June 19-20, 2010
The Local Variance Gamma Model , Peter Carr
In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we assume that the risk-neutral process for the underlying futures price is a pure jump Markov martingale and that European option prices are given at a continuum of strikes and at one or more maturities. We show how to construct a time-homogeneous process which meets a single smile and a piecewise time-homogeneous process, which can meet multiple smiles. We also show that our construction leads to partial differential difference equations (PDDE's), which permit both explicit calibration and fast numerical valuation.
When Diversification Fails: The Perils of Endogenous Correlation, Rama Cont, Columbia University,
The optimality of free markets is called into question by the presence of limited liability. Risk distortions induced by limited liability in the private sector are documented. It is noted that risk preferences are biased towards higher volatility, skewness and kurtosis coupled with an incentive to decorrelate assets from liabilities. The consequence is economically poor risk choices that are exaggerated by compensation aligned with stock market values. In such a context we introduce the concept of socially acceptable risks, operationalized by a positive expectation after distortion of the distribution function for risky cash flows.This results in a definition of capital requirements making the risks undertaken acceptable to the wider community. Enforcing such capital requirements can mitigate the perverse risk incentives introduced by limited liability provided that the set of acceptable risks is suitably conservatively defined. A careful, critical and external assessment of capital requirements is therefore essential for the efficient and proper functioning of the private sector.
The Impact of the Federal Reserve's Interest Rate Target Announcement on Stock Prices: A Closer Look at How the Market Impounds New Information, Stephen Figlewski
The Federal Reserve announces its new interest rate target while the stock market is open, at precisely 2:15 P.M. eight times a year. In the Efficient Markets model, information is impounded in prices immediately and accurately as soon as it becomes public knowledge and only the unanticipated portion moves prices. Responding accurately to news requires investors to judge how much other investors have been surprised and how their investment decisions will be affected, so how the market responds to the news generates additional information to be digested and acted upon. This suggests that the full process of returning to equilibrium can not be instantaneous. In this paper, we combine a non-model dependent procedure for extracting the market's risk neutralized probability density over future stock prices from a set of option prices, with a newly available real time options data set, in order to examine the "informational microstructure" of the stock market around Fed funds target announcements.
Double Gamma Stochastic Volatility Discrete Time Model , Ali Hirsa
Using Pseudo-Parabolic and Fractional Equations for Option Pricing in Jump Diffusion Models, Andrey Itkin
In mathematical finance a popular approach for pricing options under some L\'evy model is to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution while numerical solution brings some problems. In this paper we elaborate a new approach on how to transform the PIDE to some class of so-called pseudo-parabolic equations which are known in mathematics but are relatively new for mathematical finance. As an example we discuss several jump-diffusion models which L\'evy measure allows such a transformation. We show that using this technique one can achieve a significant speedup in option pricing as compared with a standard FFT approach.
Dynamic Mortgage Rate Replication and Risk Management of CMOs, Andrew Lesniewski
Modeling mortgage backed securities (MBSs) poses significant challenges for mathematical finance: their risks are multidimensional and involve market risk (interest rates), event risk (prepayment, default), as well as liquidity risk. Models of MBSs rely on the methods of term structure modeling and survival analysis (doubly stochastic processes). In this talk I discuss an option theoretic approach to modeling the key market quantity, namely the mortgage rate, which distills the information contained in the liquidly traded MBSs (TBAs, options on TBAs, as well as CMM forwards and swaps). I apply the results to quantifying and hedging some aspects of the risk of portfolios of complex mortgage backed securities. This is joint work with Zhouhua Li, Jared Samet, and Xiang Xia.
Capital Requirements, Acceptable Risks, Profits and US Bank Reserves, Dilip Madan
Libor Market Models with Stochastic Basis, Fabio Mercurio, Bloomberg LP
We extend the LIBOR market model to accommodate the new market practice of using different forward and discount curves in the pricing of interest-rate derivatives. Our extension is based on modeling the joint evolution of forward rates belonging to the OIS curve and corresponding spreads with FRA rates for different tenors. We consider stochastic-volatility dynamics and address the related caplet and swaption pricing problems.
Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions: Disentangling the Multi-Dimensional Variations in S&P 500 Index Options, Liuren Wu
The equity index and index volatility interact through several distinct channels. First, holding business risk fixed, an increase in the level of financial leverage raises the level of the equity volatility. Second, regardless of the level of financial leverage, a positive shock to business risk increases the cost of capital and reduces the valuation of future cash flows, generating an instantaneous negative correlation between asset returns and asset volatility. Finally, the market experiences both small continuous movements and large market disruptions. The large and negative market disruptions often generate self-exciting behaviors. The occurrence of one disruption induces more disruptions to follow, thus raising market volatility. We propose an equity index dynamics that capture all three channels of interactions through the separate modeling of the asset return dynamics and the financial leverage variation. We analyze how the different sources of variations impact the index options behaviors differently across a wide range of strikes, maturities, and calendar days.