Black-Scholes PDE

The Black-Scholes partial differential equationis the partial differentiation equation:

$rf={\displaystyle \frac{\partial f}{\partial t}}+rx{\displaystyle \frac{\partial f}{\partial x}}+{\displaystyle \frac{1}{2}}{\sigma }^2{x}^2{\displaystyle \frac{{\partial }^{2}f}{\partial x^2}},f=f(t,x),$(1)

on the domain .Its solution gives the price functionof a stock option (or any other contingent claim on a tradable asset)under the assumptions of the Black-Scholes model for prices.

The parameters $r$, $\sigma >0$ and $T>0$are, respectively, the prevailing continuously-compounded risk-freerate of interest, the volatility of the stock,and the time to maturity of the stock option.

More generally, the term Black-Scholes partial differential equationcan refer to other partial differential equationssimilar in form to equation (1), derivedin similar ways under moregeneral modelling assumptions than the most basic Black-Scholes model.

0.1 Derivation from martingale form

The Black-Scholes formula, in its martingale form,gives the theoretical value of the stock option

$V(t)=e^{-r(T-t)}{𝔼}^{\mathbb{Q}}[V(T)\mid {ℱ}_t],0\le t\le T,$(2)

where $V(T)$ is a given pay-off functionor terminal condition, assumed to be a ${𝐋}^1(\mathbb{Q})$ random variable;${ℱ}_t$ is the filtration generated by the the standard Wienerprocess $\tilde{W}(t)$under the probability measure $\mathbb{Q}$ (the “risk-neutral measure”).

Equation (2)gives the solution to the stochastic differential equations:

$V(t)$$=\mathrm{\Delta }(t)X(t)+\mathrm{\Theta }(t)M(t),M(t)=e^{rt},$(3)

$dV(t)$$=\mathrm{\Delta }(t)dX(t)+\mathrm{\Theta }(t)dM(t),dM(t)=rM(t)dt,$(4)

$dX(t)$$=rX(t)dt+\sigma X(t)d\tilde{W}(t).$(5)

From the martingale form of the result,there are two ways to derive the PDE form (1).

0.1.1 Using the Feynman-Kac formula

One way is to appeal to the Feynman-Kac formula,which states in this case, thatunder certain regularity conditionson thefunction $\psi :(0,\mathrm{\infty })\to ℝ$, thefunction $f:[0,T]\times (0,\mathrm{\infty })\to ℝ$defined by

$f(t,x)=e^{-r(T-t)}{𝔼}^{\mathbb{Q}}[\psi (X(T))\mid X(t)=x],$(6)

satisfies the partial differential equation (1)with the terminal condition

$$f(T,x)=\psi (x).$$

(The stochastic process $X(t)$is to satisfy equation (5).)

Now, equations (6) and (2)look quite similar,but with an important difference:the conditional expectation on equation (6)is with respect to the event $\{X(t)=x\}$,while equation (2)is conditioning on the filtration ${ℱ}_t$.We claim that these are actually the same thing —that is, for fixed $t$, the random variable${𝔼}^{\mathbb{Q}}[V(T)\mid {ℱ}_t]$can be written as a function of $X(t)$ —provided that the terminal condition is itself a function of $X\mathit{}\mathrm{(}T\mathrm{)}$:

$$V(T)=\psi (X(T)).$$

We will demonstrate this claim later. Assuming that the claim holds,we may thus set $V(t)=f(t,X(t))$,and the Black-Scholes partial differential equation follows fromthe Feynman-Kac formula as explained earlier.

0.1.2 Proof of Markov property

Formally, any ${ℱ}_t$-measurable stochastic process$X(t)$ thatensures $𝔼[\psi (X(T))\mid {ℱ}_t]$is always a measurable function of $X(t)$,for any Borel-measurable $\psi$,is called a Markov process,or is said to have the Markov property.Thus we want to show that the stock process $X(t)$ hasthe Markov property.For this,we will need the following explicitsolution for it at time $T$in terms of an initial condition at time $t$:

$X(T)$$=X(t)\exp \left((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma \left(\tilde{W}(T)-\tilde{W}(t)\right)\right)$

$:=X(t)g\left(\tilde{W}(T)-\tilde{W}(t)\right).$

Then

${𝔼}^{\mathbb{Q}}[V(T)\mid {ℱ}_t]$$={𝔼}^{\mathbb{Q}}[\psi (X(T))\mid {ℱ}_t]={𝔼}^{\mathbb{Q}}[\psi \left(X(t)g(\tilde{W}(T)-\tilde{W}(t))\right)\mid {ℱ}_t].$

Since $X(t)$ is ${ℱ}_t$-measurable,it may be treated as a constant $x$ while taking conditionalexpectations with respect to ${ℱ}_t$.And $\tilde{W}(T)-\tilde{W}(t)$is a random variable independent of ${ℱ}_t$ (and hence of $X(t)$).Therefore the expression that appears on the right-hand sideabove is a function of $x=X(t)$,and would be unchanged if the conditioningis changed from ${ℱ}_t$ to $X(t)$.

0.1.3 Using Itō’s formula

There is also a more direct methodof deriving equation (1)using Itō’s formula for Itō processes.

Set $V(t)=f(t,X(t))$as before.Make the assumption that $f$ is twice continuously differentiable,to be checked later; then we can apply Itō’s formula to expand $dV(t)$:(partial derivatives evaluated at $x=X(t)$)

(7)

By theorems on the uniqueness of solutions to stochastic differentialequations, the coefficients of thecorreponding $dt$ and $dX(t)$terms in equations (4) and (7)must be equal almost surely.Equating the coefficients, we find:

$\mathrm{\Delta }(t)$$={\displaystyle \frac{\partial f}{\partial x}},$(8)

$r\left(V(t)-\mathrm{\Delta }(t)X(t)\right)$$=\mathrm{\Theta }(t)rM(t)={\displaystyle \frac{\partial f}{\partial t}}+{\displaystyle \frac{1}{2}}{\sigma }^{2}X{(t)}^2{\displaystyle \frac{{\partial }^{2}f}{\partial x^2}}.$(9)

Equation (9)is essentially equation (1),except that $x$ has been replaced by $X(t)$ in some places.However,because $X(t)$ is a random variable that takes on all values on $(0,\mathrm{\infty })$,and $f$ and its derivatives are assumed to be continuous,equation (9) must hold for arbitrary values $x$substituted for $X(t)$. Hence we obtain equation (1).

To verifythe initial assumption that $f$ is twice continuously differentiable,we write formula (2)in a more explicit form (using the Markov property for $X(t)$ as before):

$f(t,x)$$=e^{-r(T-t)}{𝔼}^{\mathbb{Q}}[\psi (X(T))\mid X_t=x]$

$={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{e}^{-r(T-t)}\psi (y)p(x,y,T-t)𝑑y,$

where $p(x,y,\tau )$ is the transition density of thestock process from $x$ to $y$ over a time interval of length $\tau$.In fact, from the solution for $X(t)$,the density $p(x,\cdot ,\tau )$is the density for the log-normal random variablewhose logarithm has mean $\log x+(r-\frac{1}{2}{\sigma }^2)\tau$and variance ${\sigma }^2\tau$.By differentiation under the integral sign,we see that $f(t,x)$ must be twice continously differentiablefor and .

0.2 Comparison of the martingale and PDE forms

The martingale formulation of the result, in equation (2)is more general than the PDE formulation (1):the latter imposes extra regularity conditions,and more importantly,it can only describe prices $V(t)=f(t,X(t))$that arefunctions only of the time $t$ and the current stock price $X\mathit{}\mathrm{(}t\mathrm{)}$.In the case of the call option ($\psi (x)=x-K$ for some $K>0$),this assumption is true;but there are other sorts of contingent claimswhose pay-off depend on the history of the stock process $X(t)$ —for example, a common kind of contingent claim has a finalpay-off that depends on an averageof the stock price $X(t)$ over time $t\in [0,T]$.

However, the PDE form of the solution is often moreamenable to a numerical solution.While the expectation in equation (2)can be approximated numericallywith Monte-Carlo simulation,for low-dimensional problems,solutions based on finite-difference approximationsof PDEs are often quicker to compute than those based on Monte-Carlo.

0.3 Analytic solution

Though the solution for the Black-Scholes PDEis already known using the martingale representation (2),we can also solve the PDE directly using classical analytical methods (http://planetmath.org/AnalyticSolutionOfBlackScholesPDE).

Actually, the classical solution can be quite instructivein that it shows more or less the physical meaningbehind the PDE, and also the behavior of solutionsas the terminal condition is varied. (For example,the PDE is a transformation of a diffusion equation,and consequently its solutions are always infinitely smooth,as long as the terminal condition satisfiessome local-integrability properties.)

0.4 Quantity of stock needed to replicate option

Another important result thatmust not be left unmentioned isequation (8), derived duringthe course of showing the Black-Scholes PDE.

In the article on the Black-Scholes pricing formula,a solution for $V(t)$ was producedthat satisfied (4),but the ${ℱ}_t$-adapted process $\delta (t)$that appears in that stochastic differential equation wasobtained by appealing to an existence theorem.However, under the assumption that $V(t)=f(t,X(t))$,equation (8) gives an actual computable formula:

$$\mathrm{\Delta }(t)={\frac{\partial f(t,x)}{\partial x}|}_{x=X(t)}.$$

which is crucial in practice,for the theoretical “price”of a stock optionwould be useless if one cannot produce in reality that optionfor the stated price.

0.5 Generalizations

(To be written.)