Chapter 6 Black-Scholes Theory

6.1 The Black-Scholes Formula

For a European call option with strike $K$ and maturity $T$, the price at time $t$ with stock price $S$ is:

\[\boxed{C(t, S) = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)}\]

where: \[d_1 = \frac{\log(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma\sqrt{T-t}\]

and $\Phi$ is the standard normal CDF.

6.1.1 Derivation

Under the risk-neutral measure, stock price evolves as:

\[S(T) = S(t)\exp\left[\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma\sqrt{T-t}Z\right]\]

where $Z \sim N(0,1)$.

The call price is: \[C = e^{-r(T-t)}E^{\mathbb{Q}}[\max(S(T) - K, 0)]\]

After working through the integral (completing the square), we obtain the formula above.

6.1.2 Interpretation

\[C = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)\]

$S\Phi(d_1)$: Expected present value of stock if exercised
$Ke^{-r(T-t)}\Phi(d_2)$: Expected present value of strike payment
$\Phi(d_2)$: Risk-neutral probability option finishes in-the-money
$\Phi(d_1)$: “Delta” of the option (hedge ratio)

6.2 The Greeks

The Greeks measure sensitivities of option prices to various parameters. They’re essential for risk management and hedging.

6.2.1 Delta ($\Delta$)

\[\Delta = \frac{\partial C}{\partial S} = \Phi(d_1)\]

Interpretation: - How much the option price changes per $1 change in stock price - Hedge ratio: to hedge a short call, buy $\Delta$ shares - Ranges from 0 (deep out-of-money) to 1 (deep in-the-money)

6.2.2 Gamma ($\Gamma$)

\[\Gamma = \frac{\partial^2 C}{\partial S^2} = \frac{\phi(d_1)}{S\sigma\sqrt{T-t}}\]

where $\phi$ is the standard normal PDF.

Interpretation: - Rate of change of Delta - Measures convexity of option price - Maximum at-the-money - Hedging $\Gamma$ is expensive (requires frequent rebalancing)

6.2.3 Vega ($\mathcal{V}$)

\[\mathcal{V} = \frac{\partial C}{\partial \sigma} = S\phi(d_1)\sqrt{T-t}\]

Interpretation: - Sensitivity to volatility - Options are “long volatility” - gain value when $\sigma$ increases - Maximum at-the-money - Long-dated options have more Vega

6.2.4 Theta ($\Theta$)

\[\Theta = \frac{\partial C}{\partial t} = -\frac{S\phi(d_1)\sigma}{2\sqrt{T-t}} - rKe^{-r(T-t)}\Phi(d_2)\]

Interpretation: - Time decay - value lost per day - Usually negative for long options (you lose time value) - Accelerates as expiration approaches

6.2.5 Rho ($\rho$)

\[\rho = \frac{\partial C}{\partial r} = K(T-t)e^{-r(T-t)}\Phi(d_2)\]

Interpretation: - Sensitivity to interest rates - Usually less important than other Greeks - More relevant for long-dated options

6.2.6 The Greeks Relationship

They satisfy the Black-Scholes PDE:

\[\Theta + \frac{1}{2}\sigma^2 S^2\Gamma + rS\Delta - rC = 0\]

This connects time decay, convexity, and delta in a beautiful way.

6.3 Delta Hedging

You’re a market maker who sold a call option. How to hedge the risk?

6.3.1 The Strategy

At time $t$: Hold $\Delta(t) = \Phi(d_1)$ shares of stock
As stock moves: Continuously rebalance to maintain $\Delta(t)$ shares
If hedged perfectly with correct volatility: Earn the risk-free rate

6.3.2 Discrete Hedging P&L

Over small time $dt$, the P&L is approximately:

\[dP\&L \approx \frac{1}{2}\Gamma(dS)^2 - \Theta \, dt\]

$(dS)^2$ term: Realized variance
$\Theta$ term: Time decay

If realized volatility = implied volatility used for pricing, these balance out.

6.3.3 The Catch

In practice: - Can’t rebalance continuously (transaction costs) - Don’t know true volatility in advance - If realized vol $\neq$ implied vol, you make/lose money

This is why options trading is essentially trading volatility!

6.4 Put-Call Parity

For European options with same strike and maturity:

\[C - P = S - Ke^{-r(T-t)}\]

Proof: Both sides have payoff $S(T) - K$ at maturity. By no-arbitrage, they must have equal value today.

Uses: - Price puts from calls (or vice versa) - Identify arbitrage opportunities - Understand synthetic positions

6.5 Implied Volatility

The market quotes option prices, not volatilities. Implied volatility is the $\sigma$ that makes the Black-Scholes formula match the market price:

\[C_{\text{market}} = C_{BS}(S, K, T, r, \sigma_{implied})\]

6.5.1 The Volatility Smile

In practice, implied volatility varies with strike: - Equity markets: Volatility skew (higher for low strikes) - FX markets: Volatility smile (higher for extreme strikes)

This violates Black-Scholes assumptions! Real markets have: - Fat tails (more crashes than log-normal predicts) - Stochastic volatility - Jumps

Modern models address these issues.

6.6 American Options

American options can be exercised anytime before maturity. No closed-form formula exists!

Must solve the free boundary problem:

\[\max\left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV, \, g(S) - V\right) = 0\]

where $g(S)$ is the exercise payoff.

6.6.1 Early Exercise Premium

\[V_{American} = V_{European} + \text{Early Exercise Premium}\]

For calls on non-dividend paying stocks: Early exercise premium = 0 (never optimal to exercise early).

For puts: Early exercise can be optimal (when deep in-the-money).

6.7 Extensions

6.7.1 Dividends

If stock pays continuous dividend yield $q$:

\[C = Se^{-q(T-t)}\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)\]

where $d_1, d_2$ are modified to include $q$.

6.7.2 Exotic Options

Digital options: Pay $1 or $0
Barrier options: Activated/deactivated if stock crosses barrier
Asian options: Payoff depends on average price
Lookback options: Payoff depends on maximum/minimum

Each requires specialized techniques!

6.8 Limitations of Black-Scholes

Constant volatility: Real volatility varies and is stochastic
No jumps: Stocks can gap (earnings, news)
Log-normal distribution: Real returns have fat tails
Continuous trading: Transaction costs exist
Known parameters: Volatility and drift are unknown

Despite limitations, Black-Scholes is: - A benchmark for pricing and hedging - The foundation for more sophisticated models - Still widely used (with adjustments)

The framework is more important than the formula itself!

Stochastic Calculus: An Idiosyncratic Approach