The Quantum Black-scholes Equation

Motivated by the work of Segal and Segal in [16] on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the BlackScholes equation within the context of Hudson-Parthasarathy quantum stochastic calculus,. Our model includes stock markets described by quantum Brownian motion and Poisson process. 1. The Merton-Black-Scholes Option Pricing Model An option is a ticket which is bought at time t = 0 and which allows the buyer at (in the case of European call options) or until (in the case of American call options) time t = T (the time of maturity of the option) to buy a share of stock at a fixed exercise price K. In what follows we restrict to European call options. The question is: how much should one be willing to pay to buy such an option? Let XT be a reasonable price. According to the definition given by Merton, Black, and Scholes (M-B-S) an investment of this reasonable price in a mixed portfolio (i.e part is invested in stock and part in bond) at time t = 0, should allow the investor through a self-financing strategy (i.e one where the only change in the investor’s wealth comes from changes of the prices of the stock and bond) to end up at time t = T with an amount of (XT − K) := max(0, XT − K) which is the same as the payoff, had the option been purchased (cf. [12]). Moreover, such a reasonable price allows for no arbitrage i.e, it does not allow for risk free unbounded profits. We assume that there are no transaction costs and that the portfolio is not made smaller by consumption. If (at, bt), t ∈ [0, T ] is a self -financing trading strategy (i.e an amount at is invested in stock at time t and an amount bt is invested in bond at the same time) then the value of the portfolio at time t is given by Vt = atXt + bt βt where, by the self-financing assumption, dVt = at dXt+bt dβt. Here Xt and βt denote, respectively, the price of the stock and bond at time t. We assume that dXt = cXt dt+ σ Xt dBt and dβt = βt r dt where Bt is classical Brownian motion, r > 0 is the constant interest rate of the bond, c > 0 is the mean rate of return, and σ > 0 is the volatility of the stock. The assets at and bt are in general stochastic processes. Letting Vt = u(T − t, Xt) where VT = u(0, XT ) = (XT −K)+ it can be shown (cf. [12]) that u(t, x) is the solution of the Black-Scholes equation Date: December 3, 2008. 1991 Mathematics Subject Classification. 81S25, 91B70. 1 2 LUIGI ACCARDI AND ANDREAS BOUKAS ∂ ∂t u(t, x) = (0.5 σ x ∂ ∂x2 + r x ∂ ∂x − r) u(t, x) u(0, x) = (XT −K), x > 0, t ∈ [0, T ] and it is explicitly given by u(t, x) = xΦ(g(t, x))−K e Φ(h(t, x)) where g(t, x) = (ln(x/K) + (r + 0.5 σ) t)(σ √ t) −1 , h(t, x) = g(t, x)− σ √ t and