Friday, 9/29 - [PHOTOS]

The  Early Years of the Variance-Gamma  Process

Dilip Madan and I worked on stochastic models for returns (log-price  increments) at the University of Sydney  1980 -1988, and completed the 1990 paper during meetings while respectively at U.Md. and U.Va. The (symmetric) Variance-Gamma distribution for returns first appears in several papers in 1987.  The emphasis in the pre-1990 paper is on estimation of parameters, using historical Australian financial data , and using the simple form of characteristic function. In this paper the  early collaboaration is reviewed. Some recent estimation methodology, focussed on the probability density function is  presented.

Random walk versus Mean-reversion in Energy Commodity Prices

Investments, wealth and risk tolerance

The novel concept of forward dynamic utility is introduced. A class of such utilities is constructed by compiling variational (utility) and stochastic (market) components. The optimal asset allocation is found in closed form and under minimal model assumptions. An important emerging quantity is the local risk tolerance that solves, inversely in time, a fast diffusion equation. Solutions of the latter are presented. Joint work with Marek Musiela.

Risk tolerance and optimal portfolio choice

A new framework for utility maximization is introduced. General classes of such utilities are constructed by combining the variational utility input with the market dynamics represented by the general multidimensional Ito process. Explicit closed form expressions for optimal allocations are obtained. They depend on the coefficients of the market dynamics, on the optimal wealth level and on the utility-generated measure of risk tolerance. Direct and analytic link is established between the distribution of wealth in the future and the implicit to it current risk tolerance. Consequently, given the equilibrium market dynamics and given the views an investor formulates with respect to this equilibrium, the specification of distribution of wealth in the future provides information about the current risk tolerance and the utility function of an investor. Joint work with Thaleia Zariphopoulou.

Arbitrage Bounds for Volatility Derivatives and the Skorokhod embedding Problem

Options on realized variance are booming, although no satisfactory pricing model has emerged yet. We establish the arbitrage bounds that the price of vanilla options of the same maturity impose, under the sole assumption of continuity, by exploiting some solutions of the Skorohod embedding problem. We construct the direct (ROOT) and reverse (ROST) barrier solutions, as free boundaries of two different optimal stopping problems.

Some remarkable properties of Gamma processes

A number of remarkable properties of the Gamma processes, i.e. : realisation of their bridges, absolute continuity relationships, realisation of a gamma process as an inverse local time, the effect of a gamma process as a time change... are gathered in this paper. Some of them are put in perspective with their Brownian counterparts.

Equilibrium Asset Pricing: With Non-Gaussian Factors and Exponential Utilities

We analyse the equilibrium asset pricing implications for an economy with single period return exposures to explicit non-Gaussian systematic factors, that may be both skewed and long-tailed, and Gaussian idiosyncratic components. Investors maximize expected exponential utility and equilibrium factor prices are shown to reflect exponentially tilted prices for non-Gaussian factor risk exposures. It is shown that these prices may be directly estimated from the univariate probability law of the factor exposure, given an estimate of average risk aversion in the economy. In addition a residual form of the capital asset pricing model continues to hold and prices the idiosyncratic or Gaussian risks. The theory is illustrated on data for the US economy using independent components analysis to identify the factors and the variance gamma model to describe the probability law of the non-Gaussian factors. It is shown that the residual CAPM accounts for no more than one percent of the pricing of risky assets, while the exponentially tilted systematic factor risk exposures account for the bulk of risky asset pricing.

Saturday, 9/30 [PHOTOS]

Risk Neutral Compatibility with Option Prices

A common problem is to choose a "risk neutral" measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique "equivalent local martingale measure'' by completing the market with option prices. We do this by modeling the behavior of the stock price X, together with the behavior of the option prices for a relevant family of options which are (or can theoretically be) effectively traded. In doing so, we need to ensure a kind of "compatibility'' between X and the prices of our options, and this poses some interesting mathematical difficulties.

Pricing of Long-dated Straddles & Their Properties

In stochastic interest rate affine model with stochastic volatility, we derive the characteristic function of log swap rate under the swap measure. We then study the relationship between straddles premiums and interest rate factors. Conclusion is that the long-dated straddles are simply curve trades.

Operational Risk

This talk discusses operational risk. We discuss its economic and mathematical characterization and its estimation. The insights for this characterization originate in the corporate finance and credit risk literature. Operational risk is of two types, either (i) the risk of a loss due to the firm's operating technology, or (ii) the risk of a loss due to agency costs. These two types of operational risks generate loss processes with completely different characteristics. The mathematical characterization of these operational risks is modeled after the risk of default in the reduced form credit risk literature. We argue that because operational risk is internal to the firm, it is necessarily confounded by the net-present value of a firm's operating technology, and as such, it cannot be estimated using historical time series of the firm's balance sheet and market prices alone. Finally, we show that the inclusion of operational risk into the computation of fair economic capital (as with revised Basel II) without the consideration of a firm's NPV, will provide biased (too large) capital requirements.

Information-Based Asset Pricing

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such market factor we associate a so-called market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents noise. The noise term is modelled by an independent Brownian bridge process that spans the time interval from the present to the time at which the value of the given market factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent market information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk neutral measure, conditional on the information provided by the market filtration thus constructed. In the case where the cash flows are the random dividend payments associated with equities, an explicit model is obtained for the share-price process. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black-Scholes type. The information-based framework has another interesting consequence: it generates a natural explanation for the origin of stochastic volatility, without the need for specifying on an ad hoc basis the stochastic dynamics of the volatility. We also consider how to model the dynamics of interest rates in an information-based setting. Co-authors: Dorje C. Brody, Imperial College London, and Andrea Macrina, King's College London.

Distributional study of De Finetti's dividend problem for a general Lévy insurance risk process

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent work in the actuarial literature concerning calculations for the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than existing literature in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér-Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators. The talk is based on joint work with Z. Palmowski.

A Generic One-Factor Lévy Model for Pricing Synthetic CDOs

The one-factor Gaussian model is well-known not to fit simultaneously the prices of the different tranches of a collateralized debt obligation (CDO), leading to the implied correlation smile. Recently, other one-factor models based on different distributions have been proposed. Moosbrucker used a one-factor Variance Gamma model, Kalemanova et al. and Guegan and Houdain worked with a NIG factor model and Baxter introduced the BVG model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic one-factor Lévy model and work out the large homogeneous portfolio (LHP) approximation. Then, we discuss several examples and calibrate a battery of models to market data.

New Results for Fractional Brownian Motion

Mathematically Fractional Brownian Motion is a difficult process to define. We shall present a new approach to Fractional Brownian Motion extending recent work of Hu, Oksendal, Duncan and Pasik Duncan to include processes with Hurst parameters 0<H<1. A central tool for fractional Brownian motion is the analog of the Ito formula. We give new proofs and results. We also answer some of the criticisms of the use of fractional Brownian motion in financial modeling. This is joint work with John van der Hoek of the University of Adelaide, Australia.

Lévy driven fixed income models

First the three basic approaches to model interest rate term structures are surveyed: the Lévy forward rate model, the Lévy forward process model and the Lévy Libor or market model. In joint work with N. Koval the latter has been extended to a multi-currency setting. As an application of the multi-currency market model we derive closed form pricing formulae for cross-currency derivatives. Foreign caps and floors and cross-currency swaps are studied in detail. Numerically efficient pricing algorithms are derived. A calibration example is given for a two-currency setting (EUR, USD).

Sunday, October 1 [PHOTOS]

Limited Tax Rebates, Tax Arbitrage and Equilibrium in a Multi-period Financial Market with a General Tax Structures and Transaction Costs

Dammon and Green (1987) in their two-period model, showed that there are well-known difficulties in dealing with taxes in a general-equilibrium setting. When tax rates differ across investors, there will be tax-arbitrage and therefore, an equilibrium will fail to exist unless short-sale restrictions are imposed that prevent investors from exploiting such arbitrage opportunities. In the present paper, we introduce two frictions: limited tax rebates on capital losses and ordinary income taxes on long and short asset positions; and we assume transaction costs on asset transactions. With reasonable conditions on taxes and transaction technologies, we show that a general equilibrium exists. Consequently, these frictions are suggested as one possible explanation for why an apparent arbitrage may in fact not exist and a general equilibrium does exist. (Joint work with X. Jin)

Robust Replication of Default Contingent Claims

We show how to replicate the payoffs to a class of default-contingent claims by taking static positions in a continuum of credit default swaps (CDS) of different maturities. Although we assume deterministic interest rates and a constant recovery rate on the CDS, the replication is otherwise robust in that we make no assumptions on the process triggering default. In particular, we can robustly replicate the payoff to an Arrow Debreu security paying one dollar at a fixed date if a given entity survives to that date. As a consequence, we can determine risk-neutral survival probabilities from an arbitrarily given yield curve and CDS curve. (Joint work with Bjorn Flesaker of Bloomberg)

Characterisation of b(M) for continuous BMO martingales