In the world of finance, options are derivative contracts that provide investors with the right, but not the obligation, to buy or sell an underlying asset at a specified price within a certain period. One of the most significant developments in options pricing theory is the Black-Scholes formula, introduced by Fischer Black and Myron Scholes in 1973. In this blog post, we will delve into the mathematical derivation of the Black-Scholes formula and explore the key steps and principles behind it.
Understanding Options and the Black-Scholes Formula
The Black-Scholes formula calculates the theoretical price of a European-style option. European options can only be exercised at their expiration date. The formula is widely used by financial analysts and traders to assess the fair value of options and manage risk. Before we proceed with the derivation, let's define the key parameters involved in the formula:
- \( S \): The current price of the underlying asset
- \( K \): The strike price of the option
- \( T \): The time to expiration (in years)
- \( t \): The current time
- \( r \): The risk-free interest rate (annualized)
- \( \sigma \): The volatility of the underlying asset's returns (annualized)
The formula for the price of a European call option is given by: \[ C(S, t) = S N(d_1) - K e^{-r(T - t)} N(d_2) \] where: \[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T - t)}{\sigma \sqrt{T - t}} \] \[ d_2 = d_1 - \sigma \sqrt{T - t} \] Here, \( N(d) \) represents the cumulative distribution function of the standard normal distribution.
Derivation of the Black-Scholes Formula
Step 1: Model the Underlying Asset Price Dynamics
We assume that the price of the underlying asset, denoted by \( S_t \), follows geometric Brownian motion. This means that the change in the asset price over a small interval of time, \( dS_t \), can be modeled as: \[ dS_t = \mu S_t dt + \sigma S_t dW_t \] Here, \( \mu \) is the expected return rate, \( \sigma \) is the volatility, and \( dW_t \) represents the standard Brownian motion.
Step 2: Apply Ito's Lemma
Let \( C(S_t, t) \) be the price of the European call option. By applying Ito's lemma, we find the differential of the option price in terms of the asset price and time: \[ dC = \frac{\partial C}{\partial S_t} dS_t + \frac{1}{2} \frac{\partial^2 C}{\partial S_t^2} (dS_t)^2 + \frac{\partial C}{\partial t} dt \]
Step 3: Construct a Riskless Portfolio
We construct a portfolio \( \Pi \) by taking a long position in one call option and shorting \( \Delta \) units of the underlying asset. The value of the portfolio is: \[ \Pi = C - \Delta S_t \]
Step 4: Hedge the Portfolio and Apply No-Arbitrage
We choose \( \Delta \) such that the portfolio is risk-free by eliminating the stochastic term: \[ \Delta = \frac{\partial C}{\partial S_t} \] By applying the no-arbitrage principle, the return on a riskless portfolio must equal the risk-free rate of return: \[ \frac{d\Pi}{dt} = r \Pi \]
Step 5: Derive the Black-Scholes PDE
Substituting the expressions for \( dC \), \( dS_t \), and \( \Delta \) into the equation for the differential of the portfolio value, we get: \[ d\Pi = \left( \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C}{\partial S_t^2} \right) dt \] By applying the no-arbitrage condition, we arrive at the Black-Scholes partial differential equation (PDE): \[ \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C}{\partial S_t^2} + r S_t \frac{\partial C}{\partial S_t} - rC = 0 \]
Step 6: Solve the Black-Scholes PDE with Boundary Conditions
To solve the Black-Scholes PDE, we need to consider the appropriate boundary conditions for a European call option:
- At expiration (\( t = T \)), the option value is equal to the intrinsic value:
\[ C(S_T, T) = \max(S_T - K, 0) \]
- As the asset price approaches zero (\( S_t \rightarrow 0 \)), the option value approaches zero:
\[ \lim_{{S_t} \to 0} C(S_t, t) = 0 \]
- As the asset price approaches infinity (\( S_t \rightarrow \infty \)), the option value approaches the asset price minus the present value of the strike price:
\[ \lim_{{S_t} \to \infty} C(S_t, t) = S_t - Ke^{-r(T-t)} \]
Solving the PDE with these boundary conditions yields the Black-Scholes formula for the price of a European call option:
\[ C(S, t) = S N(d_1) - K e^{-r(T - t)} N(d_2) \]
where:
\[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T - t)}{\sigma \sqrt{T - t}} \]
\[ d_2 = d_1 - \sigma \sqrt{T - t} \]
and \( N(d) \) is the cumulative distribution function of the standard normal distribution.
Conclusion
The Black-Scholes formula revolutionized the field of options pricing and remains one of the most widely used models in finance. It provides a powerful tool for pricing European options, assessing risk, and implementing trading strategies. While the model is based on certain assumptions, it provides a valuable theoretical framework for understanding option pricing dynamics. As financial markets continue to evolve, the Black-Scholes formula remains a fundamental building block for the development of more advanced pricing models.
Note: This blog post is intended to provide an overview of the derivation of the Black-Scholes formula. It is important to understand that the Black-Scholes model is based on certain assumptions that may not always hold in real-world scenarios. Additionally, there are various types of options and financial derivatives that may require different pricing models. Always consult with a qualified financial professional before making investment decisions.