Black-Scholes Model and Option Pricing for IAI Actuarial

Understand Black-Scholes Model and Option Pricing

Key rules to remember

Share price model (GBM)

dS = μS dt + σS dW

Under the risk-neutral measure, replace μ with r (or r − q if the share pays a continuous dividend yield q).

Black-Scholes PDE (no dividends)

∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0

With continuous dividend yield q, the third term becomes (r − q)S ∂V/∂S. The drift μ does not appear.

d1 and d2

d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T

T is time to expiry in years. r is the continuously compounded risk-free rate. With dividend yield q, use r − q + σ²/2 in d1.

European call price

C = S·N(d1) − K·e^(−rT)·N(d2)

N is the standard normal cumulative distribution function. Share paying no dividends.

European put price

P = K·e^(−rT)·N(−d2) − S·N(−d1)

Or find C first and use put-call parity: P = C − S + K·e^(−rT).

Risk-neutral valuation

V = e^(−rT) × E_Q[payoff]

Under Q, S grows at r. The same price results as from the hedging argument.

Delta

Call: Δ = N(d1); Put: Δ = N(d1) − 1

Call delta lies between 0 and 1. Put delta lies between −1 and 0 (no dividends).

Gamma

Γ = φ(d1) ÷ (S σ √T)

φ(x) = e^(−x²/2) ÷ √(2π). Same for calls and puts.

Vega

ν = S φ(d1) √T

Same for calls and puts. This is the change per unit change in σ (for example 1.00 = 100 percentage points).

Call theta and rho

Θ_call = −S φ(d1) σ ÷ (2√T) − r K e^(−rT) N(d2); ρ_call = K T e^(−rT) N(d2)

Theta is per year. Divide by 365 for a daily figure if asked.

Greeks link

Θ + rSΔ + ½σ²S²Γ = rV

This is the PDE rewritten in Greeks. It is a useful check (no dividends).

How to solve Black-Scholes Model and Option Pricing questions

1. 1Read the inputs and write them down: S, K, r (continuously compounded), σ, T in years, and any dividend yield q. Convert rates and times to a consistent annual basis.
2. 2If asked to derive the PDE, set up the portfolio Π = V − ΔS. Apply Ito's lemma to V(S,t) and use dS = μS dt + σS dW.
3. 3Choose Δ = ∂V/∂S so the dW terms cancel. State that the portfolio is now riskless and must earn r, so dΠ = rΠ dt. Equate terms and show μ disappears.
4. 4If asked for risk-neutral valuation, replace μ by r, then write V = e^(−rT) E_Q[payoff] and say why this is valid (the hedging argument shows preferences are irrelevant).
5. 5For a price, compute d1, then d2 = d1 − σ√T. Look up N(d1) and N(d2) using normal tables, interpolating if needed.
6. 6Substitute into the call formula. For a put, use the put formula or put-call parity. Check that the price is above the lower bound, for example C ≥ S − K e^(−rT).
7. 7For Greeks, use the standard formulas above. Say what the result means, for example how many shares hedge the position.
8. 8State assumptions or limitations if the question asks you to comment, such as constant volatility and continuous rebalancing.

Quickest way: Fast route to a European option price and delta

Common mistakes in Black-Scholes Model and Option Pricing

• Using μ, the real-world drift, in the pricing formula or PDE.

  The share price model has μ in it, so students carry it forward.

  Fix: Remember that μ cancels in the hedging argument. In pricing, use r (or r − q).

• Using a simple annual interest rate instead of the continuously compounded rate, or using T in months.

  The question gives an effective rate or a term in months.

  Fix: Convert to a continuous rate (δ = ln(1 + i)) if needed and express T in years before computing.

• Forgetting that d2 = d1 − σ√T, or subtracting σ² instead of σ√T.

  The two d formulas look alike and are easy to confuse under time pressure.

  Fix: Always compute d1 first, then subtract σ√T. Write σ√T as a separate number.

• Getting put values wrong by using N(d1) and N(d2) instead of N(−d1) and N(−d2).

  Students adapt the call formula without changing the signs.

  Fix: Use P = K e^(−rT) N(−d2) − S N(−d1), or use put-call parity to check the answer.

• Treating put delta as positive, or saying call delta can be above 1.

  Delta is remembered as N(d1) for everything.

  Fix: Put delta = N(d1) − 1, which is negative. Call delta lies between 0 and 1 for a share with no dividends.

• Saying the hedge is a one-time trade and the portfolio stays riskless.

  The derivation uses an infinitesimal time step, so the hedge looks permanent.

  Fix: State that delta changes with S and t, so the hedge must be rebalanced continuously in theory. In practice, gamma measures how fast the hedge goes stale.

Worked examples