BESSEL PROCESSES,STOCHASTIC VOLATILITY,AND TIMER OPTIONS

Motivated by analytical valuation of timer options (an important innovation in realized variance‐based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first‐passage time problem on realized variance, and generalize the standard risk‐neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first‐passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton‐type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed.     

Static and semistatic hedging as contrarian or conformist bets

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semistatic portfolios should more properly be thought of as separate classes of derivatives, with nontrivial, model‐dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from the Carr–Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizably larger than the errors of variance‐minimizing portfolios. We explain why the exact semistatic hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance‐minimizing semistatic portfolios. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener–Hopf factors and Laplace–Fourier inversion.     

LÉVY PROCESSES INDUCED BY DIRICHLET (B‐)SPLINES: MODELING MULTIVARIATE ASSET PRICE DYNAMICS

We consider a new class of processes, called LG processes, defined as linear combinations of independent gamma processes. Their distributional and path‐wise properties are explored by following their relation to polynomial and Dirichlet (B‐)splines. In particular, it is shown that the density of an LG process can be expressed in terms of Dirichlet (B‐)splines, introduced independently by Ignatov and Kaishev and Karlin, Micchelli, and Rinott. We further show that the well‐known variance gamma (VG) process, introduced by Madan and Seneta, and the bilateral gamma (BG) process, recently considered by Küchler and Tappe are special cases of an LG process. Following this LG interpretation, we derive new (alternative) expressions for the VG and BG densities and consider their numerical properties. The LG process has two sets of parameters, the B‐spline knots and their multiplicities, and offers further flexibility in controlling the shape of the Levy density, compared to the VG and the BG processes. Such flexibility is often desirable in practice, which makes LG processes interesting for financial and insurance applications. Multivariate LG processes are also introduced and their relation to multivariate Dirichlet and simplex splines is established. Expressions for their joint density, the underlying LG‐copula, the characteristic, moment and cumulant generating functions are given. A method for simulating LG sample paths is also proposed, based on the Dirichlet bridge sampling of gamma processes, due to Kaishev and Dimitriva. A method of moments for estimation of the LG parameters is also developed. Multivariate LG processes are shown to provide a competitive alternative in modeling dependence, compared to the various multivariate generalizations of the VG process, proposed in the literature. Application of multivariate LG processes in modeling the joint dynamics of multiple exchange rates is also considered.     

CLOSED FORM PRICING FORMULAS FOR DISCRETELY SAMPLED GENERALIZED VARIANCE SWAPS

Most of the existing pricing models of variance derivative products assume continuous sampling of the realized variance processes, though actual contractual specifications compute the realized variance based on sampling at discrete times. We present a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one‐dimensional Fourier integrals. We also verify through analytic calculations the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. The proposed methodology can be applied to any affine model and other higher moments swaps as well. We examine the exposure to convexity (volatility of variance) and skew (correlation between the equity returns and variance process) of these discretely sampled generalized variance swaps. We explore the impact on the fair strike prices of these exotic variance swaps with respect to different sets of parameter values, like varying sampling frequencies, jump intensity, and width of the monitoring corridor.     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.     

The valuation of multiple stock warrants

The issue of multiple series of stock purchase warrants by the same firm is an interesting financial structure not just in America, but is common in countries such as Switzerland, Malaysia, and Singapore. This paper derives valuation formulas for multiple series of outstanding warrants. The theoretical warrant prices from this model are compared against existing models. We report a subtle slippage effect and also a cross dilution effect that cause the existing models, such as Galai‐Schneller model, to be inappropriate for pricing such classes of multiple warrants. We also provide an example to illustrate the practicality of our model. The Greeks of the model are also derived in this paper. The complexity of multiple warrants could extend to other classes of contingent securities issued by the same firm but with differing expiry terms. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:517–534, 2003     

THE AFFINE LIBOR MODELS

We provide a general and flexible approach to LIBOR modeling based on the class of affine factor processes. Our approach respects the basic economic requirement that LIBOR rates are nonnegative, and the basic requirement from mathematical finance that LIBOR rates are analytically tractable martingales with respect to their own forward measure. Additionally, and most importantly, our approach also leads to analytically tractable expressions of multi‐LIBOR payoffs. This approach unifies therefore the advantages of well‐known forward price models with those of classical LIBOR rate models. Several examples are added and prototypical volatility smiles are shown. We believe that the CIR process‐based LIBOR model might be of particular interest for applications, since closed form valuation formulas for caps and swaptions are derived.     

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.     

Analytical Valuation of American Options on Jump-Diffusion Processes

We derive analytic formulas for the value of American options when the underlying asset follows a jump-diffusion process and pays continuous dividends. They early exercise premium has a form very different form from that for diffusion processes, and this can be attributed to the discontinuous nature of the price paths. Analytical formulas are derived for several distributions of the jump amplitude.     

PSEUDODIFFUSIONS AND QUADRATIC TERM STRUCTURE MODELS

The non-Gaussianity of processes observed in financial markets and the relatively good performance of Gaussian models can be reconciled by replacing the Brownian motion with Lévy processes whose Lévy densities decay as  exp(−λ| x |)  or faster, where  λ > 0  is large. This leads to asymptotic pricing models. The leading term, P ₀, is the price in the Gaussian model with the same instantaneous drift and variance. The first correction term depends on the instantaneous moments of order up to 3, that is, the skewness is taken into account, the next term depends on moments of order 4 (kurtosis) as well, etc. In empirical studies, the asymptotic formula can be applied without explicit specification of the underlying process: it suffices to assume that the instantaneous moments of order greater than 2 are small w.r.t. moments of order 1 and 2, and use empirical data on moments of order up to 3 or 4. As an application, the bond-pricing problem in the non-Gaussian quadratic term structure model is solved. For pricing of options near expiry, a different set of asymptotic formulas is developed; they require more detailed specification of the process, especially of its jump part. The leading terms of these formulas depend on the jump part of the process only, so that they can be used in empirical studies to identify the jump characteristics of the process.     

FAST MONTE CARLO GREEKS FOR FINANCIAL PRODUCTS WITH DISCONTINUOUS PAY‐OFFS

We introduce a new class of numerical schemes for discretizing processes driven by Brownian motions. These allow the rapid computation of sensitivities of discontinuous integrals using pathwise methods even when the underlying densities postdiscretization are singular. The two new methods presented in this paper allow Greeks for financial products with trigger features to be computed in the LIBOR market model with similar speed to that obtained by using the adjoint method for continuous pay‐offs. The methods are generic with the main constraint being that the discontinuities at each step must be determined by a one‐dimensional function: the proxy constraint. They are also generic with the sole interaction between the integrand and the scheme being the specification of this constraint.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION

The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.     

Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds

In this paper, we study the excursions of Bessel and Cox–Ingersoll–Ross (CIR) processes with dimensions . We obtain densities for the last passage times and meanders of the processes. Using these results, we prove a variation of the Azéma martingale for the Bessel and CIR processes based on excursion theory. Furthermore, we study their Parisian excursions, and generalize previous results on the Parisian stopping time of Brownian motion to that of the Bessel and CIR processes. We obtain explicit formulas and asymptotic results for the densities of the Parisian stopping times, and develop exact simulation algorithms to sample the Parisian stopping times of Bessel and CIR processes. We introduce a new type of bond, the zero‐coupon Parisian bond. The buyer of such a bond is betting against zero interest rates, while the seller is effectively hedging against a period where interest rates fluctuate around 0. Using our results, we propose two methods for pricing these bonds and provide numerical examples.     

THE STOCHASTIC VOLATILITY MODEL OF BARNDORFF‐NIELSEN AND SHEPHARD IN COMMODITY MARKETS

We consider the non‐Gaussian stochastic volatility model of Barndorff‐Nielsen and Shephard for the exponential mean‐reversion model of Schwartz proposed for commodity spot prices. We analyze the properties of the stochastic dynamics, and show in particular that the log‐spot prices possess a stationary distribution defined as a normal variance‐mixture model. Furthermore, the stochastic volatility model allows for explicit forward prices, which may produce a hump structure inherited from the mean‐reversion of the stochastic volatility. Although the spot price dynamics has continuous paths, the forward prices will have a jump dynamics, where jumps occur according to changes in the volatility process. We compare with the popular Heston stochastic volatility dynamics, and show that the Barndorff‐Nielsen and Shephard model provides a more flexible framework in describing commodity spot prices. An empirical example on UK spot data is included.     

FORECASTING VOLATILITY WITH THE MULTIFRACTAL RANDOM WALK MODEL

We study the problem of forecasting volatility for the multifractal random walk model. In order to avoid the ill‐posed problem of estimating the correlation length T of the model, we introduce a limiting object defined in a quotient space; formally, this object is an infinite range log volatility. For this object and the nonlimiting object, we obtain precise prediction formulas and we apply them to the problem of forecasting volatility and pricing options with the MRW model in the absence of a reliable estimate of σ and T.     

Affine multiple yield curve models

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or Heath–Jarrow–Morton modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded overnight indexed swap rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multicurve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed‐form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.     

Semimartingale theory of monotone mean–variance portfolio allocation

We study dynamic optimal portfolio allocation for monotone mean–variance preferences in a general semimartingale model. Armed with new results in this area, we revisit the work of Cui et al. and fully characterize the circumstances under which one can set aside a nonnegative cash flow while simultaneously improving the mean–variance efficiency of the left‐over wealth. The paper analyzes, for the first time, the monotone hull of the Sharpe ratio and highlights its relevance to the problem at hand.     

ON VALUING STOCHASTIC PERPETUITIES USING NEW LONG HORIZON STOCK PRICE MODELS DISTINGUISHING BOOMS,BUSTS, AND BALANCED MARKETS

For longer horizons, assuming no dividend distributions, models for discounted stock prices in balanced markets are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new concept of a balanced market proposed here permits a distinction between such markets and unduly optimistic or pessimistic ones. A tractable example is developed and termed the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. Evidence is provided for long dated equity linked claims paying coupon for time spent by equity above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given the popularity of such claims, the resulting mispricing could possibly take some corrections. Furthermore for these new discounted models, implied volatility curves do not flatten out at the larger maturities.     

Convergence of optimal expected utility for a sequence of discrete‐time markets

We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The nth discrete‐time economy is generated by a scaled n‐step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that .