*Peter Carr, New York University and Bloomberg Quantitative Finance Research
- When: Saturday, April 16, 9:00 - 10:00 am
- Title: Robust Replication of Volatility Derivatives
- Abstract: Variance swaps now trade liquidly over the counter (OTC) on several stocks and stock indices. The rise of this market lead to the re-definition of the VIX as a synthetic variance swap. Options on realized variance are now trading OTC and options on VIX will soon be listed. This talk reviews the theory of replicating variance swaps and presents a new theory for valuing any function of realized volatility.
- When: Friday, April 15, 5:00 - 6:00 pm
- Title: Rare Events in Portfolio Credit Risk
- Abstract: The measurement of portfolio credit risk focuses on rare but significant large-loss events. We investigate rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasingly high loss thresholds; the second limiting regime is based on letting the individual loss probabilities decrease toward zero. Both limits are also based on letting the size of the portfolio increase. Our analysis reveals a qualitative distinction between the two cases: in the rare-default regime, the tail of the loss distribution decreases exponentially, but in the large-threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits. We discuss applications of these ideas to efficient credit risk simulation through importance sampling. This is joint work with Wanmo Kang and Perwez Shahabuddin.
- When: Friday, April 15, 2:30 - 3:30 pm
- Title: The determination of time-varying volatility: A tale of two time scales
- Abstract: It is a common financial practice to estimate volatility from the sum of frequently sampled squared returns. However market microstructure poses challenges to this estimation approach. This work attempts to reconcile continuous-time modeling and discrete-time samples. We propose an estimation approach that takes advantage of the rich sources in tick-by-tick data while preserving the continuous-time assumption on the underlying returns. Under our framework, it becomes clear why and where then "usual'' volatility estimator fails when the returns are sampled at the highest frequency.
- When: Friday, April 15, 3:30 - 4:00 pm
- Title: Selection of an Optimal Portfolio with Stochastic Volatility and Discrete Observations
- Abstract: The optimal selection problem for a portfolio of stock and risk-free asset in the framework of stochastic optimal control of diffusion processes is considered. We extend the classical Black-Scholes model by modeling the volatility as another stochastic process, and assume that the portfolio manager has only discrete access to the continuous-time stock prices. We implement a particle-filtering and Monte-Carlo-type algorithm that optimizes the expected future utility dynamically, under these assumptions of incomplete information and stochastic volatility.
- When: Saturday, April 15, 10:00 - 10:30 am
- Title: Assessing Credit with Equity: A CEV (Constant Elasticity of Variance) Model with Jump to Default
- Abstract: Unlike in structural and reduced-form models, we use equity as a liquid and observable primitive to analytically value corporate bonds and credit default swaps. Restrictive assumptions on the firm's capital structure are avoided. Default is parsimoniously represented by equity value hitting the zero barrier either diffusively or with a jump, which implies non-zero credit spreads for short maturities. Easy cross-asset hedging is enabled. By means of a tersely specified pricing kernel, we also make analytic credit-risk management possible under systematic jump-to-default risk.
- When: Friday, April 15, 4:30 - 5:00 pm
- Title: Sparse PCA (Principal Component Analysis) with applications in finance
- Abstract: We examine the problem of approximating, in the Frobenius-norm sense, a positive, semidefinite symmetric matrix by a rank-one matrix, with an upper bound on the cardinality of its eigenvector. This problem arises in the decomposition of a covariance matrix into sparse factors (i.e. sparse Principal Component Analysis). We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation for our problem. As the penalty in this relaxation corresponds to the L1 norm of the eigenvector, this can interpreted as a variance maximization penalized by proportional transaction costs and we detail application to portfolio hedging with fixed and proportional transaction costs.
- When: Saturday, April 16, 11:00 - 11:30 am
- Title: Portfolio Management for a cost-constrained investor
- Abstract: We examine a two-asset portfolio optimization problem where the model for the stock price can have the features of bull or bear markets, and in which there are transaction costs that might inhibit the investor from trading too often. Only partial observations are allowed. The investment is over a fixed period of time and the power utility function is used.
- When: Saturday, April 16, 11:30 am - 12:00 pm
- Title: A simulation based optimization scheme for large portfolios with transaction costs
- Abstract: Portfolio optimization problems in the presence of proportional
transaction costs belong to the class of Stochastic Control problem with singular controls. Solutions to
such problems are sought by first arguing that it is equivalent to solving a related partial differential
equation known as the Hamilton Jacobi Bellman equation (HJB). The arising HJB equation for this class of
problems is of the "free boundary" type, that is, the boundaries of the region in which the HJB is to be
solved is not pre-specified and has to be obtained as a part of the solution itself.
Moreover, the dimensionality of the HJB equation is usually equal to the number of stocks in the portfolio.
Runtimes of existing solution methods grow super exponentially with dimension - making them unsuitable even for problems with 4 stocks.
In this talk we will consider a portfolio optimization problem - that of maximizing long-term rate of growth of wealth - and describe a scheme that scales polynomially in dimension. This scheme transforms the free boundary problem to a converging sequence of fixed boundary problems and estimates the solution to each of the fixed boundary problem using simulation.
