Hélyette Geman

The Fine Structure of Asset Returns: An Empirical Investigation

We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation.

Stochastic Volatility for Lévy Processes

Three processes reflecting persistence of volatility are initially formulated by evaluating three Levy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Levy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

Changes of numeraire, changes of probability measure and option pricing

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numeraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numeraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself. The key theorem of general numeraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.

Order Flow, Transaction Clock, and Normality of Asset Returns

The goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change. Clark (1973) addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator. We extend Clarks results and find the following: (i) stochastic time changes are mathematically much less constraining than subordinators; (ii) the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns; (iii) this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions. Copyright The American Finance Association 2000.

Pricing and hedging in incomplete markets

Abstract We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures and associated floors which expected payoffs must exceed in order for the investor to consider the hedged and financed investment to be acceptable. By assuming that the liquid assets are priced so that each portfolio of assets has negative expected return under at least one measure, we derive a counterpart to the first fundamental theorem of asset pricing. We also derive a counterpart to the second fundamental theorem, which leads to unique derivative security pricing and hedging even though markets are incomplete. For products that are not spanned by the liquid assets of the economy, we show how our methodology provides more realistic bid–ask spreads.

Time Changes for Lévy Processes

The goal of this paper is to consider pure jump Levy processes of finite variation with an infinite arrival rate of jumps as models for the logarithm of asset prices. These processes may be written as time‐changed Brownian motion. We exhibit the explicit time change for each of a wide class of Levy processes and show that the time change is a weighted price move measure of time. Additionally, we present a number of Levy processes that are analytically tractable, in their characteristic functions and Levy densities, and hence are relevant for option pricing.

Pricing Options on Realized Variance

Abstract.Models which hypothesize that returns are pure jump processes with independent increments have been shown to be capable of capturing the observed variation of market prices of vanilla stock options across strike and maturity. In this paper, these models are employed to derive in closed form the prices of derivatives written on future realized quadratic variation. Alternative work on pricing derivatives on quadratic variation has alternatively assumed that the underlying returns process is continuous over time. We compare the model values of derivatives on quadratic variation for the two types of models and find substantial differences.

Interest Rate Risk Management and Valuation of the Surrender Option in Life Insurance Policies

The valuation of the prepayment option embedded in mortgages attracts the attention of practitioners and academics (see Schwartz and Torous, 1989) both because of its direct negative effect on the financial value of a bank balance sheet in case of drop in interest rates and also because of its impact on the design and pricing of mortgage-backed securities. In the same manner, life insurance policyholders may surrender their contracts and take advantage of higher yields available in the financial markets; this is a source of concern for life insurers, especially during periods of highly volatile interest rates such as have prevailed in recent years. We address the surrender option pricing problem as the valuation of a contingent claim for the insurer, where the contingency is closely related to the level of interest rates, and directly price by arbitrage the surrender option embedded in life insurance policies. A closed-form solution is derived in the case of a single-premium policy when the investment portfolio consists of a fixed-term zero-coupon bond, and the dynamics of stochastic interest rates are driven by the Heath-Jarrow-Morton (1992) model. The price of the option is computed in the case of French contracts using both the closed-form expression and Monte Carlo simulations.

From Local Volatility to Local Levy Models

We define the class of local Lévy processes. These are Lévy processes time changed by an inhomogeneous local speed function. The local speed function is a deterministic function of time and the level of the process itself. We show how to reverse engineer the local speed function from traded option prices of all strikes and maturities. The local Lévy processes generalize the class of local volatility models. Closed forms for local speed functions for a variety of cases are also presented. Numerical methods for recovery are also described.

Stochastic time changes in catastrophe option pricing

Abstract Catastrophe insurance derivatives (Futures and options) were introduced in December 1992 by the Chicago Board of Trade in order to offer insurers new ways of hedging their underwriting risk. Only CAT options and combinations of options such as call spreads are traded today, and the ISO index has been replaced by the PCS index. Otherwise, the economic goal of these instruments continues to be for insurers an alternative to reinsurance and for portfolio managers a new class of assets to invest in. The pricing methodology of these derivatives relies on some crucial elements: 1. (a) the choice of the stochastic modelling of the aggregate reported claim index dynamics (since the terminal value of this index defines the pay-off of the CAT options); 2. (b) the decision of a financial versus actuarial approach to the valuation; 3. (c) the number of sources of randomness in the model and the determination of a “martingale measure” for insurance and reinsurance instruments. We represent in this paper the dynamics of the aggregate claim index by the sum of a geometric Brownian motion which accounts for the randomness in the reporting of the claims and a Poisson process which accounts for the occurrence of catastrophes (only catastrophic claims are incorporated in the index). Geman (1994) and Cummins and Geman (1995) took this modelling for the instantaneous claim process. Our choice here is closer to the classical actuarial representation while preserving the quasi-completeness of insurance derivative markets obtained by applying the Delbaen and Haezendonck (1989) methodology to the class of layers of reinsurance replicating the call spreads. Moreover, we obtain semi-analytical solutions for the CAT options and call spreads by extending to the jump-diffusion case the method of the Laplace transform and stochastic time changes introduced in Geman and Yor (1993, 1996) in order to price financial path-dependent options through the properties of excursion theory.