Options Pricing: Black–Scholes–Merton Model

The Black–Scholes–Merton (BSM) model provides a continuous-time framework for valuing options and other derivatives.
It assumes frictionless trading, continuous hedging, and lognormally distributed asset prices.
The model forms the cornerstone of modern quantitative finance.

1. Model Assumptions

1. The underlying asset price $S_t$ follows geometric Brownian motion:

   \[dS_t = \mu S_t dt + \sigma S_t dW_t,\]

   where $\mu$ is the expected return, $\sigma$ the volatility, and $W_t$ a standard Brownian motion.

2. The risk-free rate $r$ and volatility $\sigma$ are constant.
3. No transaction costs, taxes, or short-selling constraints.
4. Trading occurs continuously, and borrowing/lending are possible at $r$.
5. Markets are arbitrage-free and complete.

2. Deriving the Black–Scholes–Merton PDE

Let the option value be $V(t,S_t)$.
By Itô’s lemma:

\[dV = V_t\,dt + V_S\,dS_t + \tfrac{1}{2}V_{SS}(dS_t)^2 = \left(V_t + \mu S_t V_S + \tfrac{1}{2}\sigma^2 S_t^2 V_{SS}\right)dt + \sigma S_t V_S\,dW_t.\]

We form a self-financing replicating portfolio consisting of:

• $\Delta$ units of the underlying asset, and
• one short option position $(-1)$.

Portfolio value:

\[\Pi = \Delta S_t - V.\]

Portfolio dynamics:

\[d\Pi = \Delta\,dS_t - dV.\]

Substitute $dS_t$ and $dV$, choose $\Delta = V_S$ to eliminate the stochastic term $dW_t$, yielding a riskless portfolio:

\[d\Pi = (-V_t - \tfrac{1}{2}\sigma^2 S_t^2 V_{SS})dt.\]

By the no-arbitrage condition, the return on the riskless portfolio must equal $r\Pi\,dt$:

\[-V_t - \tfrac{1}{2}\sigma^2 S_t^2 V_{SS} = r(\Delta S_t - V).\]

Substitute $\Delta = V_S$ and rearrange:

\[V_t + \tfrac{1}{2}\sigma^2 S_t^2 V_{SS} + rS_t V_S - rV = 0.\]

This is the Black–Scholes–Merton partial differential equation.

3. Boundary Conditions

For a European call option with strike $K$ and maturity $T$:

\[V(T,S_T) = \max(S_T - K, 0).\]

For a European put:

\[V(T,S_T) = \max(K - S_T, 0).\]

4. Solving the PDE (Risk-Neutral Valuation)

Under the risk-neutral measure $\mathbb{Q}$, the drift of $S_t$ becomes $(r - q)$:

\[dS_t = (r - q)S_t dt + \sigma S_t dW_t^{\mathbb{Q}},\]

where $q$ is the continuous dividend yield.

The discounted price process $e^{-\int_0^t r du} S_t$ is a martingale.
Hence, by risk-neutral pricing:

\[V(t,S_t) = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}\!\left[f(S_T)\,|\,S_t\right].\]

5. Closed-Form Black–Scholes–Merton Formula

For a European call and put:

\[\begin{aligned} C &= S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2), \\\\ P &= K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1), \end{aligned}\]

where

\[d_1 = \frac{\ln(S_0/K) + (r - q + 0.5\sigma^2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}.\]

Here, $N(x)$ is the cumulative distribution function of the standard normal distribution.

6. Intuition Behind $N(d_1)$ and $N(d_2)$

• $N(d_2)$: risk-neutral probability that the option will expire in the money ($S_T > K$).
• $N(d_1)$: probability-weighted delta — the present value of expected exercise under risk-neutral drift.

Thus:

• The first term, $S_0 e^{-qT}N(d_1)$, is the present value of the expected asset received upon exercise.
• The second term, $K e^{-rT}N(d_2)$, is the present value of the payment made at exercise.

7. Example Calculation

Let:

• $S_0 = 100$, $K = 100$, $r = 5\%$, $q = 2\%$,
• $\sigma = 20\%$, $T = 1$ year.

Compute:

\[d_1 = \frac{\ln(1) + (0.05 - 0.02 + 0.5\times0.2^2)}{0.2} = 0.35, \qquad d_2 = 0.15.\]

Then:

\[C = 100 e^{-0.02}(0.6368) - 100 e^{-0.05}(0.5596) = 8.91, \\ P = 100 e^{-0.05}(0.4404) - 100 e^{-0.02}(0.3632) = 6.33.\]

Hence, the call price is $8.91 and put price is $6.33.

8. Economic Insights

1. Volatility ($\sigma$):
   Higher volatility increases option value due to convexity of the payoff.

2. Interest Rate ($r$):
   Increases call values (deferring payment) and decreases put values.

3. Dividends ($q$):
   Reduce call prices (less expected growth) and increase puts.

4. Time to Maturity ($T$):
   Generally increases option value — more time, more optionality.

9. Visualization: The Black–Scholes World

The BSM framework elegantly connects stochastic calculus, PDEs, and probabilistic valuation.
Its assumptions may be simplified, but the logic — replication, hedging, and no-arbitrage — underpins all modern pricing theory.

10. Summary

• The BSM model arises from continuous-time hedging and no-arbitrage.
• It yields closed-form prices via risk-neutral expectation.
• $N(d_1)$ and $N(d_2)$ have clear probabilistic interpretations.
• The framework extends naturally to Greeks, implied volatility, and exotic options.

Next up: Greeks and Risk Management — understanding sensitivities and hedging strategies.