Seminar: Spring 2005

Sam Howison (Oxford, UK)

Standard Black-Scholes theory for barrier and American options assumes
that the barrier is sampled continuously in time, or that the option is
exercisable at any time. In practice, however, many contracts are only
sampled (resp. exercisable) at a discrete set of sampling dates. When the
number of reset dates is large, one can ask how to find approximations to
the discrete price in terms of the continuous one (or vice versa). This was
addressed for barrier options by Broadie, Glasserman and Kou using
probabilistic techniques. I shall revisit this problem using matched
asymptotic expansions, showing how to calculate the BGK correction to
higher order and how to extend it to more general models. I shall then deal
with American/Bermudan options in the same framework, showing that the
correction is an order of magnitude smaller than in the barrier case.
Mathematical topics involved include the Wiener-Hopf method, renewal theory
and (incidentally) Riemann's zeta function. A short introduction to matched
asymptotic expansions in the context of vanilla call options near
expiration will be included.

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Dmitry Kramkov (Carnegie Mellon)

The principal qualitative difference between complete and incomplete financial models is that in the latter case the unit prices of contingent claims depend, in general, on the amounts of the options we are going to buy or sell. In other words, in incomplete financial models the value of the portfolio of derivative securities changes non-linearly with their quantities.

As one can expect, the precise computations in incomplete models are really possible and, hence, some asymptotic methods are needed. In this paper we present the general methodology that allows us to compute the first-order asymptotic expansion of the unit prices of the contingent claims with respect to their quantities. Among other results we show that this first-order asymptotic expansion has some important qualitative properties for any risk-neutral economic agent if and only if there is unique maximal martingale measure from the point of view of second order stochastic dominance.

The presentation is based on a joint project with Mihai Sirbu from Columbia University.
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Rene Carmona (Princeton University)

Motivated by indifference pricing challenges, we compute the value functions (max-expected utilities of terminal wealth) for non-Markovian market models when the portfolios include derivatives written on non-tradable assets. Using standard filtering techniques, we extend the results to models including non-observable factors. We apply the results to examples from the weather and energy markets.
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Paul Glasserman (Columbia University)

The distribution of losses due to defaults in large portfolios is often computed using Monte Carlo simulation, but obtaining precise Monte Carlo estimates for small probabilities of large losses can be time consuming. Importance sampling (IS) is a general technique for improving the performance of Monte Carlo methods in estimating rare-event probabilities. The application of IS to credit risk is complicated by the mechanisms commonly used to specify the dependence between default events, particularly in the industry-standard Gaussian copula model. We present a two-step approach to IS for credit losses that takes advantage of the "factor" structure often used in credit models: we apply IS conditional on the factors and then apply IS to the factors themselves. We analyze the effectiveness of the method through asymptotics in the size of portfolio and give conditions for asymptotic optimality. We also develop IS methods for conditional expectations used to measure marginal risk contributions. This is based in part on joint work with Jingyi Li and separate work with Wanmo Kang and Perwez Shahabuddin.
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Marco Avellaneda (Courant Institute of Mathematical Sciences, NYU)

We propose a model to describe stock pinning on option expiration dates. We argue that if the open interest in a particular contract is unusually large, Delta-hedging in aggregate by floor market-makers can impact the stock price and drive it to the strike price of the option. We derive a stochastic differential equation for the stock price which has a singular drift that accounts for the price-impact of Delta-hedging. According to this model, the stock price has a finite probability of pinning at a strike. We calculate analytically and numerically this probability in terms of the volatility of the stock, the time-to-maturity, the open interest for the option under consideration and a "price-elasticity'' constant that models price impact. We also discuss recent empirical evidence, from Ni, Pearson and Poteshman (2004), which corroborates our theory.
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Lisa Goldberg (MSCI Barra, Inc)

We examine multi-name credit models from the perspective of point processes. In this context, it is natural to pursue a top down approach: the economy as a whole is modeled first. The technique of random thinning consistently generates sub-models for individual firms or portfolios.

A candidate for the top down approach is a self-exciting process, whose intensity at any time depends on the sequence of events observed up to that time. A self-exciting process incorporates the contagion observed in credit markets and avoids an ad hoc choice of copula. The familiar doubly stochastic process is at the opposite end of the spectrum in the sense that it is constructed from the bottom up: individual firm intensities are estimated and then aggregated. We rigorously analyze self-exciting and doubly stochastic processes with respect to their ability to capture contagion.

Model fitness can be tested using a deep result of Meyer (1971), which shows that any point process with continuous compensator can be transformed into a standard Poisson process by a change of time. Meyer's result allows us to extend the scope of the tests proposed by Das, Duffie & Kapadia (2004) for a doubly stochastic model.

joint with Kay Giesecke
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Ken Singleton (Stanford GSB)

This paper explores in depth the nature of the risk-neutral credit-event intensities that best describe the term structures of sovereign CDS spreads. We examine three distinct families of stochastic processes: the square-root, lognormal, and three-halves processes. These models employ different specifications of mean reversions and time-varying volatilities to fit both the distributions of spreads, and the variation over time in the shapes of the term structures of spreads. We find that these models imply very different risk-neutral probabilities that a sovereign issuer will survive over various future horizons. We also explore the use of the term structure of CDS spreads to separately identify both the loss rate in the event of default and the parameters of risk-neutral intensity process. Finally, we to attempt to shed some light on the magnitudes of default-event risk premia in sovereign markets.

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Boris Rozovsky (Center for Applied Mathematical Sciences, USC)

This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process is given under the risk neutral measure by an Ito process where interest rate and volatility are functions of an unobservable cadlag homogeneous strong Markov process. We assume also that the asset price is observed only discretely, at random times. This is an appropriate assumption when modeling high frequency financial data (e.g., tick-by-tick stock prices).

In the above setting the problem of estimation of the unobservable Markov process can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process. While quite natural, this problem does not fit into the ''standard'' diffusion or simple point process filtering frameworks and requires more technical tools. We derive the optimal recursive Bayesian filter in closed form, based on the observations of the process at random times. It turns out that the filter is given by a recursive system of deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy.

Our result establishes a novel separation principle: filtering at an observation time depends only on the posterior distribution at the previous observation time, and the filtering between the observation times is determined by the posterior distribution at the beginning of the observation interval, and a priori parameters.
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