Bachelier Model

Keith A. Lewis

January 26, 2025

Abstract

Normal stock price

The Bachelier model assumes stock price has the form F_t = f + \sigma B_t where f and \sigma are constants and B_t is standard Brownian motion, assuming zero interest rate.

The risk-neutral value of a put with strike k expiring at t is E[(k - F_t)^+] = E[(k - F)1(F\le k)] = E[k - F]P(F \le k) + \operatorname{Cov}(k - F, 1(F\le k)).

Exercise. Show if M and N are jointly normal then \operatorname{Cov}(M, g(N)) = \operatorname{Cov}(M, N) E[g'(N)].

Hint: Use E[e^{\alpha N} g(M)] = E[e^{\alpha N}] E[g(M + \operatorname{Cov}(\alpha N, M))], take a derivative with respect to \alpha, then set it equal to 0.

This shows {\operatorname{Cov}(k - F, 1(F\le k)) = \operatorname{Cov}(k - F, F) E[-\delta_k(F)] = \operatorname{Var}(F) E[\delta_k(F)]}.

Exercise Show E[\delta_a(b + cZ)] = \phi((a - b)/c)/c if \phi is the density function of Z.

Solution

If d_a(x) = 1/h for a < x \le a + h and 0 otherwise then {E[\delta_a(b + cZ)] = \lim_{h\to 0} E[d_a(b + c Z)].} Using y = (b + cz)/c = b/c + z we have \begin{aligned} E[d_a(b + c Z)] &= (1/h)\int_a^{a+h} (b + cz)\phi(z)\,dz \\ &= (1/h)\int_{b/c + a}^{b/c + a + h} cy\phi(y - b/c)\,dy \\ \end{aligned}

If g is differentiable at z = g^{-1}(a) then {E[\delta_a(g(Z))] = \phi(z)/g'(z)} since {g(x) \approx g(z) + g'(z)(x - z)} for {z\approx x}.

We have {E[\delta_k(F)] = E[\delta_k(f + \sigma\sqrt{t}Z)] = \phi(z)/\sigma\sqrt{t}}, where Z is standard normal and {z = (k - f)/\sigma\sqrt{t}} so E[(k - F_t)^+] = (k - F)\Phi(z) + \sigma\sqrt{t}\phi(z) where \Phi is the standard normal cumulative distribution function and \phi = \Phi'.

Exercise. Show E[(F_t - k)^+] = (f - k)\Phi(-z) + \sigma\sqrt{t}\phi(-z).