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Path Integral Methods for Parabolic Partial Differential Equations with Examples from Computational Finance

4. The Black-Scholes Formula versus the Feynman-Kac Representation of the Solution of the Black-Scholes PDE

In the option pricing model developed by F. Black and M. Scholes [1] the asset price is assumed to be governed by the stochastic equation

in which and are given parameters. At time the price of an European call option with payoff , which expires at time , can be identified with the expression , where is the solution of the PDE:

with boundary condition , where is a parameter representing the (constant) short-term interest rate and gives the termination payoff as a function of the price of the underlying asset. The fact that the option price must satisfy the PDE (4.2) may be seen as a consequence of Bellman's principle for optimality, but one has to keep in mind that the use of this principle in finance is somewhat subtle. Notice, for example, that the Feynman-Kac formula (3.2) links the PDE (4.2) not with the stochastic equation in (4.1), but, rather, with the stochastic equation

Therefore, the solution of the Black-Scholes equation in (4.2) can be expressed as an expected value of the form

and not as

In other words, in the context of option pricing, one must use Bellman's principle as if the predictable part in the system process has the form , regardless of the form that the predictable part in the system process actually has. This phenomenon is well known and its justification, of which there are several, can be found in virtually any primer on financial mathematics. Roughly speaking, this is caused by the fact that one must take into account the reward on average which investors collect from the underlying asset (see [5]).

Since the stochastic equation in (4.3) admits an explicit solution given by

and since this solution is simple enough, it is possible to implement the recursive procedure from (3.1) with a single iteration. This means that, in this special case, the Feynman-Kac formula in (3.2) is directly computable without the need to approximate the integral kernel of the associated Markovian semigroup. This also eliminates the need for interpolation, if all that one wants to do is compute the solution for some fixed and some fixed . Of course, this assumes that the payoff can be expressed in a way that Mathematica can understand. In our first example, we will compute , assuming that , and that the boundary condition is , where .

Here is the result.

What we have just computed was the price of an European call option with a strike price of \$40 and years to maturity, assuming that the current price of the underlying asset is \$50, that the volatility in the stock price is 20 percent/year and that the fixed interest rate is 10 percent/year. Of course, the price of such an option can be computed directly by using the Black-Scholes formula. First, we will express the Black-Scholes formula in symbolic form

and then compute the price of the same option.

One can see from this example that, from a practical point of view, the Feynman-Kac representation of the solution of the Black-Scholes PDE in Mathematica is just as good as the "explicit" Black-Scholes formula. (Ironically, the Black-Scholes formula takes somewhat longer to type, not to mention the time it might take to retrieve it from a book.) However, the Feynman-Kac representation is much more universal. Indeed, with no extra work--and no additional complications whatsoever--it allows one to compute the value of an option with a very general payoff. Consider for example an option that pays at maturity the smallest of the payoffs from a call with a strike price of \$40 and a put with a strike price of \$80.

With all other parameters kept the same as before (3 years left to maturity, 20 percent volatility and 10 percent interest rate), assuming that the stock price is \$50, the price of this option is

Obviously, the computing time is negligible, but what is much more important is that the only thing that we had to do in this case was to simply change the definition of the payoff. Here is another example.

We will conclude this section with an example of what is known as a cash-or-nothing option.

The overwhelming majority of the options that are currently traded are some combination of calls, puts, and cash-or-nothing options. The origins of this tradition are not exactly certain, but it might have something to do with the fact that these are the options for which popular and easy-to-use computing tools have been widely available. It should be clear from the examples presented in this section that, in the realm of the Black-Scholes model, essentially any reasonably defined option is, in fact, trivial to price, and that having a Black-Scholes formula is no longer necessary. Of course, the well-known binomial model does allow one to compute the value of an option with a very general payoff; however, this popular approach does require some work, and its accuracy and overall efficiency is far from satisfactory. Furthermore, this method is tied exclusively to the assumption that stock prices follow the law of a geometric Brownian motion; the technology that we are developing is not.

One must realize that, in keeping with the method outlined in Section 3, solving the PDE in (4.2) is an entirely trivial matter. The point is that the integral kernel of the associated Markovian semigroup happens to be directly computable, in the sense that it can be written in symbolic form that can be understood by Mathematica. However, our main goal is to demonstrate that this same method is just as easy to implement in much more general situations. In terms of applications to computational finance, that would mean that considerably more sophisticated models for asset values--for example, models with stochastic volatility and stochastic return of a rather general form--would be just as easy to use as the basic Black-Scholes model. The main step in this program is to develop an adequate approximation of the sample path representation of the semigroup , , for a general elliptic differential operator (see the next section).