Option Pricing Theory: A Deep Dive into Stochastic Process Models

As someone deeply immersed in the world of finance and accounting, I find option pricing theory to be one of the most fascinating and intellectually stimulating areas of study. The ability to model the price of financial derivatives using mathematical frameworks has revolutionized trading, risk management, and investment strategies. In this article, I will explore the stochastic process models that underpin option pricing theory, breaking down the concepts in a way that is both accessible and rigorous.

Understanding the Basics of Option Pricing

Before diving into stochastic processes, it’s essential to understand what options are and why pricing them is so challenging. An option is a financial derivative that gives the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a predetermined price (strike price) on or before a specified date (expiration date).

The price of an option depends on several factors, including the current price of the underlying asset, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset. The challenge lies in modeling the uncertainty of the underlying asset’s future price movements, which is where stochastic processes come into play.

The Role of Stochastic Processes in Option Pricing

Stochastic processes are mathematical models that describe the evolution of random variables over time. In the context of option pricing, stochastic processes are used to model the price movements of the underlying asset. The most widely used stochastic process in finance is the Geometric Brownian Motion (GBM), which forms the foundation of the Black-Scholes-Merton model.

Geometric Brownian Motion

Geometric Brownian Motion is a continuous-time stochastic process that assumes the price of the underlying asset follows a log-normal distribution. The key idea is that the asset’s price changes are random and driven by two components: a deterministic drift and a stochastic diffusion.

The GBM is described by the following stochastic differential equation (SDE):

dS_t = \mu S_t dt + \sigma S_t dW_t

Where:

• S_t is the price of the underlying asset at time t.
• \mu is the expected return (drift) of the asset.
• \sigma is the volatility of the asset.
• dW_t is the increment of a Wiener process (Brownian motion).

The Wiener process W_t is a continuous-time stochastic process with the following properties:

1. W_0 = 0.
2. The increments W_t - W_s are normally distributed with mean 0 and variance t - s.
3. The increments are independent of past values.

Solving the GBM Equation

To solve the GBM equation, we use Itô’s Lemma, which is a fundamental tool in stochastic calculus. Applying Itô’s Lemma to the GBM equation, we obtain the following solution:

S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)

This equation shows that the asset price at time t is a function of the initial price S_0, the drift term \mu, the volatility \sigma, and the Wiener process W_t.

The Black-Scholes-Merton Model

The Black-Scholes-Merton (BSM) model is the cornerstone of modern option pricing theory. It provides a closed-form solution for the price of a European call or put option under the assumption that the underlying asset follows a GBM.

The BSM formula for a European call option is:

C(S_t, t) = S_t N(d_1) - K e^{-r(T-t)} N(d_2)

Where:

• C(S_t, t) is the price of the call option at time t.
• S_t is the current price of the underlying asset.
• K is the strike price.
• r is the risk-free interest rate.
• T is the time to expiration.
• N(\cdot) is the cumulative distribution function of the standard normal distribution.
• d_1 and d_2 are given by:

d_1 = \frac{\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}} d_2 = d_1 - \sigma \sqrt{T-t}

The BSM model assumes that markets are efficient, there are no transaction costs, and the risk-free rate and volatility are constant. While these assumptions are not always realistic, the BSM model remains a powerful tool for understanding option pricing.

Limitations of the BSM Model

Despite its widespread use, the BSM model has several limitations:

1. Constant Volatility: The model assumes that volatility is constant, which is rarely the case in real markets.
2. European Options: The BSM model only applies to European options, which can be exercised only at expiration. American options, which can be exercised at any time before expiration, require more complex models.
3. No Dividends: The basic BSM model does not account for dividends paid by the underlying asset.

To address these limitations, researchers have developed more advanced stochastic process models, such as the Heston model and the Jump-Diffusion model.

Advanced Stochastic Process Models

The Heston Model

The Heston model, introduced by Steven Heston in 1993, is a stochastic volatility model that allows volatility to vary over time. The model assumes that the volatility of the underlying asset follows a mean-reverting square root process, which is described by the following SDEs:

dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_t^1 dv_t = \kappa (\theta - v_t) dt + \sigma \sqrt{v_t} dW_t^2

Where:

• v_t is the instantaneous variance (volatility squared) at time t.
• \kappa is the rate at which the variance reverts to its long-term mean \theta.
• \sigma is the volatility of the volatility.
• dW_t^1 and dW_t^2 are correlated Wiener processes with correlation \rho.

The Heston model captures the volatility smile, a phenomenon where implied volatility varies with the strike price and time to expiration. This makes it a more realistic model for pricing options in real markets.

The Jump-Diffusion Model

The Jump-Diffusion model, introduced by Robert C. Merton in 1976, extends the GBM by incorporating jumps in the asset price. The model assumes that the asset price follows a combination of continuous GBM and discrete jumps, which are driven by a Poisson process.

The SDE for the Jump-Diffusion model is:

dS_t = \mu S_t dt + \sigma S_t dW_t + J_t dN_t

Where:

• J_t is the jump size, which is a random variable.
• dN_t is the increment of a Poisson process with intensity \lambda.

The Jump-Diffusion model is particularly useful for pricing options on assets that are prone to sudden price movements, such as stocks during earnings announcements or commodities during supply shocks.

Practical Applications and Examples

To illustrate the concepts discussed, let’s consider an example of pricing a European call option using the BSM model.

Example: Pricing a European Call Option

Suppose we have the following parameters:

• Current stock price S_0 = \$100.
• Strike price K = \$105.
• Time to expiration T = 1 year.
• Risk-free interest rate r = 5\%.
• Volatility \sigma = 20\%.

First, we calculate d_1 and d_2:

d_1 = \frac{\ln\left(\frac{100}{105}\right) + \left(0.05 + \frac{0.2^2}{2}\right)(1)}{0.2 \sqrt{1}} \approx 0.106 d_2 = 0.106 - 0.2 \sqrt{1} \approx -0.094

Next, we use the standard normal cumulative distribution function N(\cdot) to find the values of N(d_1) and N(d_2):

N(d_1) \approx 0.542 N(d_2) \approx 0.463

Finally, we plug these values into the BSM formula:

C = 100 \times 0.542 - 105 e^{-0.05 \times 1} \times 0.463 \approx \$8.02

Thus, the price of the European call option is approximately \$8.02.

Conclusion

Option pricing theory, grounded in stochastic process models, provides a powerful framework for understanding and valuing financial derivatives. From the foundational Black-Scholes-Merton model to advanced models like Heston and Jump-Diffusion, these tools enable traders, investors, and risk managers to navigate the complexities of financial markets.