Black-Scholes-Merton Derivation (Part 3: Equating Portfolio Value & Option Value Evolutions)

This is a continuation of Black-Scholes-Merton Derivation (Part 2: Option Value Evolution).

A hedging portfolio starts with initial capital $X(0)$ and invests in the stock and money market account so that the portfolio value $X(t)$ at each time $t\in [0,T]$ is equal to $c(t,S(t))$ . This is equivalent to the condition that $e^{-rt}X(t)=e^{-rt}c(t,S(t))$ for all $t$ .

One way to impose this equality is to ensure that

$\displaystyle d(e^{-rt}X(t))=d(e^{-rt}c(t,S(t)))$ for all $t\in [0,T)$ and $X(0)=c(0,S(0))$ .

Comparing the expressions $d(e^{-rt}X(t))$ and $d(e^{-rt}c(t,S(t)))$ calculated previously in Part 1 and Part 2 respectively, we see that we require the following to hold:

$\displaystyle\begin{aligned}&\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)\\&=\left[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}$

Comparing the $dW(t)$ terms, we get $\Delta(t)=c_x(t,S(t))$ . This is known as the delta-hedging rule.

Next, comparing the $dt$ terms, we have

$\displaystyle\begin{aligned}&(\alpha-r)S(t)c_x(t,S(t))\\&=-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\end{aligned}$

Simplifying and rearranging, we get $\displaystyle rc(t,S(t))=c_t(t,S(t))+rS(t)c_x(t,S(t))+\frac{1}{2}\sigma^2S^2(t)c_{xx}(t,S(t))$

Let $x=S(t)$ , then we have derived the Black-Scholes-Merton partial differential equation:

$\boxed{\displaystyle c_t(t,x)+rxc_x(t,x)+\frac{1}{2}\sigma^2x^2 c_{xx}(t,x)=rc(t,x)}$ for all $t\in [0,T), x\geq 0$ .

The terminal condition to be satisfied is $c(T,x)=(x-K)^+$ , where $K$ is the strike price of the call option.

Reference

Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.

Reference

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