The Black-Scholes model is a mathematical model that provides a formula to calculate the theoretical price of European-style options. This model, developed by Fischer Black and Myron Scholes in 1973, is one of the foundational models in modern financial theory, particularly in the valuation of options.

6.1 Derivation of the Black-Scholes Formula

The derivation of the Black-Scholes formula involves stochastic calculus, particularly Ito’s Lemma, to describe the dynamics of stock prices and develop the partial differential equation (PDE) that governs option prices.

Here's a summary of the derivation:

1. Stock Price Dynamics: We assume that the stock price $S(t)$ follows a geometric Brownian motion, expressed as: $dS = \mu S \, dt + \sigma S \, dW$ where:

   • $\mu$ is the drift rate (expected return) of the stock,
   • $\sigma$ is the volatility of the stock,
   • $dW$ represents the Wiener process, which captures the randomness.

2. Portfolio Construction: To eliminate risk, construct a portfolio containing a long position in the option and a short position in $\Delta$ shares of stock. Let the value of the option be $V(S, t)$, where $S$ is the stock price and $t$ is time. The change in portfolio value over an infinitesimal time $dt$ is: $d\Pi = dV - \Delta \, dS$

3. Applying Ito's Lemma: By Ito’s Lemma, the differential $dV$ for the option value can be expanded as: $dV = \frac{\partial V}{\partial t} \, dt + \frac{\partial V}{\partial S} \, dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 \, dt$

4. Eliminating the Random Component: Substitute $dS$ from the stock price dynamics into the expression for $dV$, resulting in: $dV = \left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) \, dt + \frac{\partial V}{\partial S} \, dS$ Setting up a risk-neutral portfolio (where $d\Pi$ has no randomness) allows us to derive a PDE independent of $\mu$.

5. Black-Scholes PDE: After eliminating $\mu$ and the risk component, we arrive at the Black-Scholes partial differential equation: $\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0$ where $r$ is the risk-free interest rate.

6. Solution to the PDE: Solving this PDE with appropriate boundary conditions yields the Black-Scholes formula for a European call option: $C = S_0 N(d_1) - X e^{-rt} N(d_2)$ where:

   • $N(\cdot)$ is the cumulative distribution function of the standard normal distribution,
   • $d_1 = \frac{\ln(S_0 / X) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}$,
   • $d_2 = d_1 - \sigma \sqrt{T}$,
   • $S_0$ is the initial stock price,
   • $X$ is the strike price, and $T$ is the time to maturity.

This derivation hinges on constructing a risk-free portfolio and using stochastic calculus. Let me know if you’d like more on any step, or details on assumptions in this model.

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Expanded Questions

1. What assumptions underlie the Black-Scholes model, and how do they affect real-world applications