measure we have

We can solve for to get

A Taylor expansion gives

where . Thus,

Theorem 13.2 implies that when the expectation is taken with respect to a risk-neutral measure, then

where and are the arbitrage-free prices of the call and the put. Under the risk-neutral measure

where is the futures price, as given in (14.2). Thus,

We arrive at the variance formula

14.65

The variance formula (14.65) was derived in Demeterfi et al. (1999). CBOE uses the approximation

CBOE:s VIX index is defined as¹¹

14.66

where is the midpoint of the bid-ask spread for the option, and . The calculation is done for two expiration dates, and the final index value is a weighted average of these. Note that the put–call parity (14.8) gives the equality .

14.3.3.1 Martingale Derivation

The second fundamental theorem of asset pricing says that an arbitrage-free market model is complete if and only if there exists exactly one equivalent martingale measure. We have stated this theorem for the discrete time model in Theorem 13.3.

The Black–Scholes prices can be derived as a corollary of the second fundamental theorem of asset pricing: If the Black–Scholes market model is arbitrage-free and complete, then the Black–Scholes prices are the discounted expectations with respect to the equivalent martingale measure.

Shiryaev (1999, p. 710) states in a continuous time framework that if there exists a unique equivalent martingale measure, then the unique arbitrage-free price of the option is the discounted expected value of the payoff with respect to the unique equivalent martingale measure.

We give a sketch of some elements of the derivation of the Black–Scholes prices directly from the second fundamental theorem of asset pricing. Details can be found in Shiryaev (1999, p. 739).

The Black–Scholes model assumes that the stock price follows the geometric Brownian motion, as defined in (5.62). The stock price satisfies

14.67

and the bank account satisfies

where .¹² Thus,

The Girsanov's theorem was stated in (5.64). We apply Girsanov's theorem with the constant function . Then,

14.68

is a Brownian motion with respect to measure , defined by , where

Measure is the unique martingale measure that is equivalent to ; see Shiryaev (1999, p. 708). Thus, the price of the call option is

We need to find the distribution of under . From (14.68) we obtain that

Thus,

14.3.3.2 The Black–Scholes Differential Equation

Black and Scholes (1973) and Merton (1973) derived the Black–Scholes price by solving a differential equation. The Black–Scholes partial differential equation is

14.69

where is the value of the option at time , is the theta of the option, is the delta of the option, is the gamma of the option, and . When is the value of a call option, then the solution is found under the boundary condition

The price of the option is . The differential equation is solved, for example, in Shiryaev (1999, p. 746).

The Black–Scholes partial differential equation can be derived heuristically in the following way. Itô's lemma (5.61) applied to the function gives

14.70

Assume that value of the option is replicated by the portfolio

that is,

The assumptions and imply

14.71

Equating (14.70) and (14.71) gives , that is,

This means that we want a perfect replication of . Choose , which is called delta hedging. This makes to disappear and leads to the differential equation (14.69). Delta hedging has removed all uncertainty, and we have obtained a perfect hedge.

14.3.3.3 Derivation of the Prices Using the Put–Call parity

We can derive the Black–Scholes prices for the calls and puts using the put–call parity given in Section 14.1.2. The basic idea is that if we are willing to assume that the prices are expectations with respect to a log-normal distribution, then the put–call parity implies that the log-normal distribution should be the risk-neutral log-normal distribution.

We assume that the distribution of the stock price is defined by (14.49). Let us denote by the price of the call option at time . We assume that the price of the call option is equal to

and the value of the put option is equal to

Thus, using (14.50) and (14.53),

because for all . The put–call parity (14.8) implies that we have to take

14.72

Inserting (14.72) to (14.50) and (14.53) leads to (14.58) and (14.59). This derivation was noted in Derman and Taleb (2005).

14.3.4.1 Options on a Forward

Let the underlying be a futures contract with the maturity . Consider a call option with the expiration time . The payoff is , where is the strike price. The price of the call option is obtained by replacing in the Black–Scholes formula (14.58) by . This gives the price

14.73

where

The volatility is the volatility of the stock. This is called Black's formula, and it was introduced in Black (1976).

Why have we replaced with ? This is due to the fact that under the risk-neutral measure the stock price has the distribution

The pricing formula of the futures contract in (14.2) gives . Thus, the distribution of under the risk-neutral measure is

The distribution of is

where . The price of the call option is

and this expectation is calculated similarly as in (14.51).

The formula (14.73) holds when the option is subject to the stock type settlement. If the option is subject to the futures type settlement, then set in (14.73). The futures type settlement means that the gains and losses are realized daily, whereas in the stock type settlement the gain or loss is realized at the time of liquidation.

14.3.4.2 Caplets

Caplets are discussed in Section 18.3.1. A caplet is a call option on the Libor rate . The payoff of a caplet at time is

where is the principal, and is the strike. We assume that under the risk-neutral measure

14.74

where and is the forward rate, defined in (18.14). Note that at time the forward Libor rate is equal to the spot Libor rate: . The Black's formula for the price of the caplet is

14.75

where

The caplet price can be written as the expectation

where the expectation is with respect to the risk-neutral distribution in (14.74). The expectation is calculated similarly as in (14.51). The caplet price is obtained from the Black–Scholes price of a call option on a stock when is replaced by and is replaced by .

Caps are defined in Section 18.3.2. Let be the time points for the caplets on the Libor rates , where . A cap is priced by

where are the prices of the caplets.

14.3.4.3 Swaptions

Swaptions are discussed in Section 18.3.3. Let be the current time, be the expiry time, and be the maturity time of the swaption. The payoff of an European call option on a swap is given in (18.27) as

where is the equilibrium swap rate, is the strike,

is the principal, and is a zero-coupon bond. We assume that under the risk-neutral measure

where . The Black's formula for the price of the swaption is

where

The swaption price can be written as the expectation

where the expectation is with respect to the risk-neutral distribution in (14.74). The expectation is calculated similarly as in (14.51). The caplet price is obtained from the Black–Scholes price of a call option on a stock when is replaced by and is replaced by .

Note that the simultaneous log-normality for all caplets and all swaptions is not consistent because a swap rate is a linear combination of forward rates and cannot be log-normal if the underlying forward rates are.

14.3.4.4 Options on a Foreign Currency

Let be the price of a foreign currency in the domestic currency units. The payoff of an European call option on a foreign currency is given by

where is the strike, and is the expiration time. According the Garman–Kohlhagen model, under the risk-neutral measure,

where . Then the price of the call option is

where

where is the risk-free rate in the foreign currency and is the risk-free rate in the domestic currency. The put price is

The price of the call option can be written as

where the expectation is with respect to the risk-neutral measure. The expectation is calculated similarly as in (14.51).

The call price on the exchange rate is obtained from the Black–Scholes price of a call option on a stock when is replaced by the exchange rate and is replaced by .

14.3.4.5 Down-and-Out Call

A down-and-out call on stock has the payoff

where is the strike price and is the barrier. We assume that the stock price has a log-normal distribution

under the risk-neutral measure, where . The price of the down-and-out call is

where is the Black–Scholes price of the vanilla call and

where

14.4.1.1 Hedging Errors

The hedging error of the writer of the option is obtained from (13.10) by the formula

14.78

where

when we take the risk-free rate . Here is the price of the option, is the terminal value of the option, are the hedging coefficients, and are the stock prices. In (14.78) the current time is denoted by , the time to expiration is days, and hedging is done daily. The hedging can be done with a lesser frequency. When hedging is done times during the period , then

14.79

where

14.4.1.2 Historical Simulation

Let us denote the time series of observed historical daily prices by . Let us denote the time to the expiration by

(The interpretation is that the last observation is made at time , and we are interested to write an option whose expiration time is .) We construct sequences of prices:

where

for . Each sequence has length and the initial prices are always . We estimate the distribution of the hedging error from the observations

where is computed from the prices . For the estimation of the density we use both the histogram estimator and the kernel density estimator, defined in Section 3.2.2.

We consider hedging of call options. The Black–Scholes prices are given in (14.58) and the hedging coefficients are given in (14.62). The volatility is estimated using the annualized sample standard deviation, computed from the complete data. We study the effect of volatility estimation to hedging in Section 14.5, where sequential (out-of-sample) volatility estimation is studied. In this section, we use in-sample volatility estimation.

More precisely, the th hedging error is

14.80

where is the Black–Scholes price from (14.58), computed with the stock price , time to expiration , and volatility . Here is the annualized sample standard deviation computed from return data

where

. Thus, the volatility is estimated in-sample. The hedging coefficient is the Black–Scholes delta from (14.62), computed with the stock price , time to expiration , and volatility . The terminal value is .

14.4.2.1 Moneyness

Figure 14.13 shows time series of hedging errors. Panel (a) shows the case when there is no hedging, so that . Panel (b) shows the case when the hedging is done daily. Moneyness is .

Figure 14.14 shows tails plots of the hedging errors. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. We show cases of no hedging (red), hedging once (black), hedging twice (blue), and hedging 20 times (green).

Figure 14.15 shows (a) histograms of hedging errors and (b) kernel density estimates of the distribution of hedging errors. Panel (a) shows cases of no hedging (red) and daily hedging (green). Panel (b) shows additionally the cases of hedging once (black) and hedging twice (blue). Note that the red kernel density estimate is very inaccurate, because the underlying distribution is such that a large part of the probability mass is concentrated at one point.

14.4.2.2 Moneyness

Figure 14.16 shows tails plots of the hedging errors. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. We show cases of no hedging (red), hedging once (black), hedging twice (blue), and hedging 20 times (green).

Figure 14.17 shows (a) histograms of hedging errors and (b) kernel density estimates of the distribution of hedging errors when the moneyness is . Panel (a) shows cases of no hedging (red) and daily hedging (green). Panel (b) shows additionally the cases of hedging once (black) and hedging twice (blue). The red kernel density estimate is very inaccurate, because the underlying distribution is such that a large part of the probability mass is concentrated at one point.

14.4.2.3 Expected Utility

Figure 14.18 shows the estimated expected utility as a function of the hedging frequency. In panel (a) the moneyness is . In panel (b) . We apply the exponential utility function , where the risk-aversion parameter takes values (black), (red), and (blue). The expected utilities are estimated using the sample averages. Increasing hedging frequency clearly increases the expected utility. We can see that when the risk aversion is small, then the increase in the expected utility as a function of hedging frequency is much smaller than the increase when the risk aversion is large.