17.1.1 Partial differential equations

First, let us look at the following quadratic equation:

(17.1.1)

For given constants a, b and c, the goal is to obtain the value(s) of x such that ax² + bx + c = 0 is verified. We know that, depending on the values of a, b and c, the value of the discriminant b² − 4ac will decide whether the quadratic equation in (17.1.1) has one or two solutions, or even no (real) solution.

In physics and biology, it is common for a function F(x) describing a physical/biological quantity to be the solution of an equation of the form

(17.1.2)

with a, b and c being known constants. An equation like the one in (17.1.2) is called a differential equation and solving it means finding an analytical expression for F(x).

One one hand, solving a quadratic equation as (17.1.1) means looking for a number x whereas on the other hand, solving a differential equation as (17.1.2) means looking for a function F(x). In both cases, the existence of a solution depends on certain conditions. In what follows we will consider ordinary differential equations and then partial differential equations.

An ordinary differential equation (ODE) is a differential equation whose solution (if it exists) is a function of only one variable, say x, and which gives a relationship between the function F(x) itself and its derivatives F′(x), F′′(x), etc.

Example 17.1.1 A simple ODE

Let b be a known constant and let us find a function F(x) such that:

for all . It is easy to verify that, for this ODE,

is one possible solution, for any value of a.

 ◼ 

A partial differential equation (PDE) is a differential equation whose function we are looking for has two variables, say t and x. The PDE is thus expressed in terms of the partial derivatives of F, namely , , , etc.

Example 17.1.2 A simple PDE

Let us find a function F(t, x) such that:

(17.1.3)

for all . For this PDE, we can easily verify that the function

is such that its partial derivatives ∂F/∂t and ∂²F/∂x² satisfy the relationship in (17.1.3). Indeed,

Note that the function F(t, x) = 0, for all t and x, is also a solution to this PDE.

 ◼ 

In the previous examples, we found more than one solution to the same ODE and PDE, respectively. In order to guarantee the uniqueness of a solution, ODEs and PDEs require initial or final conditions, known as boundary conditions.

For the ODE in example 17.1.1, if we add an initial condition such as F(0) = 1, then

becomes the only solution.

Without getting into the details, in what follows we will specify final conditions to obtain unique solutions to our PDEs. More details below.

In most cases, it is difficult or even impossible to find an explicit solution to a PDE. In those cases, numerical methods are used to obtain approximations of the solution. However, for one class of PDEs, the one most commonly used in finance, there is a way to obtain the solution using stochastic calculus, as presented in Chapter 15; this is what we will describe next.

17.1.2 Feynman-Kač formula

The Feynman-Kač formula provides a solution to PDEs in the family of parabolic PDEs. This solution is expressed as the conditional expectation of a diffusion process sampled at a fixed time. Here is a simple version of the Feynman-Kač formula, which is adapted to the broader objectives of this chapter.

As always, assume that T > 0 is fixed. For a given parameter r > 0 and given functions a( · ), b( · ), h( · ), we seek to find a solution to the PDE

(17.1.4)

for all , with final condition F(T, x) = h(x), for all .

According to the Feynman-Kač formula, the solution to this PDE can be written as follows: for each (t, x) in the domain, we have

(17.1.5)

where {X_s, s ⩾ 0} is the diffusion process determined by the following SDE:

(17.1.6)

When trying to solve a PDE of the form given in (17.1.4), we can use the Feynman-Kač formula by following these steps:

• INPUTS: parameter r > 0 and functions a( · ), b( · ), h( · ), as given by the PDE.
  1. Find the diffusion process {X_s, s ⩾ 0} given by equation (17.1.6), using stochastic calculus, as seen in Chapter 15.
  2. For each t and x in the domain, determine the conditional probability distribution of X_T given that X_t = x.
  3. Compute the conditional expectation in equation (17.1.5).
• OUTPUT: function F(t, x).

The following example illustrates this methodology.

Example 17.1.3 Solving a simple PDE

We would like to find the solution to the PDE

with boundary condition F(T, x) = x², where σ is a given constant.

Here, our inputs are a(x) = 0 and b(x) = σ, h(x) = x² and r = 0. Consequently, looking at (17.1.6), we consider the following SDE:

whose solution is simply X_s = σW_s. Now, using h(x) = x² and r = 0 in (17.1.5), the solution is

As already computed in Chapter 14 (see equation (14.3.2)), we can write

In conclusion, the function F(t, x) = x² + σ²(T − t) is the solution to the above PDE.

For validation purposes, let us check this. First, we can verify that the final condition F(T, x) = x² + σ²(T − T) = x² is indeed equal to h(x) = x². Second, we have

Therefore,

as expected.

 ◼ 

17.1.3 Deriving the Black-Scholes PDE

Let us recall that the Black-Scholes model can be summarized as a continuous-time financial market model composed of a risk-free asset {B_t, t ⩾ 0} and a risky asset {S_t, t ⩾ 0}. The dynamics of the price of the risk-free asset can be determined by the following ODE:

(17.1.7)

whose solution is simply B_t = B₀e^rt. The dynamics of the price of the risky asset can be determined by the following SDE:

(17.1.8)

whose solution is a geometric Brownian motion: .

Let us consider a simple option with payoff V_T = h(S_T). We want to compute the no-arbitrage price of this derivative and find its replicating portfolio, as defined in Chapter 16.

We make the following assumptions: the time-t value V_t of this option can be written as F(t, S_t), where F(t, x) is a sufficiently smooth function; assume also that there exists a replicating portfolio {(Θ_t, Δ_t), 0 ⩽ t ⩽ T} for this payoff, i.e. V_T = Π_T, where

and where the only cash flows of this portfolio are at time 0 and time T. Then, by the no-arbitrage principle, we have that V_t = Π_t, for all 0 ⩽ t ⩽ T, where

is the value at time t of this replicating portfolio. By the above assumptions, we then have that F(t, S_t) = Π_t, for all 0 ⩽ t ⩽ T. At this stage, we do not have an expression for the function F nor for the replicating portfolio {(Θ_t, Δ_t), 0 ⩽ t ⩽ T}: we only assume that they exist. Our goal is to find explicit expressions.

To meet this objective, we need to find a trading strategy that is both replicating and self-financing. This is similar to what we did in the binomial model, except that we are now in a continuous-time model.

In a continuous-time model, the self-financing condition is given by the following stochastic equation:

(17.1.9)

It can be obtained as the limit of equation (11.2.3) in Chapter 11. In words, the variation in the portfolio value comes from changes in values from both assets within the portfolio.

Moreover, since F(t, S_t) = Π_t, for all 0 ⩽ t ⩽ T, we must also have that

meaning that, at each instant, the variation in the portfolio value must match the variation in the derivative’s price. This is also known as the replicating condition.

We are now ready to derive the Black-Scholes PDE. First of all, F(t, S_t) is a transformation of a diffusion process. Therefore, using Ito’s lemma, we can write

Substituting dS_t, as given by equation (17.1.8), we further have

(17.1.10)

On the other hand, since Π_t = F(t, S_t), the self-financing condition of (17.1.9) can be rewritten as

(17.1.11)

where we used the fact that Θ_tB_t = F(t, S_t) − Δ_tS_t (from the definition of the portfolio’s value) and where we have substituted the expressions for dB_t and dS_t as given in equations (17.1.7) and (17.1.8), respectively.

We have obtained two equivalent stochastic dynamics for F(t, S_t): one in equation (17.1.10) and one in equation (17.1.11). This means that for these two SDEs, the terms in front of dt must be equal and the terms in front of dW_t must be equal. Consequently, we have the following system of equations:

From the second equation, we easily find that

Then, using this in the first equation, we deduce the so-called Black-Scholes PDE or Black-Scholes equation: if the function F(t, x) is such that

(17.1.12)

with boundary (final) condition F(T, x) = h(x), where h is the payoff function, then the above system of equations is verified.

To summarize, we have obtained that the time-t value of a simple derivative with payoff V_T = h(S_T) is given by F(t, S_t), where the function F(t, x) is the solution to the Black-Scholes PDE. The PDE comes from the model assumptions (dynamics of each asset) whereas the payoff of the option provides the terminal condition that the PDE should abide to. This boundary condition assures that we have a unique solution to this PDE.

Once we have found the price F(t, S_t) for the payoff V_T = h(S_T), then the replicating portfolio {(Θ_t, Δ_t), 0 ⩽ t ⩽ T} is given by:

(17.1.13)

at every time t ∈ [0, T).

It is also important to notice from the Black-Scholes equation (17.1.12) that the parameter μ has completely disappeared. Consequently, the option price (through function F) will not/does not depend on this parameter. This is similar to the result obtained in the binomial tree: the expected stock return under the real-world probability measure, which in that case was determined by the parameter p, is not relevant to determine the no-arbitrage price of a derivative.

In conclusion, if we can find a solution F(t, x) to the Black-Scholes PDE in (17.1.12), we will have a full solution to our pricing and hedging problem: at each time t, we have

• the price of the derivative V_t = F(t, S_t);
• the replicating portfolio (Θ_t, Δ_t) given by
  

17.1.4 Solving the Black-Scholes PDE

Luckily enough, under the assumptions of the BSM model, the Black-Scholes PDE (17.1.12) can be solved using the Feynman-Kač formula. Indeed, since F(t, x) is a solution of (17.1.12), then, from equation (17.1.5), it can be written in the following form:

where {X_s, s ⩾ 0} is determined by the SDE

This is simply a geometric Brownian motion with drift r whose solution we already found earlier: X_s = X₀exp {(r − σ²/2)s + σW_s}.

Using the standard arguments, such as formula (14.3.2), we get:

(17.1.14)

This is the pricing formula for simple derivatives with payoff V_T = h(S_T) obtained with the PDE approach. Note that we then have an explicit expression for the replicating portfolio as given in (17.1.13).

Let us illustrate what we have just obtained with a forward contract.

Example 17.1.4 Forward contract

The payoff of a (long) forward contract with delivery price K is V_T = S_T − K. Then, using the pricing formula in (17.1.14) with h(x) = x − K, we get

From the formula in (14.1.2) for the moment generating function of a normal distribution, we know that

and, consequently,

This means that the time-t value of a forward contract is given by F(t, S_t) = S_t − e^{− r(T − t)}K, as we already knew from Chapter 3. In fact, this was a model-free result, so it had to be true in any model, including the Black-Scholes-Merton model.

Since we have

the replicating portfolio is given by

and

Note that it is a static strategy, i.e. it is constant with respect to time. In other words, to replicate a long forward contract, we must hold (at any time t) one unit of the underlying asset and be short e^{− rT}K units of the risk-free asset. This is also a result we had obtained in Chapter 3.

 ◼ 

Note that we have found our first solution to the Black-Scholes PDE (17.1.12): it is given by the function F(t, x) = x − e^{− r(T − t)}K. We will soon find other solutions.

17.1.5 Black-Scholes formula

Let us now specialize the above results to the case of a call option. Similar to what we did before, we will substitute the notation F(t, x) by C(t, x). In this case, the final condition become C(T, x) = (x − K)₊ and by (17.1.14) we have

Note that this expectation is the stop-loss transform of a lognormally distributed random variable. Indeed, we have

Consequently, using once again the stop-loss formula in (14.1.7) of Chapter 14, we have

where the functions d₁(t, x) and d₂(t, x) have been defined in (16.3.3) of Chapter 16.

Again, we can verify (with tedious calculations) that C(t, x) is indeed a solution to the Black-Scholes PDE (see the Greek letters and their relationship to the Black-Scholes equation (PDE) in equation (20.5.7)). We could also show that as we approach maturity, then

As a byproduct of the above verification, we get

from which we deduce the replicating portfolio of a call option:

This is consistent with the results previously derived in Chapter 16.

Finally, using the put-call parity, we can deduce the analytic expression of the put option price P(t, S_t). This can also be achieved by exploiting the linearity of the Black-Scholes PDE (see the exercises).

17.2.1 Probability measure

In elementary and more advanced probability theory, for a given sample space Ω, we define a probability measure as a real-valued mapping defined on the set of all events, i.e. subsets of Ω, with the following properties:

1. for any event E⊆Ω, we have ;
2. ;
3. for any sequence of disjoint events E₁, E₂, …⊆Ω, we have
   
   This last property is known as infinite additivity and it implies finite additivity: for any n disjoint events E₁, E₂, …, E_n⊆Ω, we also have
   

Example 17.2.1 Throwing a regular die

We throw a well-balanced die, as we did in Chapter 1. Therefore, the sample space Ω is

For this experiment, we should define by:

The mapping assigns the value 1/6 to each event. It is easy to see that this meets the three criteria necessary to be called a probability measure.

 ◼ 

In an introductory probability course, the modelling is usually done with a fundamental space Ω and only one probability measure (rarely defined in an explicit way). Unfortunately, it creates the false impression that is just a symbol with no particular meaning and/or specific definition.

Example 17.2.2 Probability measures for economic states

Assume the sample space of economic states for the upcoming year is

According to actuary A, the probability of observing these events is:

But actuary B is more pessimistic. According to her, the probability of observing these events is:

Verifying the three conditions in the definition of a probability measure presented above, we have thus defined two probability measures and for the same experiment.

 ◼ 

We say that and are equivalent probability measures if, for any event E, we have

or, equivalently,

This is often written as . More or less, two probability measures are equivalent if they agree on all impossible events, which means then that they also agree on all events of probability equal to one. For any other event, they can differ.

Example 17.2.3 Probability measures for economic states (continued)

Because actuaries A and B agree that all four events have a non-zero probability, we have that .

A third actuary (C) assesses the likelihood of each event of Ω and concludes that an economic boom is impossible. Mathematically, . Therefore is not equivalent to and is not equivalent to .

 ◼ 

17.2.2 Changes of probability measure

Let us now present a methodology leading to a change of probability measure in a given probability model.

Let be a probability measure on a sample space Ω, both designed for a given experiment. Now, choose a random variable Z such that:

1. Z ⩾ 0;
2. .

For such a given random variable Z, we can define another probability measure as follows: for any event E⊆Ω, set

(17.2.1)

where we should stress that the expectation is taken with respect to . The notation emphasizes that for each such random variable Z, there is a new probability measure attached to it. In probability theory, Z is known as the Radon-Nikodym derivative of with respect to . This is just terminology as no derivatives are really involved.

We need to make sure that meets the three conditions to be a probability measure. It is rather easy to verify that for any event E⊆Ω, we have and . A little bit more work is needed to verify that for any sequence of disjoint events E₁, E₂, …⊆Ω, we have

In conclusion, for each such random variable Z, we now have a methodology to construct a new probability measure , usually different from the existing .

Let us now compute expectations of the form with respect to the probability , i.e. compute weighted averages using the weights given by . First, by definition (17.2.1), we have

Indeed, since the random variable is a Bernoulli random variable, i.e. it takes only the values 0 and 1, then

The result follows by the definition in (17.2.1).

More generally, one can show that² for any random variable X, we have a similar relationship:

(17.2.2)

The expectation of an arbitrary random variable X computed with the probability measure is equal to the expectation with respect to where X is distorted by the random variable Z.

To successfully use the above methodology, we first need to choose a positive random variable Z with unit mean. Then, we can determine the distribution of another random variable X with respect to or, in other words, the impact of the change of measure on the distribution of X.

Example 17.2.4 Change of measure for a normal distribution

Let us consider the case of a normally distributed random variable where the notation emphasizes that the distribution is affected by the choice of the probability measure ( in this case). This statement means that: for any , we have

An equivalent characterization of the normal distribution is given by its m.g.f.: for any , we have

See Chapter 14.

Fix a real number α. Let us now find the distribution of X with respect to a probability measure defined with the random variable

For simplicity, we have chosen to write instead of .

First of all, it is easy to verify (see exercise 17.4) that Z_α is indeed such that

1. Z_α ⩾ 0;
2. .

Now, let us show that . Using the characterization of the normal distribution by its m.g.f., it suffices to verify that

for all , where stands for the expectation with respect to . Indeed, using (17.2.2), we have

where, in the last step, we used the fact that or, equivalently, that . The result follows.

 ◼ 

This last example shows that it is possible to apply a well-chosen change of measure to shift the mean of a normally distributed random variable, i.e. to move from a mean of 0 under to a mean of α under . Note that the variance is not affected by this change of measure. This will be of paramount importance for linear Brownian motions and the Black-Scholes-Merton model.

Example 17.2.5 Change of measure for an exponential distribution

Let us now consider an exponentially distributed random variable , with α > 0. This means that, with respect to , the random variable X admits the following probability density function:

Again, an equivalent characterization of the exponential distribution is given by its m.g.f. Indeed, for any λ < α, we have

(17.2.3)

It is possible to change the probability measure from to to make X an exponentially distributed random variable with a larger parameter β > α with respect to . Indeed, it suffices to use

to define .

Again, it is easy to verify (see exercise 17.5) that Z_β is such that

1. Z_β ⩾ 0;
2. .

Let us now compute the m.g.f. of X with respect to :

where in the second last step we used the formula in (17.2.3). Since

for all λ < β, we can conclude that .

 ◼ 

Likelihood ratio

One might wonder how to choose the random variable Z in order to change the probability distribution of X (with respect to the original probability ) to a prescribed distribution with respect to the probability measure .

For a continuous random variable, this is given by a ratio of probability density functions (p.d.f.). Assume that X has a p.d.f. f_X(x) with respect to and that we are looking for the change of probability measure such that we would obtain the p.d.f. f^Z_X(x) with respect to . In particular, after the change of measure, we should have the following:

This is the expectation of X with respect to .

First, we know that the -expectation of the random variable given by

is given by

In other words, we must choose:

We easily verify that Z ⩾ 0, as the p.d.f.s are positive functions and

Similarly, for discrete random variables, a similar change of probability measures, given by the ratio of probability mass functions, will allow a prescribed distribution to be obtained.

17.2.3 Girsanov theorem

The last two examples have shown the effect of a given change of measure on the distribution of a random variable X. In the Black-Scholes-Merton model, we will need to change the distribution of a whole stochastic process, namely that of a linear Brownian motion.

This situation will be taken care of by the so-called Girsanov theorem, which is essentially the analog for Brownian motions of the change of measure discussed in example 17.2.4 for normal random variables. The Girsanov theorem is the last building block needed before we can finally determine the no-arbitrage price of a derivative and obtain risk-neutral pricing formulas. In this section we present a simple version of the latter.

Consider a -standard Brownian motion W = {W_t, 0 ⩽ t ⩽ T}, i.e. a standard Brownian motion under , and for a fixed number θ define the process W^θ = {W^θ_t, 0 ⩽ t ⩽ T} by

The Girsanov theorem states that if we use the random variable

to obtain the probability measure , then W^θ = {W^θ_t, 0 ⩽ t ⩽ T} will be a -standard Brownian motion.

At first, this result can be rather confusing. Indeed, by definition W is a -SBM and since

then W^θ is a linear Brownian motion with drift θ with respect to . On the other hand, since by the Girsanov theorem W^θ is a -SBM and

then W is a linear Brownian motion with drift of − θ with respect to . We have summarized this discussion in Table 17.1.

with respect to
W is a	standard BM	linear BM
Wθ is a	linear BM	standard BM

Whereas example 17.2.4 showed how we can change the probability measure to shift the mean of a normally distributed random variable, the Girsanov theorem tells us how to change the probability measure to shift the drift of a linear Brownian motion.

17.2.4 Risk-neutral probability measures

As in the general n-period binomial model and in the trinomial tree model (see e.g. Section 13.3.2), the concept of risk-neutral probability measure is fundamental in the theory of continuous-time derivatives pricing. In any continuous-time model with a stock price process S = {S_t, 0 ⩽ t ⩽ T} and a risk-free asset price process B = {B_t, 0 ⩽ t ⩽ T}, a probability measure is called a risk-neutral probability measure if it is such that:

1. is equivalent to ;
2. the stochastic process is a -martingale, i.e. a martingale with respect to the probability measure , which means that, for all 0 ⩽ t₁ < t₂ ⩽ T,
   

The probability measure is also known as an equivalent martingale measure (EMM).

As discussed earlier in this chapter, the equivalence condition means that and have the same zero-probability events or, equivalently, the same probability-one events. Intuitively, if an event is impossible or certain with respect to , then it is still the case under the risk-neutral probability measure .

The martingale condition says that if the risky asset price S is discounted with the risk-free asset B, then, with respect to a risk-neutral probability , this discounted price does not have any remaining trend as it behaves as a martingale.

Using now a more formal definition of probability measure, recall from Section 13.3.2 that the FTAP states that:

• a market model is free of arbitrage opportunities if and only if there exists at least one risk-neutral probability measure ;
• an arbitrage-free market model is complete if and only if there exists a unique risk-neutral probability measure . Otherwise the market is said to be incomplete.

17.2.5 Risk-neutral dynamics

Let us now put all the pieces together and determine how to price derivatives in the BSM model using the risk-neutral approach. In the BSM model, the risky asset S is a geometric Brownian motion of the form

and the risk-free asset is such that

with B₀ = 1. The resulting financial market is free of arbitrage opportunities.³ Thus, according to the FTAP, if the market is arbitrage-free then there exists at least one risk-neutral probability measure under which the process given by is a martingale.

Dividing S_t by B_t we get

In other words, the discounted stock price process S/B is still a geometric Brownian motion. We know that this GBM is a -martingale if and only if (see Section 14.5.3). Therefore, if we need to find an EMM, the probability measure is not going to be our candidate. That is why we called the real-world probability measure to mark the difference.

In our quest for an EMM , let us define (using a very suggestive notation) by

(17.2.4)

and by

(17.2.5)

One has to be careful: recall from Table 17.1 that, with respect to , the new process is not a standard Brownian motion, it is a linear Brownian motion.

Using the Girsanov theorem, if we set with , i.e. if we use the following change of measure

(17.2.6)

then is a -standard Brownian motion and, consequently, is a -martingale. This last statement about the martingale property of is backed up by the work we did in Section 14.5.3.

So, the probability measure defined with the random variable in (17.2.6) is the (unique) risk-neutral probability measure in the BSM model. Indeed, it is now easy to verify that is a -martingale. By definition of the geometric Brownian motion S in the BSM model, we have

where in the second last step we used the definition of given in (17.2.4) and in the last step we used the definition of given in (17.2.5).

Since , then we can summarize our findings as: for all 0 ⩽ t ⩽ T,

We can also say that the continuous-time stochastic process {S_t, t ⩾ 0} is

• a GBM with drift coefficient under the real-world/actuarial probability measure ;
• a GBM with drift coefficient under the risk-neutral probability measure .

Note that the volatility coefficient is σ with respect to both probability measures.

17.2.6 Risk-neutral pricing formulas

Imagine a financial derivative with payoff V_T = S_T, that is a derivative providing one unit of the underlying risky asset at maturity T. To avoid arbitrage opportunities, this derivative must be worth exactly S_t, i.e. we must have V_t = S_t, at any time 0 ⩽ t ⩽ T. From the definition of a risk-neutral probability measure, we already have that

In other words, for this specific derivative, we have

for all 0 ⩽ t ⩽ T. This should be reminiscent of the risk-neutral pricing formulas obtained in the general binomial tree model (see e.g. equation (11.3.1)) and in the BSM model (see e.g. Section 16.4).

It turns out that⁴ for any European payoff V_T, simple or exotic, we have a similar risk-neutral pricing formula: for all 0 ⩽ t ⩽ T,

or, written differently,

(17.2.7)

Note that as a particular case of this pricing formula, the initial no-arbitrage price is given by

This is why the risk-neutral probability , under which we can find the no-arbitrage price of any derivative, is also called the pricing probability measure.

Note that, for a simple payoff V_T = h(S_T), the pricing formula in (17.2.7) coincides with the pricing formula in (17.1.14), obtained with the PDE approach

 Again, let us emphasize that is an artificial probability measure meant to find the price of a derivative. In other words, it is a byproduct of no-arbitrage pricing. The measure should not be used for risk management purposes such as computing likelihoods of events and scenarios.

Let us put the risk-neutral pricing formula of (17.2.7) to the test.

Example 17.2.6 Forward contract

The payoff of a forward contract with fixed delivery price K is given by V_T = S_T − K. Then by equation (17.2.7) we have

for any 0 ⩽ t ⩽ T. Since {e^{− rt}S_t, 0 ⩽ t ⩽ T} is a -martingale (by definition of ), we have

and then

as obtained a few times already in this book.

 ◼ 

17.2.6.1 Black-Scholes formula

Now, let us consider the no-arbitrage price of a call and a put option. In particular, if we consider C_T = (S_T − K)₊, then from (17.2.7) we have

for any 0 ⩽ t ⩽ T. Since S is a -geometric Brownian motion, it possesses the Markovian property as discussed in Chapter 14. Consequently, as we did in Chapter 16, we can further write

where

Applying once more the stop-loss formula in equation (14.1.7), we obtain a dynamic version of the Black-Scholes formula: for 0 ⩽ t < T,

where

By the put-call parity, we have

and we can recover the Black-Scholes formula for the price of a put option: for all 0 ⩽ t ⩽ T,

It is important to emphasize that the risk-neutral pricing formula in equation (17.2.7) can be applied to any European payoff V_T, not only simple payoffs of the form V_T = h(S_T). Consequently, we now have a powerful methodology to price exotic options such as Asian, barrier and lookback options, as seen in Chapter 7, using the risk-neutral dynamics of {S_t, t ⩾ 0} and the corresponding path-dependent payoff functional.