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Derivation of the Black-Scholes Equation

1. The Financial Setup

Consider a stock with price $S_t$ at time $t$ and a European call option on this stock, denoted by $V_t$. The goal is to create a portfolio that eliminates risk by continuously adjusting the number of shares held, denoted by ${\Delta }_t$.

2. Modeling Stock Price Movement with Brownian Motion

The stock price $S_t$ follows a stochastic process known as geometric Brownian motion:

$\mathrm{dS}(t)=\mu S(t)\mathrm{dt}+\sigma S(t)\mathrm{dW}(t)$

where $\mu$ is the drift rate, $\sigma$ is the volatility, and $\mathrm{dW}(t)$ represents a Wiener process.

3. Constructing the Hedged Portfolio

The value of the portfolio ${\Pi }_t$ is:

$\Pi (t)=\Delta (t)S(t)-V(t)$

The change in the portfolio value is:

$\mathrm{d\Pi }(t)=\Delta (t)\mathrm{dS}(t)-\mathrm{dV}(t)$

4. Applying Ito's Lemma

Using Ito's Lemma, the change in the option's value $\mathrm{dV}$ is:

$\mathrm{dV}=\frac{\partial V}{\partial t}\mathrm{dt}+\frac{\partial V}{\partial S}\mathrm{dS}+\frac{1}{2}\frac{\partial V^2}{\partial S^2}{\mathrm{dS}}^2$

Substituting $\mathrm{dS}$ and ${\mathrm{dS}}^2$ gives:

$\mathrm{dV}=(\frac{\partial V}{\partial t}+\mu S\frac{\partial V}{\partial S}+\frac{1}{2}\sigma S^2\frac{\partial V^2}{\partial S^2})\mathrm{dt}+\sigma S\frac{\partial V}{\partial S}\mathrm{dW}(t)$

5. Eliminating Risk

For the portfolio to be risk-free, set:

$\Delta =\frac{\partial V}{\partial S}$

This eliminates the stochastic term:

$\mathrm{d\Pi }=(\frac{\partial V}{\partial t}+\frac{1}{2}\sigma S^2\frac{\partial V^2}{\partial S^2})\mathrm{dt}$

6. Relating to a Risk-Free Investment

Since the portfolio is risk-free, it should earn the risk-free rate $r$:

$\mathrm{d\Pi }=r\Pi \mathrm{dt}$

Substituting $\Pi$ gives:

$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma S^2\frac{\partial V^2}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0$

7. The Black-Scholes Equation

Simplifying the above, we obtain the Black-Scholes partial differential equation:

$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma S^2\frac{\partial V^2}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0$

This work is licensed under a Creative Commons Attribution 4.0 International License.

Editor: Zacharia Maganga Nyambu

Email: nyazach@gmail.com