Course 8: Continuous-Time Models

Black-Scholes Option Pricing Model

The cornerstone of modern derivatives pricing: continuous-time stochastic models, closed-form solutions, and dynamic hedging with Greeks

Learning Objectives

1.Understand geometric Brownian motion as a continuous-time stock price model
2.Apply Itô's lemma to derive the Black-Scholes PDE via delta hedging
3.Master the Black-Scholes formulas for European calls and puts
4.Calculate and interpret the Greeks: Delta, Gamma, Vega, Theta, and Rho
5.Understand implied volatility and the volatility smile phenomenon
6.Apply Black-Scholes for practical option pricing and risk management

1. Continuous-Time Framework and Geometric Brownian Motion

Definition 1.1: Brownian Motion (Wiener Process)

A Brownian motion $W_t$ is a continuous-time stochastic process with:

$W_0 = 0$
Independent increments: $W_t - W_s$ is independent of past for $t > s$
Normal increments: $W_t - W_s \sim N(0, t-s)$
Continuous paths: $W_t$ is continuous in $t$

Properties: $\mathbb{E}[W_t] = 0$ , $\text{Var}(W_t) = t$ , $\text{Cov}(W_s, W_t) = \min(s,t)$

Remark 1.1: Intuition Behind Brownian Motion

Think of Brownian motion as the limit of a random walk as time steps become infinitesimally small:

W_t \approx \sum_{i=1}^{n} \epsilon_i \sqrt{\Delta t}, \quad \epsilon_i \sim N(0,1) \text{ i.i.d.}

Key feature: Quadratic variation $[W]_t = t$ (not zero!) unlike deterministic functions.

Definition 1.2: Geometric Brownian Motion (GBM)

Stock price $S_t$ follows a geometric Brownian motion if:

dS_t = \mu S_t dt + \sigma S_t dW_t

where:

$\mu$ is the drift (expected return)
$\sigma$ is the volatility (standard deviation of returns)
$dW_t$ is the increment of Brownian motion

Theorem 1.1: Solution to GBM

The geometric Brownian motion has the explicit solution:

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

Equivalently, log returns are normally distributed:

\ln(S_t/S_0) \sim N\left(\left(\mu - \frac{\sigma^2}{2}\right)t, \sigma^2 t\right)

Proof:

Let $Y_t = \ln S_t$ . By Itô's lemma (to be stated formally later):

dY_t = d(\ln S_t) = \frac{1}{S_t}dS_t - \frac{1}{2}\frac{1}{S_t^2}d[S]_t

Since $d[S]_t = \sigma^2 S_t^2 dt$ (quadratic variation):

dY_t = \frac{1}{S_t}(\mu S_t dt + \sigma S_t dW_t) - \frac{1}{2}\sigma^2 dt

dY_t = \left(\mu - \frac{\sigma^2}{2}\right)dt + \sigma dW_t

Integrating from 0 to $t$ :

Y_t - Y_0 = \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t

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

Exponentiating both sides yields the result.

∎

Example 1.1: Stock Price Distribution

Setup: $S_0 = \$100$ , $\mu = 0.10$ (10% annual drift), $\sigma = 0.20$ (20% volatility), $t = 1$ year

Expected value:

\mathbb{E}[S_1] = S_0 e^{\mu t} = 100 e^{0.10} \approx \$110.52

Log return distribution:

\ln(S_1/S_0) \sim N(0.10 - 0.04/2, 0.04) = N(0.08, 0.04)

Probability $S_1 > \$120$ :

P(S_1 > 120) = P(\ln(S_1/100) > \ln(1.2)) = P\left(Z > \frac{0.1823 - 0.08}{0.2}\right) \approx P(Z > 0.511) \approx 0.305

About 30.5% chance the stock exceeds $120 in one year.

Theorem 1.2: Itô's Lemma

Let $X_t$ follow the SDE $dX_t = \mu_t dt + \sigma_t dW_t$ and let $f(t, X_t)$ be twice differentiable. Then:

df = \left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2}\sigma_t^2 \frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t \frac{\partial f}{\partial x} dW_t

The $\frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2}$ term (Itô correction) arises from the non-zero quadratic variation of Brownian motion.

Remark 1.2: Why Itô's Lemma Differs from Chain Rule

In deterministic calculus: $df = f'(x)dx$

In stochastic calculus: $(dW_t)^2 = dt$ in the mean-square sense, not zero!

Taylor expansion to second order: $df \approx f_t dt + f_x dX + \frac{1}{2}f_{xx}(dX)^2$

Since $(dX)^2 = \sigma^2(dW)^2 = \sigma^2 dt$ , the second-order term survives.

2. Deriving the Black-Scholes PDE

Definition 2.1: Black-Scholes Setup

Consider a European option with payoff $\Phi(S_T)$ at maturity $T$ . The option price $V(t, S_t)$ is a function of time and stock price.

Assumptions:

Stock follows GBM: $dS_t = \mu S_t dt + \sigma S_t dW_t$
Risk-free rate $r$ is constant
No transaction costs, taxes, or dividends
Continuous trading possible
No arbitrage opportunities

Theorem 2.1: Black-Scholes Partial Differential Equation

The option price $V(t,S)$ satisfies:

\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = rV

with terminal condition $V(T, S) = \Phi(S)$ .

Proof:

Step 1: Apply Itô's lemma to $V(t, S_t)$ :

dV = \left(\frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \frac{\partial V}{\partial S} dW_t

Step 2: Construct a delta-hedged portfolio:

\Pi = V - \Delta S

where $\Delta = \frac{\partial V}{\partial S}$ is the hedge ratio. The portfolio change is:

d\Pi = dV - \Delta dS

d\Pi = \left(\frac{\partial V}{\partial t} + \mu S \Delta + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \Delta dW_t - \Delta(\mu S dt + \sigma S dW_t)

The stochastic terms cancel:

d\Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt

Step 3: No-arbitrage argument:

Since $\Pi$ is riskless (no $dW_t$ term), it must earn the risk-free rate:

d\Pi = r\Pi dt = r(V - \Delta S)dt

Equating the two expressions for $d\Pi$ :

\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = r\left(V - S\frac{\partial V}{\partial S}\right)

Rearranging yields the Black-Scholes PDE.

∎

Remark 2.1: Key Insights from the Derivation

Risk-neutral: The drift $\mu$ disappeared! Option price doesn't depend on expected return
Dynamic hedging: Continuous rebalancing of $\Delta$ eliminates risk
Volatility matters: $\sigma$ appears in the PDE and affects option value
Arbitrage-free: No-arbitrage implies the PDE; otherwise, arbitrage exists

Example 2.1: Verifying a Solution

Consider the simple case: $V(t,S) = S$ (the stock itself as a "derivative").

Compute partial derivatives:

\frac{\partial V}{\partial t} = 0, \quad \frac{\partial V}{\partial S} = 1, \quad \frac{\partial^2 V}{\partial S^2} = 0

Substitute into Black-Scholes PDE:

0 + rS \cdot 1 + \frac{1}{2}\sigma^2 S^2 \cdot 0 = rS \quad \checkmark

The stock price itself satisfies the PDE (as it should)!

Proposition 2.1: Risk-Neutral Valuation Formula

The solution to the Black-Scholes PDE can be written as:

V(t,S) = e^{-r(T-t)} \mathbb{E}^Q[\Phi(S_T) \mid S_t = S]

where $\mathbb{E}^Q$ is expectation under the risk-neutral measure where:

dS_t = rS_t dt + \sigma S_t dW_t^Q

Under $Q$ , the stock drifts at the risk-free rate $r$ (not the actual drift $\mu$ ).

3. Black-Scholes Formula for European Options

Theorem 3.1: Black-Scholes Formula

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

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

For a European put option:

P(S, t) = Ke^{-r(T-t)}\Phi(-d_2) - S\Phi(-d_1)

where

d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}

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

and $\Phi$ is the cumulative distribution function of the standard normal distribution.

Proof:

(Sketch) Under the risk-neutral measure, $S_T \sim \text{Lognormal}$ :

S_T = S_t e^{(r - \sigma^2/2)(T-t) + \sigma\sqrt{T-t}Z}, \quad Z \sim N(0,1)

Call price:

C = e^{-r(T-t)} \mathbb{E}^Q[\max(S_T - K, 0)]

C = e^{-r(T-t)} \int_{-\infty}^{\infty} \max(Se^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t}z} - K, 0) \phi(z)dz

The call is in-the-money when $S_T > K$ , i.e., $z > -d_2$ . Evaluating the integral (using properties of log-normal distribution):

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

The put formula follows from put-call parity or similar integration.

∎

Example 3.1: Calculating Black-Scholes Call Price

Setup: $S = \$100$ , $K = \$105$ , $r = 0.05$ , $\sigma = 0.20$ , $T - t = 1$ year

Step 1: Calculate $d_1$ and $d_2$ :

d_1 = \frac{\ln(100/105) + (0.05 + 0.04/2)(1)}{0.20\sqrt{1}} = \frac{-0.0488 + 0.07}{0.20} = 0.1062

d_2 = 0.1062 - 0.20 = -0.0938

Step 2: Look up normal CDF values:

\Phi(0.1062) \approx 0.5423, \quad \Phi(-0.0938) \approx 0.4626

Step 3: Calculate call price:

C = 100 \times 0.5423 - 105 \times e^{-0.05} \times 0.4626

C = 54.23 - 105 \times 0.9512 \times 0.4626 = 54.23 - 46.19 \approx \$8.04

Example 3.2: Put Price via Put-Call Parity

Using data from Example 3.1, calculate the put price:

P = C - S + Ke^{-r(T-t)}

P = 8.04 - 100 + 105 \times e^{-0.05} = 8.04 - 100 + 99.88 \approx \$7.92

Verification using put formula:

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

P = 99.88 \times (1 - 0.4626) - 100 \times (1 - 0.5423)

P = 99.88 \times 0.5374 - 100 \times 0.4577 = 53.67 - 45.77 \approx \$7.90

Slight discrepancy due to rounding; both methods give approximately $7.90-$7.92.

Remark 3.1: Interpreting the Black-Scholes Formula

Call formula: $C = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)$

$\Phi(d_2)$ ≈ risk-neutral probability that option expires in-the-money
$Ke^{-r(T-t)}\Phi(d_2)$ = discounted expected payment if exercised
$\Phi(d_1)$ = delta (hedge ratio); proportion of stock in replicating portfolio
$S\Phi(d_1)$ = value of stock position in replicating portfolio

The formula decomposes the call into: value of stock received if exercised minus present value of strike paid.

Proposition 3.1: Limiting Cases

Deep in-the-money ( $S \gg K$ ):

C \to S - Ke^{-r(T-t)} \quad (\text{intrinsic value})

Deep out-of-the-money ( $S \ll K$ ):

C \to 0

At-the-money ( $S = K$ ):

d_1 \approx \frac{(r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \quad C \approx S\left[\Phi(d_1) - \Phi(d_2)\right]

As maturity approaches ( $T - t \to 0$ ):

C \to \max(S - K, 0) \quad (\text{payoff})

4. The Greeks: Option Sensitivities

Definition 4.1: The Greeks

The Greeks measure the sensitivity of option prices to various parameters. They are essential for risk management and hedging.

Theorem 4.1: Delta (Δ)

Delta measures the rate of change of option price with respect to stock price:

\Delta = \frac{\partial V}{\partial S}

For Black-Scholes:

\Delta_\text{call} = \Phi(d_1), \quad \Delta_\text{put} = \Phi(d_1) - 1 = -\Phi(-d_1)

Interpretation: For every $1 increase in stock price, the option price increases by approximately $\Delta$ dollars.

Example 4.1: Delta Hedging

From Example 3.1, we have $d_1 = 0.1062$ , so $\Delta_\text{call} = \Phi(0.1062) \approx 0.54$

To hedge a portfolio of 1000 short calls, buy $1000 \times 0.54 = 540$ shares.

If stock rises by $1: Call loss ≈ $1000 \times 0.54 = \$540$ , Stock gain = $540 \times 1 = \$540$ → Net ≈ $0

Note: Delta changes as stock moves (see Gamma), requiring periodic rebalancing.

Theorem 4.2: Gamma (Γ)

Gamma measures the rate of change of Delta with respect to stock price:

\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}

For Black-Scholes (same for call and put):

\Gamma = \frac{\phi(d_1)}{S\sigma\sqrt{T-t}}

where $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the standard normal PDF.

Interpretation: Gamma measures the curvature of option value. High Gamma means Delta changes rapidly, requiring frequent rehedging.

Remark 4.1: Gamma and Delta-Hedging Costs

Gamma is highest for at-the-money options near expiry. This means:

Delta changes rapidly → frequent rebalancing needed
Higher transaction costs for maintaining delta-neutral portfolio
Gamma risk is significant for option sellers

Theorem 4.3: Vega (ν)

Vega measures sensitivity to volatility:

\nu = \frac{\partial V}{\partial \sigma}

For Black-Scholes (same for call and put):

\nu = S\phi(d_1)\sqrt{T-t}

Interpretation: For a 1% (0.01) increase in volatility, option price increases by $\nu \times 0.01$ .

Example 4.2: Calculating Greeks

Using Example 3.1 data: $S = \$100$ , $K = \$105$ , $r = 0.05$ , $\sigma = 0.20$ , $T-t = 1$ , $d_1 = 0.1062$

Delta: $\Delta = \Phi(0.1062) \approx 0.542$

Gamma: $\phi(0.1062) = \frac{1}{\sqrt{2\pi}}e^{-0.1062^2/2} \approx 0.3977$

\Gamma = \frac{0.3977}{100 \times 0.20 \times 1} = \frac{0.3977}{20} \approx 0.0199

Vega:

\nu = 100 \times 0.3977 \times \sqrt{1} = 39.77

If volatility increases from 20% to 21%, call price increases by approximately $39.77 \times 0.01 \approx \$0.40$ .

Theorem 4.4: Theta (Θ)

Theta measures time decay:

\Theta = \frac{\partial V}{\partial t}

For a call:

\Theta_\text{call} = -\frac{S\phi(d_1)\sigma}{2\sqrt{T-t}} - rKe^{-r(T-t)}\Phi(d_2)

Typically $\Theta < 0$ for long options: option value decreases as time passes (time decay).

Convention: Theta is often quoted as the change per day, so divide by 365.

Theorem 4.5: Rho (ρ)

Rho measures sensitivity to interest rate:

\rho = \frac{\partial V}{\partial r}

For Black-Scholes:

\rho_\text{call} = K(T-t)e^{-r(T-t)}\Phi(d_2)

\rho_\text{put} = -K(T-t)e^{-r(T-t)}\Phi(-d_2)

Interpretation: Typically the least important Greek for short-dated options, more relevant for long-dated options.

Remark 4.2: Greek Relationships

The Greeks satisfy the Black-Scholes PDE relationship:

\Theta + rS\Delta + \frac{1}{2}\sigma^2 S^2 \Gamma = rV

This means for a delta-hedged portfolio ( $\Delta = 0$ ):

\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma = rV

Implication: Gamma profits offset Theta decay for a hedged position.

Proposition 4.1: Summary of Greeks for Calls and Puts

Greek	Call	Put	Typical Sign
Delta (Δ)	$\Phi(d_1)$	$\Phi(d_1) - 1$	Call: +, Put: -
Gamma (Γ)	$\frac{\phi(d_1)}{S\sigma\sqrt{T-t}}$		Always +
Vega (ν)	$S\phi(d_1)\sqrt{T-t}$		Always +
Theta (Θ)	(Complex formula)		Usually - (long)
Rho (ρ)	$> 0$	$< 0$	Call: +, Put: -

5. Implied Volatility and the Volatility Smile

Definition 5.1: Implied Volatility

The implied volatility $\sigma_{\text{imp}}$ is the volatility parameter that, when input into the Black-Scholes formula, yields the observed market price:

C_{\text{market}} = C_{\text{BS}}(S, K, r, T-t, \sigma_{\text{imp}})

Since the Black-Scholes formula is monotonically increasing in $\sigma$ , implied volatility exists and is unique for each option.

Remark 5.1: Computing Implied Volatility

There's no closed-form solution for $\sigma_{\text{imp}}$ . Common methods:

Newton-Raphson: Use Vega = $\frac{\partial C}{\partial \sigma}$ for fast convergence
Bisection: Simple but slower, guaranteed convergence
Approximations: Various analytical approximations available

Example 5.1: Implied Volatility Calculation

Market data: Call with $S = \$100$ , $K = \$100$ , $r = 0.05$ , $T-t = 0.25$ years trades at $C_{\text{mkt}} = \$5.50$

Newton-Raphson iteration: Start with $\sigma_0 = 0.30$ (30%)

Iteration 1: $C_{\text{BS}}(\sigma_0) = \$5.23$ , $\nu = 19.77$

\sigma_1 = \sigma_0 + \frac{C_{\text{mkt}} - C_{\text{BS}}(\sigma_0)}{\nu} = 0.30 + \frac{5.50 - 5.23}{19.77} \approx 0.314

Iteration 2: $C_{\text{BS}}(0.314) = \$5.50$ ✓

Implied volatility ≈ 31.4%. Converges in 2-3 iterations typically.

Definition 5.2: Volatility Smile and Skew

The volatility smile is the pattern where implied volatility varies across strike prices:

Smile: Higher IV for both deep ITM and OTM options (equity index options)
Skew: IV decreases as strike increases (individual equity options)
Smirk: Asymmetric pattern common in equity markets post-1987 crash

Implication: Black-Scholes assumption of constant volatility is violated in practice.

Remark 5.2: Causes of Volatility Smile

Several explanations for the smile:

Fat tails: Real returns have heavier tails than log-normal distribution
Leverage effect: Stock drops → higher leverage → higher volatility
Crash fears: Market participants price in tail risk (downside protection premium)
Supply/demand: Hedging demand affects option prices

Example 5.2: Volatility Skew Pattern

Typical equity index option IVs:

Strike 90% of spot: IV ≈ 25%
Strike 100% of spot (ATM): IV ≈ 20%
Strike 110% of spot: IV ≈ 18%

Interpretation: Out-of-the-money puts (downside protection) trade at higher IV, reflecting crash fears and hedging demand.

Proposition 5.1: Practical Implications

For traders:

Use implied volatility surfaces (IV as function of strike and maturity)
Volatility trading strategies exploit IV changes independent of direction
Greeks calculated with market-implied volatility, not historical

For modelers:

Black-Scholes is a benchmark; smile suggests need for extensions
Stochastic volatility models (Heston) or local volatility models address smile
Jump-diffusion models capture fat tails

Key Takeaways

1.Geometric Brownian motion models stock prices with constant drift and volatility
2.Itô's lemma extends calculus to stochastic processes, accounting for quadratic variation
3.Black-Scholes PDE derived via delta hedging eliminates risk; option price independent of expected return μ
4.Black-Scholes formula provides closed-form European option prices using risk-neutral valuation
5.Greeks (Delta, Gamma, Vega, Theta, Rho) measure sensitivities essential for hedging and risk management
6.Implied volatility and the volatility smile reveal market expectations and model limitations

Practice Problems

Black-Scholes Model

Questions

Correct

Accuracy

Which of the following is NOT an assumption of the Black-Scholes model?

Not attempted

In the Black-Scholes formula, what does Φ(d₁) represent?

Not attempted

What is the key insight of Itô's lemma compared to standard calculus?

Not attempted

In the Black-Scholes PDE derivation, why does the drift μ disappear?

Not attempted

For an at-the-money call option, Delta is approximately:

Not attempted

Which Greek measures the option's sensitivity to volatility?

Not attempted

Gamma is highest for:

Not attempted

What is implied volatility?

Not attempted

The volatility smile suggests that:

Not attempted

For a delta-neutral portfolio, which relationship holds?

Not attempted

Under the risk-neutral measure Q, the stock drift is:

Not attempted

Which statement about Theta is TRUE?

Not attempted

The solution to dS = μS dt + σS dW is:

Not attempted

Put-call parity for European options states:

Not attempted

Which Greek is the same for both calls and puts with the same strike and maturity?

Not attempted

Frequently Asked Questions

What are the key assumptions of the Black-Scholes model?

(1) Stock price follows geometric Brownian motion with constant volatility, (2) No transaction costs or taxes, (3) Risk-free rate is constant, (4) No dividends, (5) Markets are efficient (no arbitrage), (6) European options only.

Why is the Black-Scholes formula so important?

It provides a closed-form solution for option prices, revolutionized derivatives markets, and introduced the concept of dynamic hedging. Despite its limitations, it remains the industry standard and foundation for more advanced models.

What is the most important Greek for hedging?

Delta is typically most important as it measures price sensitivity and is used for delta-neutral hedging. However, Gamma (curvature) and Vega (volatility risk) are also crucial for comprehensive risk management.

What is implied volatility?

Implied volatility is the volatility parameter that makes the Black-Scholes price equal to the market price. It represents the market's expectation of future volatility and varies across strikes (volatility smile/skew).

Can Black-Scholes be used for American options?

No, the closed-form formula only applies to European options. For American options, numerical methods (binomial trees, finite differences, Monte Carlo) are required, though American calls on non-dividend stocks equal European calls.