This primer explains the Black-Scholes-Merton model and its assumptions, focusing on the pricing of European put options. It covers key concepts and demonstrates how the formula is applied in practice.
Introduction & Assumptions
The Black-Scholes-Merton model is a cornerstone of mathematical finance. In this discussion, we provide a primer on the topic and apply the formula to price a European put option.
The model relies on the following assumptions.
- Frictionless Markets/Zero Transaction Costs
- Unlimited borrowing/lending at a constant risk free rate “r”
- No restrictions on short selling
- No arbitrage opportunities
Derivation
Stock Price Dynamics
To commence, we assume the dynamics of the stock, S, follow a geometric Brownian Motion and hence can be specified as follows.
dS = \mu S \, dt + \sigma S \, dW
\)
Dynamics of the Derivative Security
We assume the option price is a function of the price of the stock and time hence let V(s,t) denote the price of the derivative security.
Firstly, using Ito’s Lemma we have the following.
\begin{align*}
(dW)^2 &= dt \\
(dW)(dt) &= 0 \\
(dt)^2 &= 0
\end{align*}
\) \(
\begin{aligned}
(dS)^2 &= (\mu S\, dt + \sigma S\, dW)^2 \\
&= \mu^2 S^2\, (dt)^2 + 2\mu \sigma S^2\, dt\, dW + \sigma^2 S^2\, (dW)^2 \\
&= \sigma^2 S^2\, dt
\end{aligned}
\)
We can then substitute the expressions for dS and its squared term into the equation for dV.
Applying Ito’s lemma again we can now view the dynamics of V through its relationship with the partial derivatives of S and T.
\begin{align*}
dV &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS)^2 \\
dV &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} (\mu S\,dt + \sigma S\,dW) + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2\,dt
\end{align*}
\)
Rearranging gives us the following expression.
\begin{align*}
dV &= \left[ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial S} \mu S + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 \right] dt + \left[ \frac{\partial V}{\partial S} \sigma S \right] dW
\end{align*}
\)
We can now observe this equation has the same form as the price process of the Stock which for clarity is decomposed into a drift term (containing dt) and a diffusion term (containing dW).
Replicating Portfolio
Our aim now is to select a portfolio that mirrors the returns of V.
We select a portfolio that corresponds to being long (δV/δS) units of the underlying stock and short 1 unit of the derivative security V.
Hence the value of our portfolio can be expressed as below.
\begin{align*}
\Pi &= \frac{\partial V}{\partial S} S – V
\end{align*}
\)
This implies the following.
\begin{align*}
d\Pi &= \frac{\partial V}{\partial S}(dS) – dV \\
d\Pi &= \frac{\partial V}{\partial S} [\mu S\,dt + \sigma S\,dW] – \left[ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial S} \mu S + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 \right] dt – \left[ \frac{\partial V}{\partial S} \sigma S \right] dW \\
d\Pi &= \frac{\partial V}{\partial S} [\mu S\,dt] – \left[ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial S} \mu S + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 \right] dt \\
d\Pi &= -\left[ \frac{\partial V}{\partial t} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 \right] dt
\end{align*}
\)
Note this portfolio has no stochasticity and is riskless. Through No Arbitrage/Law of One Price the return on the portfolio under the risk neutral measure must be equal to the riskless rate “r” which is assumed to be exogenously given and constant.
This implies the following.
\begin{align*}
d\Pi &= r\Pi\,dt \\
-[\frac{\partial V}{\partial t} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2]\,dt &= r\Pi\,dt \\
-[\frac{\partial V}{\partial t} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2] &= r\left[ \frac{\partial V}{\partial S} S – V \right] \\
-\frac{\partial V}{\partial t} – \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 – r\left[ \frac{\partial V}{\partial S} S – V \right] &= 0 \\
\mathbf{\frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 – rV} &= 0
\end{align*}
\)
Which is the Black Scholes Partial Differential Equation.
Application for a European Put
In this section we will apply the formula given to price a European Put.
A put option is a type of derivative which grants the buyer the option but not the obligation to sell 1 share at a predetermined strike price. The qualifier European denotes that the option may only be exercised at expiry. As such, the holder will only exercise the option if St< K
The payoff of the option at strike price ,K, and Stock price (at expiry) ST is stated below.
\((K – S_T)^+ = \max(K – S_T, 0) =
\begin{cases}
K – S_T, & \text{if } K > S_T \\
0, & \text{otherwise}
\end{cases}
\)
The payoff gives us the boundary condition which we can then use to find a solution to the PDE derived in the previous section.
The PDE can be solved using analytical techniques, though we omit the full derivation here. In practice, numerical methods such as finite difference schemes or Monte Carlo simulations are often used to approximate solutions.
The solution is given below.
\begin{align*}
P(S_t, t) &= K e^{-r(T – t)} \Phi(-d_2) – S_t \Phi(-d_1) \\
\\
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}
\end{align*}
\)
P(Sₜ, t): The price of the European put option at time t, based on the underlying asset price Sₜ.
Sₜ: The price of the underlying asset at time t.
K: The strike (exercise) price of the option.
T: Expiration date of the option
σ: The volatility of the underlying asset, which is the standard deviation of returns.
r: The continuously compounded risk-free interest rate.
Φ(·): The cumulative distribution function (CDF) of the standard normal distribution.
Sources/References
Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy