Convergence Of Minimum Entropy Option Prices For Weakly Converging Incomplete Market Models

We study weak convergence of a sequence of (approximating) asset price modelsSnto a limiting modelS: bothSnandSare multi-dimensional asset price processes with some physical probability measuresPnresp.P, and a natural notion of process convergence is the weak convergence of the induced path probability measures, denoted by (Sn|Pn)resp.(S|P), on the abstract topological space of possible asset price trajectories.For the purpose of no-arbitrage pricing of options or more general derivatives on the model assets, there are two different aspects of this convergence: (i) convergence under the given physical probability measures,(Sn|Pn) → (S|P)and (ii) convergence under suitably chosen equivalent martingale measures (EMM) relevant for pricing derivatives,(Sn|Qn) → (S|Q). A simple example is the convergence of a sequence of discrete-time binomial models to the Black–Scholes model (geometric Brownian motion), where the model markets are complete and hence the choice ofQnresp.Qis unique. This example and the general case of complete limit markets(S|P)have been studied in [1].In contrast we have several choices forQnresp.Qwhen all the model markets are incomplete. A natural choice is the minimum entropy EMM [2, 3], defined as the (unique) EMMRminimizing the relative entropyH(R|P)to the physical measureP, among all EMMs. We prove the following: given that the approximating models converge under the physical measures,(Sn|Pn) → (S|P)— with some mild assumptions onPand on the minimum entropy EMMsRnforSnresp.RforS— entropy number convergence implies weak convergence of the minimum entropy option price processes:H(Rn|Pn) → H(R|P)implies(Sn|Rn) → (S|R). Several rigorous examples illustrate the result; cf. also [4].