Lars Stentoft

Convergence of the Least Squares Monte Carlo Approach to American Option Valuation

In a recent paper, Longstaff and Schwartz (2001) suggest a method to American option valuation based on simulation. The method is termed the Least Squares Monte Carlo (LSM) method, and although it has become widely used, not much is known about the properties of the estimator. This paper corrects this shortcoming using theory from the literature on seminonparametric series estimators. A central part of the LSM method is the approximation of a set of conditional expectation functions. We show that the approximations converge to the true expectation functions under general assumptions in a multiperiod, multidimensional setting. We obtain convergence rates in the two-period, multidimensional case, and we discuss the relation between the optimal rate of convergence and the properties of the conditional expectation. Furthermore, we show that the actual price estimates converge to the true price. This provides the mathematical foundation for the use of the LSM method in derivatives research.

Option Pricing Using Realized Volatility

In the present paper we suggest to model Realized Volatility, an estimate of daily volatility based on high frequency data, as an Inverse Gaussian distributed variable with time varying mean, and we examine the joint properties of Realized Volatility and asset returns. We derive the appropriate dynamics to be used for option pricing purposes in this framework, and we show that our model explains some of the mispricings found when using traditional option pricing models based on interdaily data. We then show explicitly that a Generalized Autoregressive Conditional Heteroskedastic model with Normal Inverse Gaussian distributed innovations is the corresponding benchmark model when only daily data is used. Finally, we perform an empirical analysis using stock options for three large American companies, and we show that in all cases our model performs significantly better than the corresponding benchmark model estimated on return data alone. Hence the paper provides evidence on the value of using high frequency data for option pricing purposes.

Multivariate option pricing with time varying volatility and correlations

In recent years multivariate models for asset returns have received much attention, in particular this is the case for models with time varying volatility. In this paper we consider models of this class and examine their potential when it comes to option pricing. Specifically, we derive the risk neutral dynamics for a general class of multivariate heteroskedastic models, and we provide a feasible way to price options in this framework. Our framework can be used irrespective of the assumed underlying distribution and dynamics, and it nests several important special cases. We provide an application to options on the minimum of two indices. Our results show that not only is correlation important for these options but so is allowing this correlation to be dynamic. Moreover, we show that for the general model exposure to correlation risk carries an important premium, and when this is neglected option prices are estimated with errors. Finally, we show that when neglecting the non-Gaussian features of the data, option prices are also estimated with large errors.

Value Function Approximation or Stopping Time Approximation: A Comparison of Two Recent Numerical Methods for American Option Pricing using Simulation and Regression

In Longstaff and Schwartz (2001) a method for American option pricing using simulation and regression is suggested, and since then the method has rapidly gained importance. However, the idea of using regression and simulation for American option pricing was used at least as early as in Carriere (1996). In the present paper we provide a thorough comparison of these two methods and relate them to the work of Tsitsiklis and Van Roy (2001). Although the methods are often considered to be similar this analysis allows us to point out an important but often overlooked difference between the methods. The paper further shows that due to this difference it is possible to provide arguments favoring the method suggested in Longstaff and Schwartz (2001). Finally, the paper compares the methods in a realistic numerical setting and shows that the practitioner does well in choosing the method of Longstaff and Schwartz (2001) instead of the methods of Carriere (1996) or Tsitsiklis and Van Roy (2001) for American option pricing.

Seasonality in Economic Models

Seasonality has been a major research area in economics for several decades. The paper asses the recent development in the literature on the treatment of seasonality in economics, and divides it into three interrelated groups. The first group, the Pure Noise Model, consists of methods based on the view that seasonality is noise contaminating the data or more correctly contaminating the information of interest for the economists. The second group, the Time Series Models, treats seasonality as a more integrated part of the modeling strategy, with the choice of model being data driven. The third group, Economic Models of Seasonality, introduces economic theory, i.e. optimizing behavior into the modeling of seasonality.

Option Pricing with Asymmetric Heteroskedastic Normal Mixture Models

This paper uses asymmetric heteroskedastic normal mixture models to fit return data and to price options. The models can be estimated straightforwardly by maximum likelihood, have high statistical fit when used on S&P 500 index return data, and allow for substantial negative skewness and time varying higher order moments of the risk neutral distribution. When forecasting out-of-sample a large set of index options between 1996 and 2009, substantial improvements are found compared to several benchmark models in terms of dollar losses and the ability to explain the smirk in implied volatilities. Overall, the dollar root mean squared error of the best performing benchmark component model is 39% larger than for the mixture model. When considering the recent financial crisis this difference increases to 69%.

Bayesian option pricing using mixed normal heteroskedasticity models

While stochastic volatility models improve on the option pricing error when compared to the Black-Scholes-Merton model, mispricings remain. This paper uses mixed normal heteroskedasticity models to price options. Our model allows for significant negative skewness and time varying higher order moments of the risk neutral distribution. Parameter inference using Gibbs sampling is explained and we detail how to compute risk neutral predictive densities taking into account parameter uncertainty. When forecasting out-of-sample options on the S&P 500 index, substantial improvements are found compared to a benchmark model in terms of dollar losses and the ability to explain the smirk in implied volatilities.

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.

Refining the least squares Monte Carlo method by imposing structure

The least squares Monte Carlo method of Longstaff and Schwartz has become a standard numerical method for option pricing with many potential risk factors. An important choice in the method is the number of regressors to use and using too few or too many regressors leads to biased results. This is so particularly when considering multiple risk factors or when simulation is computationally expensive and hence relatively few paths can be used. In this paper we show that by imposing structure in the regression problem we can improve the method by reducing the bias. This holds across different maturities, for different categories of moneyness and for different types of option payoffs and often leads to significantly increased efficiency.

Measuring Longevity Risk: An Application to the Royal Canadian Mounted Police Pension Plan

An employer that sets up a defined benefit pension plan promises to periodically pay a certain sum to each participant starting at some future date and continuing until death. Although both the future beneficiary and the employer can be asked to finance the plan throughout the beneficiarys career, any shortcoming of funds in the future is often the employers responsibility. It is therefore essential for the employer to be able to predict with a high degree of confidence the total amount that will be required to cover its future pension obligations. Applying mortality forecasting models to the case of the Royal Canadian Mounted Police pension plan, we illustrate the importance of mortality forecasting to value a pension funds actuarial liabilities. As future survival rates are uncertain, pensioners may live longer than expected. We find that such longevity risk represents approximately 2.8 percent of the total liability ascribable to retired pensioners (as measured by the relative value at risk at the 95th percentile) and 2.5 percent of the total liabilities ascribable to current regular contributors. Longevity risk compounds the model risk associated with not knowing what is the true mortality model, and we estimate that model risk represents approximately 3.2 percent of total liabilities. The compounded longevity risk therefore represents almost 6 percent of the pension plans total liabilities.