Finance & Options

Lesson 8 of 12

The Black-Scholes Formula: N(d₂) as a Probability

Take the risk-neutral expectation of the call payoff and the formula falls out, with N(d₂) as the risk-neutral chance of exercise and N(d₁) as the same chance with paths weighted by the stock.

Price as an expectation

Risk-neutral pricing says an option is worth its discounted average payoff, with the average taken under Q\mathbb Q, where the stock grows at the risk-free rate: dS=rS dt+σS dWdS = rS\,dt + \sigma S\,dW. Solving that SDE gives the stock at expiry in closed form:

ST=S0exp⁡((r−12σ2)T+σT Z),Z∼N(0,1).S_T = S_0 \exp\Big(\big(r - \tfrac12\sigma^2\big)T + \sigma\sqrt T\,Z\Big), \qquad Z \sim N(0,1).

The call pays ST−KS_T - K when ST>KS_T > K and nothing otherwise, so split the expectation on that event:

C=e−rTEQ[(ST−K)+]=e−rTEQ[ST 1{ST>K}]−Ke−rT Q(ST>K).C = e^{-rT}E^{\mathbb Q}\big[(S_T - K)^+\big] = e^{-rT}E^{\mathbb Q}\big[S_T\,\mathbf 1_{\{S_T > K\}}\big] - Ke^{-rT}\,\mathbb Q(S_T > K).

The first piece is the stock you receive if you exercise. The second is the strike you pay if you exercise. Each one turns into one term of C=S0N(d1)−Ke−rTN(d2)C = S_0N(d_1) - Ke^{-rT}N(d_2), and the rest of this lesson computes them one at a time.

Running example for the whole lesson: S0=K=100S_0 = K = 100, σ=20%\sigma = 20\%, T=1T = 1 year, r=5%r = 5\%. The answer we are heading for is a call worth about $10.45.