Black-Scholes Option Pricing

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Definition

The Black-Scholes option pricing formula gives the no-arbitrage price of a European call option as a function of current stock price SS, strike KK, time to expiration tt, riskless discount factor RFR_F, and σ\sigma (standard deviation of the log compound return of the stock):

C=S⋅N(z∗+σ)−K⋅RF−1⋅N(z∗)C = S \cdot N(z^* + \sigma) - K \cdot R_F^{-1} \cdot N(z^*)

where z∗=[ln⁡(S/K)+ln⁡RF−12σ2]/σz^* = [\ln(S/K) + \ln R_F - \tfrac{1}{2}\sigma^2] / \sigma and N(⋅)N(\cdot) is the standard normal cumulative distribution function (CDF). The Black-Scholes value is a function solely of observables — no preference parameters appear.

Key Ideas

How It Works

Delta Hedge and PDE (Black-Scholes 1973)

Assume stock follows geometric Brownian motion (GBM): dS=αS dt+vS dzdS = \alpha S\,dt + vS\,dz. Consider a portfolio: long 1 share, short 1/w11/w_1 calls, where w1=∂w/∂xw_1 = \partial w/\partial x. The portfolio value is x−w/w1x - w/w_1; its return has zero covariance with the market (the dzdz terms cancel exactly), so by no-arbitrage it must earn rr. Writing out the return over dtdt via Itô's lemma and imposing =r dt= r\,dt gives:

12v2x2w11+rxw1+w2−rw=0(7)\tfrac{1}{2}v^2 x^2 w_{11} + r x w_1 + w_2 - rw = 0 \tag{7}

With boundary condition w(x,t∗)=max⁡(x−c,0)w(x,t^*) = \max(x-c, 0) and substitution y=ln⁡x+(r−12v2)(t∗−t)y = \ln x + (r - \tfrac{1}{2}v^2)(t^*-t), τ=v2(t∗−t)\tau = v^2(t^*-t), equation (7) reduces to the heat equation ∂F/∂τ=∂2F/∂y2\partial F/\partial\tau = \partial^2F/\partial y^2. The standard convolution solution gives:

w(x,t)=xN(d1)−ce−r(t∗−t)N(d2)w(x,t) = xN(d_1) - ce^{-r(t^*-t)}N(d_2)

where d1=[ln⁡(x/c)+(r+v2/2)(t∗−t)]/(vt∗−t)d_1 = [\ln(x/c)+(r+v^2/2)(t^*-t)]/(v\sqrt{t^*-t}) and d2=d1−vt∗−td_2 = d_1 - v\sqrt{t^*-t}.

Corporate Liabilities as Options (Black-Scholes 1973)

Let xx be the total market value of the firm's assets. With debt of face value BB maturing at t∗t^*:

This reframes all corporate securities as contingent claims on firm assets, independent of their priority structure, and directly precedes Merton's (1974) structural credit model.

From the Stochastic Discount Factor

Let Zt′=YtRF−1/E(Yt)Z'_t = Y_t R_F^{-1} / E(Y_t) be the normalized pricing kernel (risk-neutral probability kernel). The call price is:

C=E ⁣[(St−K)+Zt′]=E ⁣[(St−K)Zt′ ∣ St≥K]C = E\!\left[(S_t - K)^+ Z'_t\right] = E\!\left[(S_t - K) Z'_t \,\big|\, S_t \geq K\right]

When (St,Yt)(S_t, Y_t) are jointly lognormal, the Stein–Rubinstein lemma evaluates the resulting bivariate integral:

Cov(x,g(y))=E[g′(y)] Cov(x,y)for bivariate normal (x,y)\text{Cov}(x, g(y)) = E[g'(y)]\,\text{Cov}(x, y) \quad \text{for bivariate normal } (x,y)

Breaking the expectation into two integrals over the bivariate lognormal density and applying this identity yields the Black-Scholes formula analytically. See Stochastic Discount Factor for the full framework.

Put-Call Parity

The put payoff rewrites as max⁡(0,K−S∗)=max⁡(0,S∗−K)−S∗+K\max(0, K - S^*) = \max(0, S^* - K) - S^* + K. Applying the valuation operator gives:

P=C−(S−D)+K⋅RF−1P = C - (S - D) + K \cdot R_F^{-1}

where DD is the present value of dividends through expiration. Holds by pure arbitrage regardless of the distributional assumption on S∗S^*; puts are manufactured from calls in practice (Rubinstein 1987).

Binomial Method (Cox-Ross-Rubinstein 1979)

If the underlying can move only up or down by known percentages each discrete period, a portfolio of the index and cash exactly replicates the option. Working backward from expiration payoffs gives a unique option value at each node; for American options, replace the holding value with the exercise value whenever exercising is optimal. In the limit as the period length shrinks to zero, the binomial price converges to Black-Scholes.

Extensions

Why It Matters

Open Questions

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