Quasi-Monte Carlo Methods in Finance with Application to Optimal Asset Allocation

Diplomarbeit von Mario Rometsch

Introduction:

Portfolio optimization is a widely studied problem in finance. The common question is, how a small investor should invest his wealth in the market to attain certain goals, like a desired payoff or some insurance against unwished events.

The starting point for the mathematical treatment of this is the work of Harry Markowitz in the 1950s. His idea was to set up a relation between the mean return of a portfolio and its variance. In his terminology, an efficient portfolio has minimal variance of return among others with the same mean rate of return. Furthermore, if linear combinations of efficient portfolios and a riskless asset are allowed, this leads to the market portfolio, so that a linear combination of the risk-free asset and the market portfolio dominates any other portfolio in the mean-variance sense.

Later, this theory was extended resulting in the CAPM, or capital asset pricing model, which was independently introduced by Treynor, Sharpe, Lintner and Mossin in the 1960s. In this model, every risky asset has a mean rate of return that exceeds the risk-free rate by a specific risk premium, which depends on a certain attribute of the asset, namely its _. The so-called _ in turn is the covariance of the asset return normalized by the variance of the market portfolio. The problem of the CAPM is its static nature, investments are made once and then the state of the model changes. Due to this and other simplifications, this model was and is often not found to be realistic.

An impact to this research field were the two papers of Robert Merton in 1969 and 1971. He applied the theory of Ito calculus and stochastic optimal control and solved the corresponding Hamilton-Jacobi-Bellman equation. For his multiperiod model, he assumed constant coefficients and an investor with power utility. Extending the mean-variance analysis, he found that a long-term investor would prefer a portfolio that includes hedging components to protect against fluctuations in the market. Again this approach was generalized by numerous researchers and results in the problem of solving a nonlinear partial differential equation.

The next milestone in this series is the work by Cox and Huang from 1989, where they solve for ‘Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process’. They apply the martingale technique to get rid of the nonlinear PDE and rather solve a linear PDE. This, with several refinements, is nowadays a standard method for asset allocation in a complete market.

Approximately at the same time in 1991, Ocone and Karatzas published two pav pers, and, in which they extend the Clark-Haussmann representation formula to Wiener functionals under an equivalent measure. They use the martingale method for selecting the optimal terminal wealth resp. optimal consumption and then apply the representation theorem to derive the optimal portfolio strategy. This expression involves expectations of random variables depending on the interest rate, the market prices of risk and unspecified derivatives of these state variables.

Because of this unspecification, the hedging terms of the resulting portfolio do not have an explicit form and are pretty difficult to evaluate numerically. Therefore, the recent literature focused more on models with state variable specifications, for which closed-form solutions are available, for example, or on specifications which, as mentioned above, rely on dynamic programming.

One main part of this thesis is now to present the latest work of Detemple, Garcia and Rindisbacher, and, in which they tie up the ideas from Ocone and Karatzas. They solve for optimal portfolio allocation in a complete market model and derive explicit expressions for the hedging terms. The optimal portfolio shares are found to be expectations of random variables, which allow a simulation-based approach.

In opposite to the dynamic programming approach, this method is capable of handling complex, nonlinear dynamics for a large number of state variables.

When then the optimal portfolio shares need to be calculated, this basically reduces to a high-dimensional integration over the unit hypercube. Perhaps because of its easy and straightforward implementation, this is most often done with Monte Carlo methods. Niederreiter notes that the official emergence of the Monte Carlo method was a paper of Nicholas Metropolis and Stanislaw Ulam in 1949, but at that time, pseudo-random algorithms had been used for years in secret projects of the U.S.

Department of Defense, like the Manhattan project. Since then, these methods have become the most widely used tools for computing high-dimensional integrals. Although they have appealing features like weak regularity conditions, all these Monte Carlo or pseudo-random methods suffer from one severe shortcoming: their convergence order stems from the central limit theorem, hence they converge very slowly.

The quasi-Monte Carlo method now tries to overcome this hitch while keeping the applicability to high-dimensional integration at the same time. This is achieved by sampling from very careful chosen deterministic points. The methods and algorithms then are not called pseudo-random but quasi-random. The term ‘quasi-Monte Carlo’ method appeared the first time in a technical report from 1951, and Niederreiters outstanding monograph cites many works and applications to problems like the numerical solution of integral equations or integro-differential equations. Quasi-Monte Carlo applications to problems from the finance setting came up in the nineties with work from Paskov, L´Ecuyer et al. This topic composes the second part of this work, in which we will present some concepts from the quasi-Monte Carlo theory to problems that emerge in mathematical finance.

In this thesis, we derive the optimal portfolio formula on the basis of the proofs in resp. The main contribution is then the application of some quasi- Monte Carlo methods for its computation. Using a simple model with exact solution, different schemes were tested and showed, that the substitution of the pseudo-random number generator for the quasi-random number generator requires special care.

This thesis is organized as follows. In Chapter 1, we present some concepts from the quasi-Monte Carlo theory and show, how these methods can be applied to price a simple financial derivative, an arithmetic call option. In Chapter 2 we follow the lecture notes and present an introduction to the ‘stochastic calculus of variations’ or Malliavin Calculus, which will culminate in the definition of the Malliavin derivative and the derivation of the Clark-Ocone formula.

The reader wishing a quick start to the asset allocation problem may thus jump directly to Chapter 3. There, we will introduce the financial market model and formulate the optimization problem. We will follow the work of Detemple, Garcia and Rindisbacher resp. and derive the optimal portfolio formula. However, for this we will need the calculation rules from Chapter 2, so if skipped, the optimal portfolio formula has to be taken as granted. The implementation in Chapter 4 will then apply these formulas to a simple model with explicit solution, and we show, how the quasi- Monte Carlo methods from Chapter 1 can be used to improve the efficiency over plain Monte Carlo methods.

Contents:

Chapter 2 Malliavin Calculus In this chapter, we will now introduce the theory of stochastic calculus of variations. We will follow the lecture notes [Øks97] in this chapter. The intention will be to define the Malliavin derivative, to derive the Clark-Ocone-Haussmann-formula and to familiarize with these instruments. A more general but also more abstract approach can be found in the book [Nua06] or in the ebook [¨ Us04]1.

A starting point is the orthogonal expansion of square-integrable, measurable random variables in terms of iterated Itˆo integrals, that we will study now.