Andrew Papanicolaou

A Regime-Switching Heston Model for VIX and S&P 500 Implied Volatilities

Volatility products have become popular in the past 15 years as a hedge against market uncertainty. In particular, there is growing interest in options on the VIX volatility index. A number of recent empirical studies examine whether there is significantly greater risk premium in VIX option prices compared with S&P 500 option prices. We address this issue by proposing and analyzing a stochastic volatility model with regime switching. The basic Heston model cannot capture VIX implied volatilities, as has been documented. We show that the incorporation sharp regime shifts can bridge this shortcoming. We take advantage of Fourier methods to make the extension tractable, and we present a fit to data, both in times of crisis and relative calm, which shows the effectiveness of the regime switching.

Filtering and Portfolio Optimization with Stochastic Unobserved Drift in Asset Returns

We consider the problem of filtering and control in the setting of portfolio optimization in financial markets with random factors that are not directly observable. The example that we present is a commodities portfolio where yields on futures contracts are observed with some noise. Through the use of perturbation methods, we are able to show that the solution to the full problem can be approximated by the solution of a solvable HJB equation plus an explicit correction term.

Filtering the Maximum Likelihood for Multiscale Problems

Filtering and parameter estimation under partial information for multiscale diffusion problems are studied in this paper. The nonlinear filter converges in the mean-square sense to a filter of reduced dimension. Based on this result, we establish that the conditional (on the observations) log-likelihood process has a correction term given by a type of central limit theorem. We prove that an appropriate normalization of the log-likelihood minus a log-likelihood of reduced dimension converges weakly to a normal distribution. In order to achieve this we assume that the operator of the (hidden) fast process has a discrete spectrum and an orthonormal basis of eigenfunctions. We then propose to estimate the unknown model parameters using the reduced log-likelihood, which is beneficial because reduced dimension means that there is significantly less runtime for this optimization program. We also establish consistency and asymptotic normality of the maximum likelihood estimator. Simulation results illustrate our ...

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.

Pairs Trading of Two Assets with Uncertainty in Co-Integration's Level of Mean Reversion

This paper considers a stochastic control problem derived from a model for pairs trading under incomplete information. We decompose an individual assets drift into two parts: an industry drift plus some additional stochasticity. The extra stochasticity may be unobserved, which means the investor has only partial information. We solve the control problem under both full and partial informations for utility function U(x) = x1−γ/(1 − γ), and we make comparisons. We show the existence of stable solution to the associated matrix Riccati equations in both cases for γ > 1, but for 0 < γ < 1 there remains potential for infinite value functions in finite time. Also, we quantify the expected loss in utility due to partial information, and present a numerical study to illustrate the contribution of this paper.

Dimension Reduction in Discrete Time Portfolio Optimization with Partial Information

This paper considers the problem of portfolio optimization in a market with partial information and discretely observed price processes. Partial information refers to the setting where assets have unobserved factors in the rate of return and the level of volatility. Standard filtering techniques are used to compute the posterior distribution of the hidden variables, but there is difficulty in finding the optimal portfolio because the dynamic programming problem is non-Markovian. However, fast time scale asymptotics can be exploited to obtain an approximate dynamic program (ADP) that is Markovian and is therefore much easier to compute. Of consideration is a model where the latent variables (also referred to as hidden states) have fast mean reversion to an invariant distribution that is parameterized by a Markov chain

Extreme-Strike Comparisons and Structural Bounds for SPX and VIX Options

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Singular Perturbation Expansion for Utility Maximization with Order-ε Linear Price Impact

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Analysis of VIX Markets with a Time-Spread Portfolio

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Numerical Solution of Stochastic Differential Equations with Jumps in Finance

represents the regime-state of the market and reverts to its own invariant distribution over a much longer time scale. Data and numerical examples are also presen...