Option Pricing

Black-Scholes/Merton

The classic Black-Scholes/Merton formula for the spot value of a call option is v_0 = s N(d_1) - ke^{-rt} N(d_2) where N is the standard normal cumluative distribution function, s is the spot price, k is the call strike, r is the risk-free continuously compounded interest rate, t is the time in years to expiration, d_1 = (\log(s/k) + (r + σ^2/2)t)/σ\sqrt{t}, and d_2 = d_1 - σ\sqrt{t}.

Option delta is the derivative of value with respect to the underlying. It is true that ∂_s v_0 = ∂v_0/∂s = N(d_1), but d_1 and d_2 involve s so one needs to show s ∂_s N(d_1) - ke^{-rt} ∂_s N(d_2) = 0. Plowing through the calculations involved is a ritual we all perform when first learning the theory.

Their Nobel Prize winning work showed how to replicate the payoff of an option by dynamically hedging it with the underlying. The value of an option is the cost of setting up the initial hedge. It is not trivial to show the value is the expectation of the option payoff under some probability measure. This is why Nobel Prizes are awarded.

Fischer Black simplified this formula by expressing it in terms of forward values. v_t = f N(d_1) - k N(d_2), where f = se^{rt} is forward price and v_t = v_0 e^{rt} is the forward value of the option. In this case d_1 = (\log(f/k) + σ^2t/2)/σ\sqrt{t} and d_2 = d_1 - σ\sqrt{t} which eliminates the parameter r. Letting s = σ\sqrt{t} eliminates t.

We will skip the theory of stochastic differential equations, Ito’s lemma, self-financing portfolios, and other dainty mathematical machinery required to prove their result. Let’s fast-forward to calculating expected values and derivatives with respect to model parameters.

Put Formula

The forward value of a put with strike k is v = E[\max\{k - F\}], where F is the random value of the underlying at expiration. We let 1_A denote the characteristic function with 1_A(x) = 1 if x\in A and 1_A(x) = 0 if x\notin A.

\begin{aligned} v &= E[\max\{k - F\}] \\ &= E[(k - F)1(F \le k)] \\ &= kP(F\le k) - E[F 1(F \le k)] \\ &= kP(F\le k) - E[F]E[F/E[F] 1(F \le k)] \\ &= kP(F\le k) - fP^s(F \le k)] \\ \end{aligned} where f = E[F] and P^s is defined by dP^s/dP = F/E[F].

Exercise. If F > 0 then P^s is a probability measure.

The (forward) value of an option paying \nu(F) at expiration is v = E[\nu(F)]. If the option pays shares instead of currency its value is E[F\nu(F)] = E[F]E[F/E[F] \nu(F)] = f E^s[\nu(F)], where E^s is the expectation under share measure P^s.

Recall the moment generating function of a random variable X is E[e^{sX}] and its cumulant is \kappa(s) = \log E[e^{sX}].

Exercise. If F > 0 then F = fe^{sX - \kappa(s)} where s^2 = \operatorname{Var}(\log F) and X has mean 0 and variance 1.

Black Model

In Black’s model, the forward at expiration is F_t = fe^{σB_t - σ^2t/2}, where B_t is standard Brownian motion. The forward value of a call option is the expected value of the call payoff at expiration v_t = E[\max\{F_t - k, 0\}] The expiration t can be subsumed into the vol s = σ\sqrt{t} so F_t = F = fe^{sX - s^2/2} where X is standard normal. The only fact we use about Brownian motion is B_t is normal with mean 0 and variance t.

Since (F - k)^+ = \max\{F - k, 0\} = (F - k) 1(F\ge k), \begin{aligned} v &= E[\max\{F - k, 0\}] \\ &= E[(F - k) 1(F\ge k)] \\ &= E[F 1(F\ge k)] - kE[1(F\ge k)] \\ &= f E[e^{sX - s^2/2} 1(F\ge k)] - kP(F\ge k) \\ \end{aligned}

Exercise. Show F\ge k if and only if -X \le d_2.

Exercise. Use E[e^{sX - s^2/2} g(X)] = E[g(X + s)] to show E[e^{sX - s^2/2} 1(F\ge k)] = P(Fe^{s^2}\ge k).

Exercise. Show Fe^{s^2}\ge k if and only if -X\le d_1.

This establishes the Black formula for the forward value of an option since -X has the same distribution as X.

For any differentiable function ν, ∂_f E[ν(F)] = E[ν'(F) ∂_f F] = E[ν'(F) e^{sX - s^2/2}] = E[ν'(Fe^{s^2})] so ∂_f v = E[1(Fe^{s^2}\ge k)] = P(Fe^{s^2}\ge k) = N(d_1). This establishes the formula for option delta without any turmoil. Option values and greeks for any positive underlying can be calculated in a similar fashion.

Share Measure

Let F be the price of the underlying instrument at option expiration. The forward value of an option paying ν(F) in some currency at expiration is E[ν(F)]. If F is positive we can also consider the payoff in terms of shares of F, ν_s(F) = ν(F)F/E[F]; if we receive ν_s(F) shares of F at expiration we can convert those at price F to ν(F) in the currency.

Share measure for positive underlyings is defined by E_s[ν(F)] = E[ν_s(F)] = E[ν(F) F/E[F]]. Note F > 0 and E_s[1] = 1 so share measure is a probability measure. It shows up in the formula for valuing a call \begin{aligned} E[(F - k)^+] &= E[(F - k)1(F\ge k)] \\ &= E[F1(F\ge k)] - k P(F\ge k) \\ &= fP_s(F\ge k) - k P(F\ge k). \end{aligned} Every positive random variables F can be written F = f e^{s X - κ(s)} where X is a random variable with mean 0 and variance 1 and κ(s) = \log E[e^{sX}] is the cumulant of X. Note f = E[F] and s^2 = \operatorname{Var}(\log F).

Exercise. Clearly \log(F/E[F]) = m + sX for some random variable X with mean 0 and variance 1. Show E[F] = f implies m = -κ(s).

If we let ε_s(x) = e^{s x - κ(s)}, so F = fε_s(X), this can be written E_s[ν(F)] = E[ν(F) ε_s(X)] and we see share measure is the Esscher transform. The cumulative distribution of F under this measure is P_s(F\le y) = P_s(X\le x) = E[1(X\le x) e^{sX - κ(s)}] where x = x(y) = ε_s^{-1}(y/f) = (\log y/f + κ(s))/s is the moneyness of y. Note ∂_x ε_s(x) = ε_s(x) s, ∂_s ε_s(x) = ε_s(x) (x - κ'(s)), and ε_s(x(y)) = y/f.

Put and Call

A put option pays ν(F) = (k - F)^+ = \max\{k - F,0\} at expiration and has value p = E[(k - F)^+]. A call option pays ν(F) = (F - k)^+ at expiration and has value c = E[(F - k)^+]. Note (F - k)^+ - (k - F)^+ = F - k is a forward with strike k so all models satisfy put-call parity: c - p = f - k. Call delta is ∂_f c = ∂_f p + 1 and call gamma equals put gamma ∂_f^2 c = ∂_f^2 p. We also have ∂_s c - ∂_s p = 0 because forwards are independent of vol so call vega equals put vega.

The value of a put is p = E[(k - F)^+] = kP(F\le k) - f P_s(F\le k).

Put delta is ∂_f p = E[-1(F\le k)ε_s(X)] = -P_s(F\le k).

Gamma for either a put or call is ∂_f^2 p = E[δ_k(F)ε_s(X)^2] = E_s[δ_k(F)ε_s(X)] where δ_k is a point mass at k.

Vega for a put or call is ∂_s p = E[-1(F\le k) F (X - κ'(s))] = -f E_s[1(F\le k) (X - κ'(s))].

Black Model

In the Black modes F = fe^{sX - s^2/2} where X is standard normal. Recall if X is standard normal then E[g(X) e^{sX}] = e^{s^2/2}E[g(X + s)]. Using g(x) = 1 we see κ(s) = s^2/2. Using g(X) = 1(X\le x) we get Φ_s(x) = P(X + s \le x) = Φ(x - s) and ∂Φ_s(x)/∂s = -φ(x - s) = -φ_s(x).

Put value is p = k Φ(x(k)) - f Φ(x(k) - s), where x(k) = \log(k/f)/s + s/2.

Put delta is ∂_f p = -Φ_s(x(k)) = -Φ(x(k) - s). Gamma is ∂_f^2 p = φ_s(x(k))/fs = φ(x(k) - s)/fs Vega is ∂_s v = -f ∂_s Φ_s(x(k)) = fφ_s(x(k)) = fφ(x(k) - s).

Digital

A digital put has payoff ν(F) = 1(F \le k) and a digital call has payoff ν(F) = 1(F > k). Since 1(F \le k) + 1(F > k) = 1 we have digital put-call parity p + c = 1 where p is the digital put value and c is the digital call value: p = P(F \le k), c = P(F > k) = 1 - p.

Digital put delta is ∂_f p = -E[δ_k(F)ε_s(X)] = -E_s[δ_k(F)]

Digital gamma is ∂_f^2 p = E[δ'_k(F)ε_s(X)^2] = E_s[δ'_k(F) ε_s(X)].

Digital vega is ∂_s p = -E[\delta_k(F)F(X - s)] = -f E_s[δ_k(F) (X - κ(s))].

Parameters

The Black-Scholes/Merton values and greeks can be calculated in terms of the parameters f and s using the chain rule. For example, the Black model takes X to be standard normal and vol s = σ \sqrt{t} where σ is the standard Black volatilty and t is time in years to expiration. In this case standard vega is ∂_σ E[ν(F)] = ∂_s E[ν(F)] ∂_σ s = ∂_s E[ν(F)]\sqrt{t}.

The Black-Merton/Scholes model uses spot prices instead of forward. If a risk-free bond has realized return R = e^{rt} over the period, the value of the underlying at expiration is U = Rue^{sX - κ(s)}. Since F = U we have f = Ru. The spot value of the option is v_0 = E[ν(U)]/R. We have ∂_u v_0 = E[ν'(U) ∂_u U]/R = E[ν'(F) ∂_f F ∂_u f]/R = E[ν'(F) ∂_f F R]/R = ∂_f v. Spot and forward delta are equal but the spot gamma is ∂_u^2 v_0 = ∂_u ∂_f v = ∂_f^2 v ∂_u f = ∂_f^2 v R. Spot vega is ∂_s v_0 = E[ν'(U) ∂_s U]/R = E[ν'(F) ∂_s F/R = ∂_s v/R.