Chapter 15: The Black-Scholes-Merton Model

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Historical Context

The Black–Scholes–Merton model (1973) revolutionized finance. Fischer Black, Myron Scholes, and Robert Merton developed a closed-form solution for European option pricing. Scholes and Merton received the 1997 Nobel Prize in Economics (Black had passed away in 1995).

Lognormal Property of Stock Prices

From Chapter 14, stock prices follow geometric Brownian motion. The log return distribution:

$ln⁡ST∼N[ln⁡S0+(μ−σ22)T,σ2T]\ln S_T \sim N\left[\ln S_0 + \left(\mu - \frac{\sigma^2}{2}\right)T, \sigma^2 T\right]$

This is the lognormal property: the stock price $S_T$ has a lognormal distribution. This has three important implications:

Stock prices are always positive ( $S_T > 0$ )
The distribution is skewed to the right
Mean, median, and mode are different

The Distribution of the Rate of Return

The continuously compounded return over $T$ years:

$\frac{1}{T} \ln\left(\frac{S_T}{S_0}\right) \sim N\left(\mu - \frac{\sigma^2}{2}, \frac{\sigma^2}{T}\right)$

As $T$ increases, the standard deviation of the return falls (longer horizons are less variable on a per-year basis), but the standard deviation of the terminal price grows.

Volatility

Volatility $σ\sigma$ is the standard deviation of the continuously compounded return per annum. It's the only unobservable parameter in the Black–Scholes–Merton model.

Estimating from Historical Data

$σ=s⋅N\sigma = s \cdot \sqrt{N}$

where $s$ is the standard deviation of daily log returns and $N$ is the number of trading days per year (usually 252):

$\sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (u_i - \bar{u})^2}, \quad u_i = \ln\left(\frac{S_i}{S_{i-1}}\right)$

Typical volatilities: 15-60% per annum for individual stocks, 10-30% for stock indices.

The Black–Scholes–Merton Differential Equation

The key idea: construct a riskless portfolio consisting of a short position in one derivative and a long position in $∂f/∂S\partial f/\partial S$ shares of stock. Since it's riskless, it must earn the risk-free rate.

This leads to the Black–Scholes–Merton PDE:

$∂f∂t+rS∂f∂S+12σ2S2∂2f∂S2=rf\frac{\partial f}{\partial t} + rS\frac{\partial f}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = rf$

Remarkable property: The equation does NOT contain $μ\mu$ (the expected return on the stock). This means option prices are independent of expected returns — they depend only on $r$ , $σ\sigma$ , $S$ , $t$ , and the boundary conditions.

Risk-Neutral Valuation

Because $μ\mu$ drops out of the PDE, we can price options as if the world is risk-neutral:

Expected return on all assets = $r$
Discount expected payoffs at $r$

This is a pricing tool, not a statement about the real world.

Black–Scholes–Merton Pricing Formulas

For a European call on a non-dividend-paying stock:

$S_0 \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)$

For a European put:

$\cdot e^{-rT} \cdot N(-d_2) - S_0 \cdot N(-d_1)$

where:

$d1=ln⁡(S0/K)+(r+σ2/2)TσTd_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}$

$d2=ln⁡(S0/K)+(r−σ2/2)TσT=d1−σTd_2 = \frac{\ln(S_0/K) + (r - \sigma^2/2)T}{\sigma \sqrt{T}} = d_1 - \sigma \sqrt{T}$

$N (x)$ = cumulative standard normal distribution function.

Interpretation of the formula:

$N(d_2)$ = risk-neutral probability that the call will be exercised
$S_0 N(d_1)$ = present value of the expected stock price, conditional on exercise
$K e^{-rT} N(d_2)$ = present value of the expected strike payment

Implied Volatility

The implied volatility of an option is the value of $σ\sigma$ that makes the Black–Scholes–Merton price equal to the market price. It cannot be solved analytically — must use numerical methods.

The implied volatility is forward-looking (the market's expectation) versus historical volatility which is backward-looking.

Dividends

Continuous Dividend Yield

When the stock pays a continuous dividend yield $q$ :

$c = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$

$d1=ln⁡(S0/K)+(r−q+σ2/2)TσTd_1 = \frac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma \sqrt{T}}$

Known Cash Dividends

Replace $S_0$ with $S0−PV(dividends)S_0 - PV(\text{dividends})$ in the formulas. This applies only to European options (American options with dividends require special treatment).

The VIX Index

The CBOE Volatility Index (VIX) is computed from the prices of S&P 500 options. It is a measure of the market's expectation of 30-day volatility. Often called the "fear index" — it tends to spike during market turmoil.