Jean-Guy Simonato

Pricing Discretely Monitored Barrier Options by a Markov Chain

Barrier options have become commonplace in the option market, and a variety of other financial contracts may also be thought of in terms of barrier options. But the existence of a price barrier can significantly complicate the option valuation problem when volatility is time-varying, or the barrier itself moves over time, or the barrier is only monitored at discrete intervals. In this article, Duan et al. present a new Markov chain technology for pricing barrier options that readily handles all of these problems. Out-and-in options can be valued within their framework even when volatility follows a GARCH process and a discretely monitored time-varying barrier is present.

Approximating the GJR-GARCH and EGARCH option pricing models analytically

In Duan, Gauthier and Simonato (1999), an analytical approximate formula for European options in the GARCH framework was developed. The formula is however restricted to the nonlinear asymmetric GARCH model. This paper extends the same approach to two other important GARCH specifications GJR-GARCH and EGARCH. We provide the corresponding formulas and study their numerical performance. keywords: Option pricing, EGARCH, GJR-GARCH, analytical approximation Duan is with Rotman School of Management, University of Toronto; Gauthier and Simonato are with HEC Montreal; Sasseville is a Ph.D. candidate at the Kellog Graduate Business School. Duan, Gauthier and Simonato acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), Les Fonds pour la Formation de Chercheurs et l’Aide a la Recherche du Quebec (FCAR) and from the Social Sciences and Humanities Research Council of Canada (SSHRC). Duan also acknowledges support received as the Manulife Chair in Financial Services.

A simulation-and-regression approach for stochastic dynamic programs with endogenous state variables

We investigate the optimum control of a stochastic system, in the presence of both exogenous (control-independent) stochastic state variables and endogenous (control-dependent) state variables. Our solution approach relies on simulations and regressions with respect to the state variables, but also grafts the endogenous state variable into the simulation paths. That is, unlike most other simulation approaches found in the literature, no discretization of the endogenous variable is required. The approach is meant to handle several stochastic variables, offers a high level of flexibility in their modeling, and should be at its best in non time-homogenous cases, when the optimal policy structure changes with time. We provide numerical results for a dam-based hydropower application, where the exogenous variable is the stochastic spot price of power, and the endogenous variable is the water level in the reservoir.

A Markov Chain Method for Pricing Contingent Claims

This chapter introduces the use of a time-homogenous Markov chain for the valuation of options. In the computational finance literature, three categories of numerical techniques have been explored extensively for the valuation of contingent claims. The first category involves the use of a lattice structure to approximate the price movement of the underlying asset under the risk neutral probability measure and then computes the price of a contingent claim as a discounted expected payoff. The lattice approach essentially discretizes both time and state in a particular way. The second category is the finite difference/element approach. This technique numerically solves the partial differential equation that the value function of a contingent claim must obey under the no-arbitrage condition. The third category is the Monte Carlo method, which simulates the system under the risk-neutral probability measure so that the expectation of a contingent payoff can be approximated by the sample average.

Dynamic portfolio choices by simulation-and-regression

Simulation-and-regression methods have been recently proposed to solve multi-period, dynamic portfolio choice problems. In the constant relative risk aversion (CRRA) framework, the value function recursion vs portfolio weight recursion issue was previously examined in van Binsbergen and Brandt [24] and Garlappi and Skoulakis [14]. We revisit this issue in the context of an alternative simulation-and-regression algorithmic approach which does not rely on Taylor series approximations of the value function. We find that, in this context and for the CRRA example examined here, both approach are capable of obtaining precise results, but that the portfolio weight recursion variant of the algorithm provides more accurate results for a similar level of computational complexity, especially for problems with long maturities and large risk-aversion levels. HighlightsDynamic portfolio choices are computed with a simulation-and-regression approach.Unlike the current literature, our approach does not use Taylor series.We examine and compare two computational alternatives within this framework.We find that both alternatives can achieve precise results, one being more robust.

New Warrant Issues Valuation with Leverage and Noisy Equity Values

The empirical analysis of new warrant issues in the context of a structural model of the firm typically assumes the absence of debt and a perfect equity pricing model. We examine here an approach relaxing these two assumptions. The proposed approach develops simple analytical expressions for the prices of warrant, debt and equity in the presence of leverage. An empirical strategy is proposed to implement the model with equity prices containing model errors. An illustration with a recent warrant issue deal between Bank of America and Berkshire Athaway is provided.

Approximating the Multivariate Distribution of Time-Aggregated Stock Returns Under GARCH

An approach to approximate the multivariate distribution of time-aggregated stock returns in the GARCH context is developed here. The approach yields a one time-step simulation procedure as opposed to a multiple time-step simulation required in such a context. For this purpose, the exact moment formulas for the time-aggregated return under a QGARCH process are combined with multivariate non-normal simulation procedures using as inputs, the first four moments and correlation structure of the unknown target distribution. Estimation and simulation results are presented for a portfolio of 30 stocks from the Dow Jones Industrial Average index. The results reveal that the proposed simulation method can generate random numbers with moments and correlations agreeing with the targets. Using value at risk computations for different horizons and probabilities, we show that the percentiles of portfolios return distributions computed with the proposed approach provide good approximations of benchmark values obtained from a multi-step simulation.