Café Math : Black-Scholes Equation

Hi today I want to talk about the most famous equation of quantitative finance, the celebrated and Nobel prize winner known as the Black-Scholes-Merton equation, $$ \frac{\partial}{\partial t} V + rS \,\frac{\partial}{\partial S} V + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2}{\partial S^2} V = r V $$

A european style call option is a financial contract between two participants, one of the participant (option seller) gives the right to the other participant (option buyer) to buy an underlying assert at some prescribed price (the strike price) until some day (the maturity of the option).

The formula relates the price of a call european option $V$ to the price of its underlying asset $S$. This price $S$ is modeled by a geometric brownian motion of drift $\mu$ and diffusion coefficient $\sigma$. We already talked of this model in this blog, its Itô stochastic equation is, $$ d S = \mu S\,dt + \sigma S\,dX $$ where $X$ is a standard brownian motion. The $r$ appearing in the Black-Scholes equation is the risk free interest rate to which some cash in your portfolio can grow. For banks this corresponds to the interest rate of US treasury bonds more or less.

There are some assumptions made on the market in order to make the Black-Scholes equation relevant and most of them are never satisfied in real life. For example, zero transaction cost, being able to rebalance the hedging portfolio continuously etc... Albeit that, one can overcome most of this difficulties by changing the equation a bit and incorporate counter-terms that provides the necessary corrections. A more serious critique which is most often made can't be addressed this simply. It's precisely the *gaussian* nature of the geometric brownian motion used to model the price of the underlying asset. The problem is that the geometric brownian motion model.

For the story, this has been invented several times buy clever individuals, and has been successfully used by professor Ed. Thorp and by some wall street options traders for years before it got published by Black, Scholes and Merton. Sadly Merton died before the Nobel prize were attributed to his two collaborators.

Let $\Pi$ be the value of a portfolio containing one european option and a negative quantity $\Delta$ of the underlying asset. We have,
$$
\Pi = V - \Delta S.
$$
Of course the price of the option depends on time $t$ and the value of the underlying $S$. If we consider the variation of the value of this portfolio, by Itô lemma we get,
\begin{align*}
d\Pi &= dV - \Delta dS \\
&= \frac{\partial}{\partial t} V dt + \frac{\partial}{\partial S} V dS + \frac{1}{2} \frac{\partial^2}{\partial S^2} V dS^2 - \Delta dS \\
&= \frac{\partial}{\partial t} V dt + \frac{\partial}{\partial S} V dS + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2}{\partial S^2} V dt - \Delta dS
\end{align*}
As $dS^2 = \sigma^2 S^2 dt$. The value of this portfolio is fairly random as its total differential contains stochastic terms (terms in $dS$) but this source of randomness can be eliminated by choosing,
$$
\Delta = \frac{\partial}{\partial S} V.
$$
Doing that, the two random terms cancel and one gets,
$$
d\Pi = \left( \frac{\partial}{\partial t} V + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2}{\partial S^2} V \right) dt
$$
and this expression is an ordinary differential equation, free from stochastic terms. In financial terms we say the portfolio $\Pi$ is *risk free*. As such, by the *non-arbitrage principle*, it must satisfy the differential equation of any risk free asset which is,
$$
d\Pi = r \Pi \, dt.
$$
From this we get that,
\begin{align*}
\frac{\partial}{\partial t} V + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2}{\partial S^2} V &= r \Pi \\
&= rV - r\Delta S \\
&= rV - r S \frac{\partial}{\partial S} V
\end{align*}
And voila,
$$
\frac{\partial}{\partial t} V + rS \,\frac{\partial}{\partial S} V + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2}{\partial S^2} V = r V
$$
the Black-Scholes formula.

(to be continued)

Samuel VIDAL posted 2012-04-29 21:03:38

Thank you very much Talulu ;-) I hope I can post more on this interesting subject very soon.

This is crystal clear. Thanks for taking the time!