Convergence of Binomial Large Investor Models and General Correlated Random Walks

Abstract

This thesis studies the problem of option pricing via replication by a large investor whose trading affects the stock price. We formulate and solve this question first in a binomial setting. Then we consider a suitably scaled sequence of such binomial large investor models and prove their convergence towards a continuous-time diffusion. This requires that we analyze carefully both the convergence of the large investor’s strategy functions and the stochastic process of the underlying fundamentals. The convergence of the latter is derived from a new convergence result for general correlated random walks. In each single time step, we model the stock price as a function of time, some fundamentals and the large investor’s stock holdings, and we assume that the fundamentals describe a random walk. We analyze in detail the price mechanism which models how the large investor’s trades affect prices and elaborate on the importance of a “fair” price system as a theoretical benchmark. This can be used to define implicit transaction losses and the real value of a large investor’s portfolio. We derive conditions which prevent paper-value and real-value arbitrage opportunities for the large investor and show the existence and uniqueness of a replication strategy for a given contingent claim. As a consequence of its feedback on the stock price, this strategy is in general only given implicitly by a fixed point theorem. To study the convergence of a sequence of binomial large investor models, we rescale the fundamentals as in Donsker’s theorem. In a first step, we then show that the convergence of the large investor’s strategy functions is implied by their convergence at maturity. The limit function is identified as the solution of a non-linear final value problem. By a suitable strategy transform, this can be simplified to a perturbation of a linear problem in a “fair” market. We then prove the convergence in distribution of the binomial large investor models under two different regimes of martingale measures. Because the transition probabilities for the fundamentals under these measures typically depend on the large investor’s stock holdings before and after his trade, we have to extend classical convergence results to a setting with general correlated random walks. For general correlated random walks, the direction of the next move depends on time, the current position and the direction of the previous move. Using Donsker’s scaling, we prove the convergence in distribution of a sequence of such walks towards a diffusion limit, and we explicitly identify the diffusion coefficients. It turns out that in comparison to the classical case, both volatility and drift are reinforced due to the correlation between the increments of the discrete walks. In particular, we obtain a convergence result for existing large investor models from the literature. Moreover, our study highlights the importance and influence of the choice of price mechanism.