Numerical option pricing in the presence of bubbles

Abstract

Abstract. For the standard Black-Scholes equation, there is a unique solution of at most polynomial growth, towards which any reasonable numerical scheme will converge. However, there are financial models for which this uniqueness does not hold, for instance in the case of models for financial bubbles and certain stochastic volatility models. We present a numerical scheme to find the solution corresponding to the option price given by the risk-neutral expectation in the presence of bubbles.

On optimal arbitrage

“…Can an iterative method be constructed which converges to the minimal solution of the parabolic differential inequality (1.1), U (0, ·) ≡ 1 and is numerically implementable [possibly as in Ekström, Von Sydow and Tysk (2008)]? How about a Monte Carlo scheme that computes the quantity U (T, x) of (9.22) by generating the paths of the diffusion process Y(·), then simulating the probability Q x [T > T ] that Y(·) does not hit the boundary of the nonnegative orthant by time T , when started at Y(0) = x ∈ (0, ∞) n ?…”

Strict local martingale deflators and valuing American call-type options

“…First one needs to find the solution of (2.4) with strictly sublinear growth in its first variable (which is unique).Subtracting this value from the stock price gives the American option price. The approximation method described by Theorem 2.2 of[7] can also be used to compute e. (The idea is to approximate e by a sequence of functions that are unique solutions of Cauchy problems on bounded domains.) Thanks to fact that e is of strictly sublinear growth, (2.3) gives lim x→∞ a(x, t)/x = 1…”

Dupire's Equation for Bubbles

“…The fact that C satisfies the given boundary conditions also follows from the put-call parity (12) and the boundary behaviour of P . Finally, the proof of the uniqueness of the solutions to equation (6) within the given class also shows uniqueness of solutions with a bounded difference to e −rT K. This translates directly to uniqueness for equation (5) for bounded functions. This finishes the proof of Theorem 2.2.…”

“…However, it follows directly from (3) and (4) that the functions C and P are convex in the strike price K. Thus we note thatP does not coincide with P for models with bubbles, so there is no uniqueness of solutions to equation (6) in the class of functions of at most linear growth.…”

“…Financial bubbles have been studied extensively over the last few years, see for instance [4], [6], [7], [9], [11] and [12]. It has been suggested to use models in which the underlying discounted price process is a strict local martingale under the pricing measure.…”