16.1.1 Risk-free asset

In the BSM model, there is an asset B = {B_t, 0 ⩽ t ⩽ T} from which investors can earn an interest rate of r. This asset is said to be risk-free because capital and interest is repaid with certainty: there is no default. Thus, r > 0 is known as the risk-free rate and it is assumed to be continuously compounded, i.e. for 0 ⩽ t ⩽ T,

As before, we set B₀ = 1. Note that, we could also have fixed B_T = 1 instead of B₀, in which case B would be modeling a zero-coupon bond.

16.1.2 Risky asset

In the BSM model, the price of the risky asset (often an index or a stock) S = {S_t, 0 ⩽ t ⩽ T} evolves according to a geometric Brownian motion

(16.1.1)

where W = {W_t, 0 ⩽ t ⩽ T} is a standard Brownian motion (with respect to the probability measure ) and S₀ is a known quantity (initial asset price).

It follows from Section 14.5 that

i.e. for each time 0 < t ⩽ T, the random variable S_t follows a lognormal distribution with parameters and σ²t (with respect to the actuarial probability measure).

Finally, note that the BSM model can be fully specified by four parameters: S₀, μ, r and σ > 0.

We can also express the model for the risky asset in terms of log-returns. On any given year, the log-return is such that

i.e. the continuously compounded annual return is normally distributed with mean and variance σ². As a result, σ is called the volatility of the log-return.

 In many textbooks, μ is interpreted as the expected annual return on the asset. This is because the expected asset price in the BSM model is

However, the expected annual log-return is

To avoid any confusion with other textbooks and unless stated otherwise, we will interpret μ as the mean (annual) real rate of return on the asset.

Usual description of the BSM model

One may wonder why the BSM model is based on a geometric Brownian motion with drift instead of μ. This is because, in the literature, the dynamics of the risky asset is usually given by the following stochastic differential equation:

More details on stochastic differential equations can be found in Chapter 15. It turns out that the solution to this equation is the process defined in equation (16.1.1), which is a geometric Brownian motion with drift .

Moreover, it provides a financial interpretation to parameter μ: it is then the expected asset price is S₀e^μt.

Recall from the binomial tree model that in order to price derivatives with risk-neutral formulas, we had to contrast between probabilities p and q for each up-move. More formally, we had to distinguish between the real-world or actuarial probability measure and the artificial risk-neutral probability measure obtained as a by-product of an equation rearrangement. A similar situation will occur in the BSM model.

At this point, it is important to note that the notation is used to emphasize that at each time t > 0, the risky asset price S_t really follows a lognormal distribution with parameters and σ²t. Any computations involving the likelihood that the risky asset price S reaches a given level over the next year, or that it crashes, have to be done with this real-world distribution.

16.1.3 Derivatives

We now introduce a third asset in our market, namely a European derivative, whose value is given by the stochastic process V = {V_t, 0 ⩽ t ⩽ T}. The price of the derivative at inception is given by V₀ which is a constant that will be determined later. As the value of the derivative depends upon the underlying asset S, then, for each time 0 < t ⩽ T, the value V_t is random. In particular, the derivative matures at time T, so the random variable V_T represents the payoff of this derivative.

One of our main objectives is to compute the no-arbitrage price of derivatives in the BSM model, i.e. to determine V_t, for each time 0 ⩽ t < T, given a payoff V_T. In other words, we will compute the initial price V₀ and characterize the price V_t, at any time 0 < t < T, using no-arbitrage arguments.

This chapter will focus on simple European derivatives, i.e. derivatives whose payoff only depends on the final asset price S_T:

where g( · ) is a function specified in the derivative’s contract. As seen previously in this book, examples include a call option with strike price K, i.e.

or a put option, i.e.

Exotic path-dependent options, those whose payoff depends on the risky asset price at several times preceding maturity, will be treated later in Chapter 18.

As in discrete-time models, instead of using V = {V_t, 0 ⩽ t ⩽ T}, we will use C = {C_t, 0 ⩽ t ⩽ T} and P = {P_t, 0 ⩽ t ⩽ T} for the price process of a call option and a put option respectively, in which cases we clearly have

when their common strike price is K.

Example 16.1.1 Probability of expiration in the money

A 3-year call option, with a strike price of $120, has been issued on a stock. If the initial stock price is $100 and it is assumed the stock price evolves in a BSM model with mean return of 7% and volatility of 30%, calculate the probability that the call option expires in the money.

We have K = 120 and T = 3 years, so the payoff is given by

The probability that the option expires in the money is , which in this case is equal to .

In this BSM model, we also have that μ = 0.07, σ = 0.3 and S₀ = 100. Then,

or, equivalently,

Finally, we can write

where we used the fact that

 ◼ 

16.2.1 Second look at the binomial model

Suppose we have a given time horizon [0, T] in mind, with 0 being the initial time and T the end of the investment period (expressed in years). In an n-period binomial tree model, we divide this time interval [0, T] in n periods: (0, T/n), (T/n, 2T/n), …, ((n − 1)T/n, T). Then, there are n + 1 time points: 0, T/n, 2T/n, …, (n − 1)T/n, T. In Chapter 11, we put a significant amount of effort into characterizing the corresponding stock price process

with S⁽ⁿ⁾₀ = S₀, as the initial price is not affected by the partition of the interval [0, T]. Here, we have added the superscript (n) to emphasize that we are in an n-period binomial tree model.

For each k = 1, 2, …, n, we can write²

where the random variable I_k keeps track of the number of upward movements in the trajectory after k periods. In other words, I_k can be viewed as the number of successes in k trials. Therefore, I_k follows a binomial distribution with parameters k and p (w.r.t. to ), and then, for a given j = 0, 1, 2, …, k, we have

However, note that the random variable S⁽ⁿ⁾_kT/n does not follow a binomial distribution. It is a random variable whose probability mass function is linked to that of a binomial distribution, namely the distribution of I_k. Figure 16.1 illustrates an eight-period binomial tree (left panel) along with the probability mass function of the asset price at time k = 6 (right panel).

The collection of random variables S = {S₀, S⁽ⁿ⁾_T/n, S⁽ⁿ⁾_2T/n, …, S⁽ⁿ⁾_{(n − 1)T/n}, S⁽ⁿ⁾_T} is a discrete-time stochastic process that can also be written as

for each k = 1, 2, …, n, where U_k ∈ {u, d} (up and down factors) with probability p and 1 − p, respectively. Note that we can also write

for each k = 1, 2, …, n. These last two equations describe how the asset price evolves over time.

In conclusion, the risky asset price should be viewed as a stochastic process, with each possible path in the tree being a realization of this process. Figure 16.2 illustrates four different paths taken by the asset price in a 20-period binomial tree.

16.2.2 Convergence of the binomial model

Suppose we represent the evolution of the price of a financial index over 1 year, using a binomial model with monthly time steps. After each month, the price can take one of two possible values. Thus, after a year, there will be 13 possible values. More generally, after k months, there are k + 1 possible values for the random variable S⁽¹²⁾_k/12, where k = 0, 1, …, 12. But what happens if we want to squeeze in additional up and down moves? In other words, what would happen if we were to have daily time steps or even hourly time steps?

Said differently, for a fixed investment horizon [0, T], we will look at what happens if the number of time steps increases, i.e. if we let n go to infinity. If we increase the number of time steps in the binomial model, the mesh of realizations will increase.

For a given investment horizon, i.e. for a fixed T, the probability distribution of the asset price S⁽ⁿ⁾_kT/n, which is a discrete distribution for each n, will slowly converge to a continuous distribution, as n goes to infinity. To what distribution exactly?

Since

taking the log on both sides, we get

where X⁽ⁿ⁾_k = ln (S⁽ⁿ⁾_kT/n), for each k = 1, 2, …, n. In other words, X⁽ⁿ⁾ = {X⁽ⁿ⁾_k, k = 0, 1, …, n} is given by X⁽ⁿ⁾₀ = ln (S₀) and, for each k = 1, 2, …, n, by

The latter process is a random walk because its increments ε_i = ln (U_i), where i = 1, 2, …, n, are independent and identically distributed random variables taking their values in {ln (u), ln (d)}. However, it is a non-symmetric random walk.

We know from Section 14.2 that a standard Brownian motion is obtained as the continuous-time limit of symmetric random walks. Similarly, non-symmetric random walks will converge to a linear Brownian motion. Consequently, the non-symmetric random walks given by X⁽ⁿ⁾ = {X⁽ⁿ⁾_k, k = 0, 1, …, n} will converge to the linear Brownian motion

used in the BSM model.

Figure 16.3 illustrates how paths in the binomial model resemble paths of a geometric Brownian motion as the number of time steps increases, for a fixed time horizon [0, T].

16.2.3 Formal proof

In this section, we show how the distribution of the risky asset price in the binomial model converges to a lognormal distribution as the number of time steps increases. Let us focus on the convergence of the final price S⁽ⁿ⁾_T. The same work would be necessary to prove the convergence of each S⁽ⁿ⁾_kT/n, for k = 1, 2, …, n − 1.

For the convergence to go through, we need to let the up and down factors depend also on n. It would not make sense to increase the number of time steps, for a given time horizon, without adapting the size of the jumps. We did the same thing when proving that some symmetric random walks were converging to a standard Brownian motion. Therefore, the up and down factors are denoted by u_n and d_n, emphasizing that they depend on the number of time steps.

Recall that we can write

where the random variable I_n records the number of up moves in the full trajectory of the underlying asset’s price. As discussed above, we can further write

The random variable I_n follows a binomial distribution with parameters n and p_n, i.e. , where

Hence, by the Central Limit Theorem,³ we deduce that

or, more precisely,

for any real numbers a < b.

On the other hand, since , we have

We are interested in the convergence of , which can also be rewritten as

where

With the specifications of the Jarrow-Rudd tree (see Chapter 11), we have

where μ and σ are the BSM model parameters. Then it can be shown that

Consequently,

(16.2.1)

This means that, at the limit, the final asset price S_T follows a lognormal distribution with parameters ln (S₀) + (μ − σ²/2)T and σ²T (with respect to the real-world or actuarial probability measure ).

Note that we have obtained the limiting distribution for a single random variable S_T. We could repeat the above procedure for any time 0 < t ⩽ T and get

Putting all these random variables together, we get the linear Brownian motion of the BSM model, namely the stochastic process

or, equivalently, the geometric Brownian motion

defined in equation (16.1.1).⁴

16.2.4 Risk-neutral probabilities

Recall that in the binomial model, for any given simple derivative, we were able to find its replicating portfolio and then rewrite its value as an expectation using new probability weights q and 1 − q known as risk-neutral probabilities. It is important to emphasize that the risk-neutral (conditional) probability q had nothing to do with the real-world (conditional) probability p: it is only a convenient way to rewrite the value of the replicating portfolio as an expectation.

This approach tells us that whenever we need to find the no-arbitrage price of a derivative, and only then, we are allowed to use the artificial binomial distribution with parameters n and q_n in order to write the option price as a discounted (with r) expectation. We should insist that q_n has no other actuarial or financial meaning.

As the convergence of the binomial model to the BSM model does not depend on the specific value taken by p, we deduce that the underlying binomial distribution with parameter q_n for the log asset price will also converge to a normal distribution, but with slightly different parameters. Following from equation (16.2.1), we summarize the results as follows.

The risk-neutral distributions of the random variables ln (S⁽ⁿ⁾_T/S₀), which depend on the (conditional) probability q_n of an up-move, converge to the risk-neutral distribution of ln (S_T/S₀), as n tends to infinity. Therefore,

where we use the notation to emphasize that this is a risk-neutral distribution (rather than a real-world distribution). This is the equivalent in the BSM model of using q rather than p in the binomial model.

We can repeat the above procedure for any time 0 < t ⩽ T and get at the limit

or, equivalently,

at each time 0 < t ⩽ T.

16.3.1 Limit of binomial models

As above, assume the maturity of the call option is T and split the time interval [0, T] into n sub-intervals of length T/n. We want to find the no-arbitrage price of a vanilla call option. Recall from Chapter 11 that in an n-period binomial tree, the initial price of a call option with payoff (S⁽ⁿ⁾_T − K)₊ can be written as a binomial sum:

where

See equation (11.3.5) in Chapter 11. In many cases, i.e. when the realization of S⁽ⁿ⁾_T is not big enough, the first terms of this summation are all equal to zero. In other words, if we define

then we can write

where we have distributed the term inside the first summation.

Then, let us define the incomplete binomial summation, for integers k ⩽ m and real numbers 0 < θ < 1, by

Therefore, the call option price can be written as

where q*_n = uq_ne^{− r(T/n)}. It is easy to show that 1 − q*_n = d(1 − q_n)e^{− r(T/n)} and 0 < q*_n < 1.

Intuitively, since the incomplete binomial summation Φ is a (survival) probability for binomial distributions, by the Central Limit Theorem, we expect it to converge to a similar probability for normal distributions.

Letting C₀ = lim_{n → ∞}C⁽ⁿ⁾₀, then

and one can show that we obtain

(16.3.1)

with

This is the classical Black-Scholes pricing formula for the initial price C₀ of a European call option in the Black-Scholes-Merton model.

Figure 16.4 illustrates the convergence of the call option price in a Cox-Ross-Rubinstein (CRR) binomial tree as a function of the number of time steps. The plot shows an oscillatory pattern around C₀ = 13.26967658, where the wave amplitude goes to 0 as n → ∞.

Example 16.3.1 Price of a call option

A share of stock currently trades for $65. The mean annual log-return on this stock is 6% and the volatility of the log-return is 27%. Calculate the Black-Scholes price of a 6-month 62-strike call option if the risk-free rate is 3%.

We have S₀ = 65, K = 62, T = 0.5, σ = 0.27, r = 0.03. Then

with corresponding probabilities

Using the Black-Scholes formula in equation (16.3.1), the price of this call option is thus

 ◼ 

We shall note that μ, the mean (annual) real rate of return on the risky asset, does not appear in the Black-Scholes formula. This is because the Black-Scholes formula can be obtained as the limit of binomial prices which were obtained by first replicating the call option payoff. No matter what is the expected return on the stock, the replicating strategy will replicate the option payoff in any scenario. This parameter is irrelevant for option pricing.

Example 16.3.2 Probability of expiring in the money

Your investment bank has issued the call option of example 16.3.1. Your boss asks about the probability that the option will end up in the money at maturity.

We are thus asked to compute

where S₀ = 65. Using the fact that

we have

We have to be careful here: we should not compute , i.e. using the distribution

This risk-neutral distribution has no real-world meaning and should be used for option pricing purposes only.

 ◼ 

Using the put-call parity in equation (6.2.4) of Chapter 6, we have

Substituting the Black-Scholes formula of equation (16.3.1), we get

Since N(x) − 1 = −N( − x) for all , we get

Reorganizing yields the desired result:

(16.3.2)

This is the classical Black-Scholes pricing formula for the initial price P₀ of a European put option.

We could also define P⁽ⁿ⁾₀ as the time-0 price of a put option in an n-period binomial tree and verify that

converges to

as n tends to infinity.

Example 16.3.3

In a BSM model, you are given:

• S₀ = 45;
• r = 0.05;
• σ = 0.25.

Compute the price of a 3-month at-the-money put option.

We can find:

Using the Black-Scholes formula in equation (16.3.2), the price of this put option is

 ◼ 

16.3.2 Stop-loss transforms

The attentive reader will have noticed that the Black-Scholes formulas of equations (16.3.1) and (16.3.2) have a familiar look. Indeed, the call option price is related to the stop-loss transform of a lognormally distributed random variable. In equation (14.1.7) of Chapter 14, we obtained the following identity: for and a real number a > 0,

Moreover, we know from the binomial model that

converges, as n → ∞, to

In other words, the no-arbitrage price of a call option is the discounted stop-loss transform of S_T using its risk-neutral distribution:

or, equivalently,

Using this lognormal random variable with its risk-neutral distribution along with the above formula for the stop-loss transform yields the Black-Scholes formula for the call option, that is

with

The same can be done for a put option; see exercise 14.1 in Chapter 14.

16.3.3 Dynamic formula

As the price of the risky asset changes over time, the prices of call and put options also change. We intend to determine how to calculate those option prices when the underlying asset price moves from S₀ to S_t at time t.

In an n-period binomial tree, the random variable V_kT/n is the no-arbitrage price after k steps of a derivative with payoff V_T. To find this price, we can compute recursively the risk-neutral conditional expectation

for the corresponding nodes. This is equivalent to applying equation (11.3.1), i.e. it is as if we were issuing, at time kT/n, a derivative maturing one period later with random payoff V_{(k + 1)T/n}, given that the “initial price” of the risky asset is S_kT/n. Let us emphasize once more that this risk-neutral expectation was obtained as a rewriting of the value of the replicating portfolio for this derivative. At that point, only elementary algebra was needed.

Furthermore, applying iterated expectations, we find that

We can interpret this expectation as if we were issuing, at time kT/n, a derivative maturing n − k periods later with payoff V_T, given that the “initial price” of the risky asset is S_kT/n. In other words, when we observe the value S_kT/n after k periods, we can determine the value V_kT/n by analyzing the sub-tree corresponding to the dates kT/n, (k + 1)T/n, …, T with a root corresponding to the observed value of S_kT/n. Figure 16.5 illustrates a 20-period binomial tree conditional upon observing S_10T/n: a 10-period binomial sub-tree is highlighted in the figure for a given root.

In the BSM model, the argument is similar. Indeed, suppose now that at any time 0 ⩽ t < T, we observe that the underlying asset price is equal to S_t. Then, the time interval [t, T] can be split into n sub-periods and the evolution of the asset price is approximated by a binomial tree. We can determine what happens when n tends to infinity using the same arguments as in Section 16.2.2.

We deduce that, in the BSM model, given the value of S_t, we have the following conditional risk-neutral probability distributions

or, equivalently,

Consequently, the time-t Black-Scholes formula, where 0 ⩽ t < T, is thus

(16.3.3)

where

Note that d₁(0, S₀) = d₁ and d₂(0, S₀) = d₂ as used in formula (16.3.1).

Using the put-call parity relationship at time t, we deduce that the price of a put option at time t is

(16.3.4)

Example 16.3.4 Call and put prices at intermediate times

For a stock currently trading at $34, you have:

• σ = 0.32;
• μ = 0.09;
• r = 0.04.

In this BSM model, calculate the initial price of 6-month 32-strike call and put options. Determine the price of these options 2 months later, in the scenario that the stock price drops to $31.

Using S₀ = 34, K = 32 and T = 0.5 in equations (16.3.1) and (16.3.2), we get

Let us now consider the following scenario. Suppose that, 2 months later, the price of the stock goes down to S_2/12 = 31. Using t = 2/12 and T = 0.5 in equations (16.3.3) and (16.3.4), we get

These last two option prices are realizations of the corresponding random variables given that S_2/12 = 31. If we were to consider another scenario, i.e. another value for S_2/12, then the values of C_2/12 and P_2/12 would be different.

 ◼ 

16.4.1 Forward contracts

We know from Chapter 3 that the payoff of a forward contract on a stock is g(S_T) = S_T − K. Applying the risk-neutral pricing formula in (16.4.1), we deduce that the value of this forward is given by

at any time 0 ⩽ t ⩽ T.

Using the fact that

together with formula (14.1.5) for the mean of a lognormal distribution, we get

In conclusion, the price process of a forward contract is given by

for every time 0 ⩽ t ⩽ T. As expected, we have recovered the model-free formula of equation (3.2.2), which was obtained with replication arguments and without any assumption on the distribution of S_T.

16.4.2 Binary options

Recall from Chapter 6 that there are two classes of binary options: cash-or-nothing and asset-or-nothing binary options. Binary call options pay whenever S_T ⩾ K whereas binary put options pay whenever S_T < K.

More precisely, binary call options with maturity date T admit the following payoffs (up to a multiplicative constant): for a strike price K > 0, we have

• cash-or-nothing call: ;
• asset-or-nothing call: .

Let the stochastic process represent the price process of a cash-or-nothing call option and that of an asset-or-nothing call option. Note that if we buy one unit of the asset-or-nothing call and (short-)sell K units of the cash-or-nothing call, then the payoff of our position is equal to

which is the payoff of a vanilla call option. By the no-arbitrage assumption, we have the same relationship regarding time-t prices, that is

for all 0 ⩽ t ⩽ T. As a consequence, it suffices to obtain an expression for and then, together with the Black-Scholes formula in (16.3.3), we will automatically obtain an expression for .

For a cash-or-nothing call, using the risk-neutral pricing formula (16.4.1), we have

for all 0 ⩽ t ⩽ T. Using the lognormal risk-neutral distribution of S_T|S_t, as given in (16.4.2), together with the c.d.f. of the lognormal distribution as seen in Section 14.1.2, we can write

Consequently, the time-t value of this cash-or-nothing call is given by

(16.4.3)

As mentioned above, since

using the Black-Scholes formula in (16.3.3) together with the pricing formula for a cash-or-nothing call in (16.4.3), we get the following pricing formula for an asset-or-nothing call: for all 0 ⩽ t ⩽ T,

(16.4.4)

Example 16.4.1 Price of binary call options

In the BSM model, you are given:

• S₀ = K = 75;
• r = 0.05;
• σ = 0.25.

Calculate the initial price of 3-month binary call options.

Since we have

then, using the pricing formulas in (16.4.3) and in (16.4.4), we get

 ◼ 

Recall that binary put options with maturity date T and strike price K admit the following payoffs:

• cash-or-nothing put: ;
• asset-or-nothing put: .

Using parity relationships, it is easy to obtain expressions for the time-t prices of these options (see exercise 16.5). More precisely, for each 0 ⩽ t ⩽ T, we will obtain

Example 16.4.2 Price of binary put options

In the same BSM model as the previous example, calculate the initial price of 3-month binary put options.

Using the above formulas, we obtain

 ◼ 

16.4.3 Gap options

Recall from Chapter 6 that gap call and put options are options to buy or sell an asset for a strike price of K only if the underlying asset price at maturity is greater or less than a trigger price H.

First, a gap call option has a payoff given by

We know this payoff can be decomposed as a long position in one unit of an asset-or-nothing call with strike price H and a short position in K units of a cash-or-nothing call with strike price H. Mathematically,

Therefore, by linearity, and using the pricing formulas in (16.4.4) and in (16.4.3) for the time-t prices of binary call options, we get

where

This formula is very similar to the Black-Scholes formula for a vanilla call option as given in (16.3.3). In fact, when K = H, they are equal.

Example 16.4.3 Price of a gap call option

A share of stock currently trades for $35 and the volatility of the log-returns on this stock is 28%. If the risk-free rate is 3%, calculate the initial Black-Scholes price of a 6-month gap call option with strike price $35 and trigger price $37.

We have σ = 0.28, r = 0.03, T = 0.5, S₀ = K = 35 and H = 37. If we compute d₁(0, S₀; H) and d₂(0, S₀; H) with these quantities, then we get

Therefore, the initial price of this gap call option is

 ◼ 

Second, a gap put option has a payoff given by:

In other words, it corresponds to a long position in K units of a cash-or-nothing put with strike price H and a short position in one unit of a cash-or-nothing put with strike price H.

Using this parity relationship together with the formulas for binary put options, the price at time t of a gap put option is

Again, this pricing formula is very similar to the Black-Scholes formula for a vanilla put option as given in (16.3.4) and the two formulas coincide when K = H.

16.5.1 Stock price

From a financial point of view, we expect the value of a call option to increase if S₀ increases. Indeed, for a fixed strike price K, a greater stock price S₀ increases the intrinsic value of the call, which in turn increases the price. It should be the opposite for a put option.

Let us verify this mathematically. We can show that⁶

and then, using the put-call parity P₀ = C₀ − S₀ + e^{− rT}K, it is easy to deduce that

Since the standard normal c.d.f. N( · ) is a probability, its value lies between 0 and 1 and then we get

and

As anticipated, the call option value increases with the initial stock price because , i.e. C₀ is a non-decreasing function of S₀. Similarly, the put option value decreases with the initial stock price because .

16.5.2 Strike price

Because it is the price at which we buy or sell the underlying asset at maturity, the strike price K has the opposite effect, compared with the initial stock price, on the up-front premium. Indeed, if K₁ ⩽ K₂, then

in all possible scenarios, i.e. for any value of the random variable S_T. By the usual no-arbitrage arguments, it is easy to verify that this also holds true at any time t prior to maturity.

In conclusion, if we increase the strike price, then we reduce the initial value of the call option and increase that of the put option. Note that we did not use the Black-Scholes formulas, so this conclusion is model-free: it holds in any market model.

16.5.3 Risk-free rate

From a financial perspective, if a long call option ends up in the money at maturity, the holder will pay K (and she will receive a share of the stock). An increase in the risk-free interest rate reduces the present value of K and therefore increases the value of this call option. For a long put option, it is the opposite.

We can verify that

In both cases, as we are multiplying positive quantities, we can further deduce that

In words, an increase in the risk-free rate increases the value of the call and decreases the value of the put.

16.5.4 Volatility

Volatility in general is a measure of the range of outcomes of a random variable. Higher volatility means there is a wider spectrum of scenarios in which call and put options can end up in the money. Therefore, more volatility increases the values of call and put options.

We can also compute the partial derivatives of C₀ and P₀ with respect to σ to get:

where φ(x) is the p.d.f. of the standard normal distribution. As the partial derivatives are positive, it confirms that increasing the volatility increases both the call and put option prices.

16.5.5 Time to maturity

We want to determine how the time to maturity T affects an option price. Intuitively, the time to maturity impacts at least two elements:

1. A longer time to maturity increases the range of outcomes of asset prices. Indeed, it is intuitive to think that the distribution of stock prices after 1 year will be wider than the possible stock prices after 1 month.
2. A longer time to maturity decreases the present value of future cash flows. Indeed, one dollar paid in a year is worth less than one dollar paid in a month, or K paid in a year is better than K paid tomorrow.

In the case of call options, those effects add up, thus increasing the call option price. As for the put option, the effects go in opposite directions.

Differentiating C₀ with respect to T and then using the put-call parity, we get

Since ∂C₀/∂T is the sum of two positive quantities, we can conclude that the call value increases with T. In other words, a greater time to maturity increases the call option price. For a put option, we cannot make such a definitive conclusion. However, Figure 16.6 suggests that it is also increasing.

16.6.1 Trading strategies/portfolios

In a continuous-time setup, a trading strategy (or portfolio) on the investment horizon [0, T] is a pair of stochastic processes

where

• Θ_t is the number of units of the risk-free asset B held in the portfolio at time t;
• Δ_t is the number of units of the risky asset S held in the portfolio at time t.

As in the discrete-time setup, (Θ_t, Δ_t) can be chosen with specific investment goals in mind. For the rest of this chapter, it will be for replication purposes.

For a given trading strategy {(Θ_t, Δ_t), 0 ⩽ t ⩽ T}, its value at time t is given by Π_t where

This clearly depends on the risky asset price S_t and on the (possibly) random number of units Δ_t and Θ_t. Therefore, Π = {Π_t, 0 ⩽ t ⩽ T} is a stochastic process and it is called the portfolio value process.

In discrete-time financial models, we encountered several times the notion of a self-financing strategy and found many equivalent definitions:

• a strategy that does not involve deposits/withdrawals between inception and maturity;
• the portfolio value resulting from such a strategy is the same before and after rebalancing;
• the variation in the portfolio value comes only from variations in the risky asset and risk-free asset prices.

In a continuous-time model, a self-financing strategy can be defined heuristically as a pair (Θ_t, Δ_t) such that

(16.6.1)

for all t ⩾ 0, i.e. the value of the strategy is the same just before rebalancing (t − ) and just after rebalancing (t). A rigorous definition of a self-financing strategy requires knowledge of stochastic calculus. The interested reader is referred to Section 17.1 for more details.

We seek to replicate the payoff of simple derivatives using a dynamically updated strategy {(Θ_t, Δ_t), 0 ⩽ t ⩽ T}. A strategy {(Θ_t, Δ_t), 0 ⩽ t ⩽ T} is said to be a replicating strategy for the payoff V_T if:

• it is self-financing;
• their final values coincide, meaning that Π_T = V_T in each possible scenario.

Since a replicating strategy is self-financing (by definition), which means it generates no cash flows except at time 0 and time T (just like the derivative it replicates), then by the no-arbitrage assumption we can conclude that the corresponding portfolio value process Π mimics the derivative price at each time point t and in each possible scenario. In other words, if {(Θ_t, Δ_t), 0 ⩽ t ⩽ T} is a replicating strategy for V_T, then we have

for all 0 ⩽ t ⩽ T. In particular, the initial values are equal: V₀ = Π₀ = Δ₀S₀ + Θ₀B₀. In conclusion, the price of the derivative coincides with the value or cost of the replicating portfolio.

In a general continuous-time model, it is not clear whether each payoff admits a replicating portfolio or not. However, as the Black-Scholes-Merton model can be obtained as the limit of binomial trees, it is also a complete market model, i.e. all derivatives can be replicated.

16.6.2 Replication for call and put options

If we want to find the replicating portfolio of a call option, then we must find a self-financing portfolio {(Θ_t, Δ_t), 0 ⩽ t ⩽ T} such that

in each possible scenario, at any time 0 ⩽ t ⩽ T.

The Black-Scholes formula is famous for many reasons, one of them being that it yields an explicit representation of the replicating portfolio within the expression for C_t. Indeed, since we have

then, by identification, we directly obtain that

(16.6.2)

are the number of units of each asset needed to replicate a call option.⁷

Said differently, at any given time t, in order to replicate the value of the call, we must be

• long N(d₁(t, S_t)) units of S;
• short units of B.

From the properties of the normal c.d.f., we further deduce that 0 ⩽ Δ_t ⩽ 1 and thus we should be holding a fraction of the underlying asset to replicate the call option. This is similar to what we obtained in the binomial tree model. Moreover, as the call option gets further into the money (or out of the money), then d₁(t, S_t) increases (or decreases) so that Δ_t approaches 1 (or 0).

This replicating portfolio for a call option, i.e.

requires to be able to trade continuously. Indeed, the number of units Δ_t = N(d₁(t, S_t)) and change continuously with t and S_t, because d₁(t, S_t) and d₂(t, S_t) change continuously with t and S_t. In fact, they are never constant for any time interval. Thus, at every instant, the replicating portfolio must be rebalanced to keep matching the call option price.

Example 16.6.1 Replicating portfolio for a call option

Assume a stock, currently trading for $100, behaves according to a geometric Brownian motion with mean return of 6% and volatility of 30%. If the risk-free rate is 3%, find the replicating strategy of a 1-year at-the-money call option:

• at inception;
• after 6 months, in the scenario where S_0.5 = 115;
• and just before maturity, in the scenario where S_{1 −} = 107.

At inception, we have d₁ = 0.25, d₂ = −0.05 and thus, N(d₁) = 0.598706326 and N(d₂) = 0.480061194. Therefore, the Black-Scholes formula in equation (16.3.1) tells us that this call option price is 13.2833084. The corresponding replicating strategy is given by

at inception. This position is valid at time 0 only and will change immediately after.

In the chosen scenario, the stock price is S_0.5 = 115 after 6 months. Then we can compute d₁(0.5, 115) and d₂(0.5, 115), and find that

Again, this replicating portfolio is only valid at time t = 0.5 (after 6 months) and in this scenario. Note that they will change immediately after.

Moreover, it is interesting to realize that in this scenario, we have Δ₀ = 0.5987 < 0.7983 = Δ_0.5. This is because the call option is no longer at the money at time 0.5 in this scenario; it is then in the money because we have S₀ = K = 100 < 115 = S_0.5.

Suppose now that at (or one millisecond prior to) maturity, the stock price is equal to 107. We know that the payoff of the option will be 107 − 100 = 7. Then, the replicating portfolio must have a balance of $7.

To compute the positions slightly prior to maturity, we must look at the behavior of d₁ and d₂ when t → T. Because S₁ > K, then d₁ → ∞ and d₂ → ∞ when t → T. In this scenario, we get

and the replicating portfolio value just before maturity is

as expected.

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Given that the time-t Black-Scholes formula for a put is given by

then again by identification, we get that

(16.6.3)

are the number of units required in each asset to replicate a put option. In this case, at any given time t, in order to replicate the price of the put, we must be

• short N( − d₁(t, S_t)) units of S;
• long units of B.

As for the replicating portfolio of a call option, the replicating strategy for a put option requires continuous rebalancing.

Again, from the properties of the normal c.d.f., we deduce that − 1 ⩽ Δ_t ⩽ 0 and thus we should be selling a fraction of the underlying asset to replicate the put option. Moreover, as the put option gets further into the money (or out of the money), then d₁(t, S_t) decreases (or increases) so that Δ_t approaches − 1 (or 0).

Example 16.6.2 Replicating portfolio for a put option

Consider a BSM model such that

• S₀ = 75;
• σ = 0.25;
• r = 0.04.

A 1-year at-the-money put option is issued. Find the replicating strategy for this put option:

• at inception;
• after 6 months, in the scenario where S_0.5 = 72;
• and just before maturity, in the scenario where S_{1 −} = 77.

At inception, using the Black-Scholes formula, we have that the put option price is equal to 5.93699. The corresponding replicating portfolio is

Again, this position is valid at time 0 only and will change immediately after.

In the chosen scenario, 6 months later, the stock price has gone down and the put option is then in the money. Using the formulas in (16.6.3) at time t = 0.5 and in the scenario where S_0.5 = 72, we get

As the put option gets more into the money, we must short more stocks: from 0.3878 unit to 0.5117 unit.

Finally, just before maturity, the stock price is worth $77. In this scenario, it is clear the put option expires unexercised as

Therefore, as t gets closer to T = 1, the values of d₁ and d₂ get closer to ∞, which means that

So, the put option price just before maturity is also equal to zero and in this scenario the option expires out of the money.

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16.6.3 Replication for simple derivatives

We now focus on how to replicate simple derivatives, not just call and put options. Unfortunately, finding the replicating portfolio of general simple options will not be as easy as for calls and puts. To help us with this matter, we will relate once more with the binomial model.

Procedure

First of all, let us consider a simple option with payoff V_T = g(S_T). We know from Section 16.4 that, for all 0 ⩽ t < T, the time-t value of this option is given by V_t = F(t, S_t), where

with

For call and put options, we will now use the notation C_t = C(t, S_t) and P_t = P(t, S_t) where

Second, we know from equation (11.2.5) in Chapter 11 that for a derivative with payoff V_T,

is the number of units of the risky asset that an investor should hold, at time step k and in node j, in order to replicate the derivative value. This formula is very specific to the binomial model because it involves discrete time steps. However, it tells us that the delta of a derivative is the ratio of the variation in the derivative value over the variation in the underlying asset value.

Therefore, we can expect that for a very small quantity ϵ, the delta of a simple derivative will be such that

If we take the limit when ϵ → 0, formally we will obtain the partial derivative of F with respect to its second variable:

(16.6.4)

Here, the notation ∂F/∂S simply means taking the first-order partial derivative of the deterministic function F with respect to its second variable.⁸

The formula for Δ_t given in equation (16.6.4) is known as the delta of the option (see also Chapter 20). It is the number of units of the risky asset S to hold at time t in the replicating portfolio in order to match the value of the simple option with payoff V_T = g(S_T).

Now that we have the option price V_t and the delta Δ_t at any time t, we can completely specify the replicating portfolio. Indeed, since we want the replicating portfolio to be such that

then we must have

This is the number of units of the risk-free asset B to hold at time t in the replicating portfolio in order to match the value of the simple option. In conclusion, we now have fully described the replicating portfolio⁹ for a simple derivative with payoff V_T = g(S_T).

To summarize, the replicating procedure for a simple derivative with payoff V_T = g(S_T) is as follows:

1. Identify the function of two variables F( ·, ·) such that .
2. Compute ∂F/∂S, i.e. the first-order partial derivative of F( ·, ·) with respect to its second variable.
3. Set .
4. Set
   

It is important to emphasize that depending on the derivative, this replicating portfolio might need to be continuously rebalanced (or not).

Forward

A forward contract can be considered as a simple derivative with payoff of g(S_T) = S_T − K. The first step is to identify the function of two variables F( ·, ·) such that . We have already found that

Then, the partial derivative ∂F/∂S with respect to its second variable is constant and equal to 1, i.e.

Consequently,

In other words, the replicating portfolio of a forward contract should always be long one unit of the risky asset, at any time t and in any scenario.

Finally,

for all 0 ⩽ t < T. Since B₀ = 1, then at time 0 we should borrow the present value of K (by investing in the risk-free asset) and hold on to this position.

Note that both expressions for Δ_t and Θ_t are constant, i.e. they do not depend on the value of t nor of S_t. In other words, the replicating strategy of a forward contract is a static replicating strategy. Such trading strategies are also called buy-and-hold strategies because no rebalancing is needed. This is consistent with the model-free results obtained in Chapter 3.

Call option

A call option is a simple derivative with payoff g(S_T) = (S_T − K)₊. The Black-Scholes formula for a call option gives us

with

Following from Section 16.5.1, we directly get

as obtained in equation (16.6.2).

Finally,

as obtained in equation (16.6.2).

16.6.4 Delta-hedging strategy

The above procedure tells us how to perfectly hedge, i.e. how to replicate, a simple payoff in the BSM model. However, for many derivatives, replication requires continuous rebalancing to maintain the corresponding replicating portfolio, which is obviously impossible in practice. When trading can only occur discretely, i.e. daily, weekly or monthly, then attempting to replicate a payoff will inevitably yield errors for most derivatives. This is the case for vanilla call and put options. Few derivatives can be replicated using a static strategy, as for forward contracts, or a discretely rebalanced trading strategy.

Whenever the objective is to manage as much as possible the risk of a derivative, taking into account the practical constraints of trading, we say we are hedging (rather than replicating) the derivative. We will come back to hedging in Chapter 20.

The discrepancy between the price of the derivative and the value of its hedging portfolio is known as the hedging error. This is illustrated in the following example.

Example 16.6.3 Hedging error over a month

An investment bank sells an at-the-money call option with maturity T = 1 year and exercise price K = $1000. Using a Black-Scholes-Merton model with volatility σ = 0.2 and risk-free rate r = 0.04 for the evolution of the underlying stock price, let us implement a monthly rebalanced hedging strategy for this call option by discretizing the Black-Scholes replicating strategy of equation (16.6.2). Let us also compute the hedging error made over 1 month in each of the following two scenarios:

1. if S_1/12 = $975;
2. if S_1/12 = $1075.

Note that S₀ = $1000. At time 0, let us compute the number of units as given in (16.6.2). In this case, since

we have

In other words, at time 0, we buy 0.6179114 shares of the underlying stock and short-sell 518.6609 units of the risk-free asset (this also means that we borrow $518.6609 at time 0). Using the Black-Scholes formula, we see that C₀ = 99.2505, which coincides with the initial value of this hedging portfolio:

Now, no matter what the value of S_1/12 will be, 1 month later our loan will be worth 518.6609 × e^0.04/12 = 520.3927. Since no trading is allowed between time 0 and time 1/12, we should not expect the portfolio and the option to have the same value at time t = 1/12 (1 month later). On one side, the portfolio will be worth

On the other side, using the Black-Scholes formula in (16.3.3), we know that the option will be worth

where

Now, if we consider the scenario where S_1/12 = 975, then our position in the risky asset would be worth 0.6179114 × 975 = 602.4636, the portfolio value would be

while the option price would be C(1/12, 975) = 79.57933. In this specific scenario, the bank would be lucky: the hedging error will be negative, i.e. there will be an overflow of 82.07096 − 79.57933 = $2.49. The hedging portfolio is worth more than the call option on the market.

In the other scenario, that is if S_1/12 = 1075, the portfolio value would be

while the option price would be C(1/12, 1020) = 145.4774. In this case, the hedging portfolio would be worth less than the option price, for an hedging error (loss) of $1.62.

 ◼ 

As in the previous example, we will call delta-hedging strategy the trading strategy obtained by discretizing the Black-Scholes replicating portfolio, i.e. whose positions are based upon the procedure of Section 16.6.3. It acknowledges the fact that, in practice, trading can only occur discretely.

The delta-hedging strategy for a call option is defined as follows. Assume that the portfolio can be rebalanced only at the (equidistant) time points t₀, t₁, t₂, …, t_{n − 1} such that 0 = t₀ < t₁ < t₂ < … < t_{n − 1} < T. Even if prices {S_t} and {B_t} are still following the continuous-time Black-Scholes-Merton dynamics, we want to set up a piecewise buy-and-hold trading portfolio. At each of these trading dates t_i, we will observe the realization of and then choose such that:

1. buy/sell units of the risky asset S;
2. invest the remaining portfolio balance in the risk-free asset.

In other words, at the beginning of the (i + 1)-th period we must make sure to hold units of the risky asset. Then, the hedging portfolio’s remaining balance is invested at the risk-free rate.

For each t_i, we hold on to the quantities from time t_i (beginning of period) up to time t_{i + 1} (end of period), and repeat for i = 0, 1, 2, … until t_{i + 1} = T. Mathematically,

Consequently, the corresponding portfolio value process evolves continuously and is given by

This is indeed a discretization of the Black-Scholes replicating strategy. To delta-hedge any other simple derivative, we only need to replace by the corresponding , as given in (16.6.4).

In this case, the portfolio is self-financing by construction: no funds are injected or withdrawn from the portfolio at any of the trading dates. More precisely, at each of these trading dates t_i, we will observe the realization of and then we will choose such that

Since we want , it implies that

or, equivalently,

for each i = 0, 1, …, n − 1.

The value of the portfolio and the derivative it is tracking are only compared at maturity. In a self-financing hedging portfolio, hedging errors are rolled over until maturity. This is illustrated in the following examples.

Example 16.6.4 Rebalancing a delta-hedging portfolio

Let us come back to example 16.6.3 and see how to rebalance the portfolio after 1 month in the scenario where S_1/12 = 975.

We saw, in the previous example, that in this scenario the value of the portfolio after 1 month (before rebalancing) is $82.07096, split into a loan of $520.3927 and 0.6179114 units of the risky asset. This is the amount available for investing over the next month. Recall that this is more than the option price. However, since we are implementing a delta-hedging strategy, we do not withdraw this excess from the portfolio.

Given that the asset price went down to 975, we should trade to make sure we now own

units of the underlying asset. In this scenario, this means selling 0.6179114 − 0.5615934 = 0.056318 unit of this asset for a net income of 0.056318 × $975 = $54.91005.

This amount is then used to partially reimburse the loan: we should now have a short position in the risk-free asset worth 520.3927 − 54.91005 = $465.4826. Since this risk-free asset is now worth

we should be short 465.4826/1.003339 = 463.9335 units.

In conclusion, after rebalancing we have:

We can verify that the portfolio value before and after rebalancing is the same:

 ◼ 

It is important to remember that for most derivatives, a delta-hedging strategy is not a replicating strategy: hedging errors will occur. These errors are solely due to discretization because we cannot trade continuously. However, in principle, if we rebalance the portfolio more often, the hedging errors should decrease. We will come back to this issue in Chapter 20.

Example 16.6.5 Delta-hedging until maturity

A 4-month at-the-money European call option is issued. You are using a BSM model with parameters μ = 0.07, r = 0.04 and σ = 0.2. The underlying asset price is S₀ = 25 and the call option price is given by the Black-Scholes formula. Calculate the profit or loss resulting from delta-hedging this option monthly until maturity in the following scenario:

At inception

We first need to find the option price at inception. Using the Black-Scholes formula with a maturity of T = 4/12, we get d₁ = 0.17 (rounded to two decimals), N(d₁) = 0.5675 and thus C₀ = 1.2635.

Consequently, over the time interval , we must hold 0.5675 share of stock for a total value of 25 × 0.5675 = 14.1875. But since we received only $1.2635 from selling the call option, we will borrow the difference, i.e. 14.1875 − 1.2635 = 12.9240 (recall that B₀ = 1). The delta-hedging portfolio over the time interval is

After 1 month

In our scenario, after a month, the stock price will increase to S_1/12 = $26 and hence the portfolio value (before rebalancing) will be

Now, we should hold N(d₁(1/12, 26)) = 0.7054 share of stocks over the time interval . After rebalancing, the portfolio value must still be worth 1.7878 since it must be self-financing. Therefore, solving for Θ_1/12 in

we find that

in this scenario.

After 2 months

After 2 months, the stock price in this scenario is now $28 and hence the portfolio value (before rebalancing) is

Now, we should hold N(d₁(2/12, S_2/12)) = 0.9345 share of stocks over the time interval . After rebalancing, since the portfolio value must still be 3.1433 (self-financing condition), solving for Θ_2/12 in

we get

At maturity

Repeating the procedure at times 3/12 and 4/12, we get that the delta-hedging portfolio is worth 3.9004 at the option maturity whereas the payoff is 29 − 25 = 4. In this scenario, there is a hedging error (loss) of 0.1.

The following table summarizes the computations:

t	T − t	St	Bt			Πt
0	4/12	25	1.0000	0.5675	−12.9240	1.2635
1/12	3/12	26	1.0033	0.7054	−16.4975	1.7878
2/12	2/12	28	1.0067	0.9345	−22.8697	3.1434
3/12	1/12	27	1.0101	0.9222	−22.5409	2.1320
4/12	0	29	1.0134			3.9004

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