Partial Differential Equations (PDEs) and Stochastic Differential Equations (SDEs) are both essential mathematical tools in quantitative finance. They are interconnected, and their relationship can be best understood through the link between PDEs and SDEs known as the Feynman-Kac theorem. In this answer, I will present an overview of both PDEs and SDEs, discuss the connection between the two, and provide an example to illustrate the relationship.
PDEs in Quantitative Finance:
Partial Differential Equations arise in various option pricing models in finance. One of the most well-known PDEs in finance is the Black-Scholes equation, which provides an analytical pricing formula for European call and put options. The Black-Scholes PDE is expressed as:
$$\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} - rV = 0,$$
where V(S,βt) represents the option price, S is the underlying asset price, Ο is the asset volatility, r is the risk-free interest rate, and t is the time to expiry.
SDEs in Quantitative Finance:
On the other hand, Stochastic Differential Equations describe the dynamics of financial asset prices, where the randomness is typically attributed to the Brownian motion, W(t). For example, the most commonly used SDE for stock price dynamics is the Geometric Brownian Motion (GBM), which is represented as:
dS(t)β=βΞΌS(t)dtβ
+β
ΟS(t)dW(t),
where ΞΌ represents the drift term, and Ο represents the volatility.
The Connection between PDEs and SDEs:
The Feynman-Kac theorem establishes a strong relationship between PDEs and SDEs. The theorem states that under certain conditions, the solution to a PDE can be expressed as the expectation of a function of the solution to an associated SDE. In other words, if we can find an SDE that models the financial asset price dynamics, we can use the relationship provided by the Feynman-Kac theorem to solve the corresponding PDE for the option price.
Suppose we have the following PDE:
$$\frac{\partial u}{\partial x} + \sum_{i=1}^{n} a_i(x)\frac{\partial u}{\partial x_i} + \frac{1}{2}\sum_{i,j=1}^{n} b_{ij}(x)\frac{\partial^2 u}{\partial x_i \partial x_j} = 0,$$
with terminal condition u(x,βT)β=βf(x), where u is the option price, x is the state vector, ai(x) and bij(x) are coefficients, and f(x) is the payoff function. According to the Feynman-Kac theorem, under some regularity conditions, the solution to this PDE can be expressed as
u(x,βt)β=βE[f(XT)|Xt=x],
where Xt is a vector process satisfying the following system of SDEs:
$$dX_i(t) = a_i(X_t) dt + \sum_{j=1}^{n} \sqrt{b_{ij}(X_t)} dW_j(t), \quad i = 1, \dots, n.$$
Example β European Call Option using Black-Scholes PDE and GBM SDE: Consider a European call option with payoff function f(S)β=βmaxβ(Sβ ββ K,β0), where K is the strike price, and the underlying asset price dynamics follow the Geometric Brownian Motion (GBM) given as:
dS(t)β=βrS(t)dtβ
+β
ΟS(t)dW(t).
By applying the risk-neutral valuation and the Feynman-Kac theorem, the option price represented by V(S,βt) can be expressed as an expected value:
V(S,βt)β=βeβ
ββ
r(Tβ
ββ
t)E[max(S(T)βK,0)|S(t)=S],
where S(T) is the underlying asset price at maturity time T. This expectation can be solved using Monte Carlo simulation or other numerical methods.
Simultaneously, the option price can also be found by solving the Black-Scholes PDE with the boundary condition V(S,βT)β=βmaxβ(Sβ ββ K,β0). As a result, we can observe how the solutions of the stochastic differential equation and the partial differential equation are interconnected through the Feynman-Kac theorem.
In conclusion, PDEs and SDEs play a crucial role in quantitative finance and are intrinsically linked through the Feynman-Kac theorem. PDEs typically describe option pricing models like the Black-Scholes equation, whereas SDEs model the randomness exhibited in the underlying asset price dynamics. The connection between PDEs and SDEs allows for the solution of one framework to be expressed in terms of the other.