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Observations on the Theory and Empirics of Mathematical Finance

[PDS] Feynman-Kac Representation of BSM PDE

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In practice, most people price financial derivatives by Monte Carlo simulation. However, when Black-Scholes-Merton (BSM) gave their famous (or notorious) option pricing formula they came up with that after solving a Partial Differential Equation (PDE). So, just from that point of view, it is not immediately obvious that price obtained via Monte Carlo simulation should be the same as achieved by solving the PDE.

One can, of course, come up with the same result by approaching the option pricing problem from a probabilistic point of view – what is known as the ‘risk-neutral’ method – according to which option price is the discounted expected payoff (which ultimately justifies pricing options by Monte Carlo simulation, in turn relying on the Law of Large Numbers).

So while there are these two very theoretically sound approaches to option pricing, it would be good to know if there is an underlying mathematical connection between the two approaches. It turns out there is, and in fact the result that there is a connection between the two precedes much of the development of option pricing theory.

This result was given by the famous physicist Richard Feynman and probabilist Mark Kac who showed that solution to parabolic (read ‘nice’) PDEs is intimately related to conditional expectations. In what follows we lay out that connection for the specific case of BSM PDE.

Feynman-Kac Representation of BSM PDE

Given the the drift rate \mu and the volatility \sigma, the Geometric Brownian Motion (GBM) for the stock price process S(t) is given by:

dS(t) = \mu S dt + \sigma S dX(t)

where dX(t) represents the increment of a standard Brownian Motion X(t). The above SDE for the stock price process can be said to be in the ‘real world probability measure’.

Then, given a financial derivative, say, a Call Option, C(t, S(t)), Ito’s lemma gives us the Stochastic Differential Equations (SDEs) for C(t, S(t)) as:

\displaystyle dC = \Big( \frac{\partial C}{\partial t}+ \mu S\frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}\Big)dt + \sigma S \frac{\partial C}{\partial S} dX(t)

Setting up a hedging portfolio with one unit in C(t, S(t)) and -\Delta units of the stock S with \Delta = \frac{\partial C}{\partial S} gives us Black-Scholes-Merton PDE with the drift \mu replaced by the risk-free rate r, i.e.:

\displaystyle \frac{\partial C}{\partial t}+ r S \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = rC

After delta hedging has ‘removed’ the risk of the portfolio of one unit in C and \Delta units in stock S, the ‘right’ stock price process to consider is the one in the `risk-neutral’ measure, as:

dS(t) = r S dt + \sigma S dX(t)

(Girsanov theorem implies that the diffusion term in the GBM for stock prices does not change when we move from the `real world’ to a ‘risk-neutral world’.)

Feynman-Kac representation of SDEs tell us that PDEs of the BSM kind have an equivalent probabilistic representation. That is, Feynman-Kac assures that one can solve for the price of the derivative C(t, S(t)) by either discretizing the BSM PDE using Finite Difference methods, or by exploiting the probabilistic interpretation and using Monte Carlo methods.

With this as the backdrop we are now set to write the Feynman-Kac representation for the specific case of BSM PDE.

We start with considering the following functions:

\begin{aligned} Z_1(\tau) &= e^{-r(\tau - t)} \\ Z_2(\tau)&= C(\tau, S(\tau))\end{aligned}

and their differentials:

\begin{aligned} dZ_1(\tau) &= -re^{-r(\tau - t)} d \tau\\ dZ_2(\tau)&= dC(\tau, S(\tau))\end{aligned}

Recall that since in BSM PDE the risk has been `hedged away’, our stock price process S(\tau) is in the risk-neutral world, and that is why the drift term is r (instead of \mu) in the SDE for C.

Next we consider the differential of the product Z_1Z_2, i.e. d(Z_1Z_2):

\begin{aligned} d(Z_1 Z_2) &= Z_2 dZ_1 + Z_1 dZ_2 \\& = -rCe^{-r(\tau - t)} d \tau + e^{-r(\tau - t)} dC\\ &=-rCe^{-r(\tau - t)} d \tau + e^{-r(\tau - t)} \Bigg[ \Big( \frac{\partial C}{\partial \tau}+ rS\frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}\Big)d\tau + \sigma S \frac{\partial C}{\partial S} dX(\tau) \Bigg]\end{aligned}

where the last step in the above equation follows directly PDE for C from Ito’s lemma. Now BSM PDE tells us that:

\displaystyle \frac{\partial C}{\partial t}+ r S \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = rC

With this we can simplify d(Z_1Z_2) as:

\displaystyle \begin{aligned} d(Z_1Z_2) &= -rCe^{-r(\tau - t)} d\tau + e^{-r(\tau - t)}\Bigg[\underbrace{\Big( \frac{\partial C}{\partial \tau}+rS\frac{\partial C}{\partial S}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}\Big)}_{rC} d\tau + \sigma S \frac{\partial C}{\partial S} dX(\tau) \Bigg] \\&= -rCe^{-r(\tau - t)} d\tau+e^{-r(\tau - t)} \Bigg[rC d\tau + \sigma S \frac{\partial C}{\partial S} dX(\tau) \Bigg] \\&= -rCe^{-r(\tau - t)} d\tau + rCe^{-r(\tau - t)} d\tau + e^{-r(\tau - t)}\sigma S \frac{\partial C}{\partial S} dX(\tau) \\&=e^{-r(\tau - t)}\sigma S \frac{\partial C}{\partial S} dX(\tau)\end{aligned}

That is, the change in the function Z_1Z_2 is a driftless SDE. We integrate both sides to give:

\displaystyle \int_{t}^{T}{d(Z_1Z_2)} =\int_{t}^{T}{e^{-r(\tau - t)}\sigma S \frac{\partial C}{\partial S} dX(\tau)}

Then taking expectations of both sides w.r.t the filtration F_t at time t and using the fact that stochastic integrals are martingales (RHS of the equation below) gives:

\displaystyle \begin{aligned} E\big[\int_{t}^{T}{d(Z_1Z_2)}\big\rvert F_t \big] &= E\big[\int_{t}^{T}{e^{-r(\tau - t)}\sigma S \frac{\partial C}{\partial S} dX(\tau)}\big\rvert F_t \big] \\ E \big[Z_1(T)Z_2(T, S(T)) - Z_1(t)Z_2(t, S(t)) \big\rvert F_t\big] &= 0 \\ \mbox{or } Z_1(t)Z_2(t, S(t)) &=E\big[Z_1(T)Z_2(T, S(T)) \big\rvert F_t\big] \end{aligned}

Substituting back the value of original expressions for Z_1 and Z_2 at time t and T, i.e. Z_1(t) = e^{-r(t - t)} and Z_2(t) = C(t, S(t)) gives:

\begin{aligned} e^{-r(t - t)}C(t, S(t)) &= E\big[e^{-r(T - t)})C(T, S(T))\big\rvert F_t\big] \\ \Rightarrow C(t, S(t)) &= E\big[e^{-r(T - t)})C(T, S(T))\big\rvert F_t\big] \end{aligned}

and we are done!

That is, BSM PDE implies that the price of the derivative C(t, S(t)) at time t is equivalent to the discounted value of the expected payoff at expiration (time T). This is the famous Feynman-Kac representation.

And this is why pricing derivatives via Finite Difference methods (by discretizing the PDE) is mathematically equivalent to pricing them using  Monte Carlo methods (taking expectations).

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Written by Vineet

February 15, 2013 at 9:42 pm

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