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.     

Good jump,bad jump,and option valuation

I develop a new class of closed‐form option pricing models that incorporate variance risk premium and symmetric or asymmetric double exponential jump diffusion. These models decompose the jump component into upward and downward jumps using two independent exponential distributions and thus capture the impact of good and bad news on asset returns and option prices. The empirical results show that the model with an asymmetric double exponential jump diffusion improves the fit on Shanghai Stock Exchange 50ETF returns and options and provides relatively better in‐ and out‐of‐sample pricing performance.     

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.     

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.     

Laguerre Series for Asian and Other Options

This paper has four goals: (a) relate ladder height distributions to option values; (b) show how Laguerre expansions may be used in the computation of densities, distribution functions, and option prices; (c) derive some new results on the integral of geometric Brownian motion over a finite interval; and (d) apply the preceding results to the determination of the distribution of the integral of geometric Brownian motion and the computation of Asian option values. The usual fixed‐strike options on the average are treated, as well as options with payoffs expressed in terms of one over the average of the underlying security, which this author calls “reciprocal Asian options.” In all cases the underlying asset is represented by geometric Brownian motion, the averages are performed continuously, and the options are of European type.     

ON ARBITRAGE AND DUALITY UNDER MODEL UNCERTAINTY AND PORTFOLIO CONSTRAINTS

We consider the fundamental theorem of asset pricing (FTAP) and the hedging prices of options under nondominated model uncertainty and portfolio constraints in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value process is a local super‐martingale under these measures. We also get the nondominated optional decomposition with constraints. From this decomposition, we obtain the duality of the super‐hedging prices of European options, as well as the sub‐ and super‐hedging prices of American options. Finally, we get the FTAP and the duality of super‐hedging prices in a market where stocks are traded dynamically and options are traded statically.     

Option pricing in the moderate deviations regime

We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.     

The pricing of foreign currency options under jump‐diffusion processes

In this article, the authors derive explicit formulas for European foreign exchange (FX) call and put option values when the exchange rate dynamics are governed by jump‐diffusion processes. The authors use a simple general equilibrium international asset pricing model with continuous trading and frictionless international capital markets. The domestic and foreign price level are introduced as state variables that contain jumps caused by monetary shocks and catastrophic events such as 9/11 or Hurricane Katrina. The domestic and foreign interest rates are stochastic and endogenously determined in the model and are shown to be critically affected by the jump risk of the foreign exchange. The model shows that the behavior of FX options is affected through the impact of state variables and parameters on the nominal interest rates. The model contrasts with those of M. Garman and S. Kohlhagen (1983) and O. Grabbe (1983), whose models have exogenously determined interest rates. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:669–695, 2007     

A maximal affine stochastic volatility model of oil prices

This study develops and estimates a stochastic volatility model of commodity prices that nests many of the previous models in the literature. The model is an affine three‐factor model with one state variable driving the volatility and is maximal among all such models that are also identifiable. The model leads to quasi‐analytical formulas for futures and options prices. It allows for time‐varying correlation structures between the spot price and convenience yield, the spot price and its volatility, and the volatility and convenience yield. It allows for expected mean‐reversion in the short term and for an increasing expected long‐term price, and for time‐varying risk premia. Furthermore, the model allows for the situation in which options' prices depend on risk not fully spanned by futures prices. These properties are desirable and empirically important for modeling many commodities, especially crude oil. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:101–133, 2010     

ASSET PRICE BUBBLES IN INCOMPLETE MARKETS*

This paper studies asset price bubbles in a continuous time model using the local martingale framework. Providing careful definitions of the asset's market and fundamental price, we characterize all possible price bubbles in an incomplete market satisfying the “no free lunch with vanishing risk (NFLVR)” and “no dominance” assumptions. We show that the two leading models for bubbles as either charges or as strict local martingales, respectively, are equivalent. We propose a new theory for bubble birth that involves a nontrivial modification of the classical martingale pricing framework. This modification involves the market exhibiting different local martingale measures across time—a possibility not previously explored within the classical theory. Finally, we investigate the pricing of derivative securities in the presence of asset price bubbles, and we show that: (i) European put options can have no bubbles; (ii) European call options and discounted forward prices have bubbles whose magnitudes are related to the asset's price bubble; (iii) with no dividends, American call options are not exercised early; (iv) European put‐call parity in market prices must always hold, regardless of bubbles; and (v) futures price bubbles can exist and they are independent of the underlying asset's price bubble. Many of these results stand in contrast to those of the classical theory. We propose, but do not implement, some new tests for the existence of asset price bubbles using derivative securities.     

An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

We introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univpm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm.it/recchioni/finance . © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009     

Static hedging and pricing of exotic options with payoff frames

We develop a general framework for statically hedging and pricing European‐style options with nonstandard terminal payoffs, which can be applied to mixed static–dynamic and semistatic hedges for many path‐dependent exotic options including variance swaps and barrier options. The goal is achieved by separating the hedging and pricing problems to obtain replicating strategies. Once prices have been obtained for a set of basis payoffs, the pricing and hedging of financial securities with arbitrary payoff functions is accomplished by computing a set of “hedge coefficients” for that security. This method is particularly well suited for pricing baskets of options simultaneously, and is robust to discontinuities of payoffs. In addition, the method enables a systematic comparison of the value of a payoff (or portfolio) across a set of competing model specifications with implications for security design.     

A Discrete‐Time Intertemporal Asset Pricing Model: GE Approach with Recursive Utility

This paper studies the equilibrium characterization of asset pricing in a discrete‐time Lucas exchange economy (Lucas 1978) with the intertemporal recursive utility function of Epstein and Zin (1989). A general formulation of equilibrium asset pricing is presented. It is shown that risk aversion of a certainty equivalent corresponds to risk aversion in the intertemporal asset pricing model. The discrete‐time analogue of Ma's (1993) option pricing formula is derived in an i.i.d. environment, with which we prove an observational nonequivalence theorem in distinguishing the differences of the betweenness recursive utility functions and the expected utility functions. Additionally, when the consumption growth rate follows a first-order Markov process, it is shown that the observational nonequivalence result holds for Kreps–Porteus expected utility. Finally, as by-products, this paper also contains derivations of closed-form formulas for the aggregate equity (with endogenously determined yields), the term structure of interest rates, and European call options on the aggregate equity in a Markov setting.     

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.     

Option pricing with orthogonal polynomial expansions

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein–Stein, and Hull–White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier‐transform‐based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.     

A Multiphase,Flexible, and Accurate Lattice for Pricing Complex Derivatives with Multiple Market Variables

With the rapid growth and the deregulation of financial markets, many complex derivatives have been structured to meet specific financial goals. Unfortunately, most complex derivatives have no analytical formulas for their prices, particularly when there is more than one market variable. As a result, these derivatives must be priced by numerical methods such as lattice. However, the nonlinearity error of lattices due to the nonlinearity of the derivative's value function could lead to oscillating prices. To construct an accurate, multivariate lattice, this study proposes a multiphase method that alleviates the oscillating problem by making the lattice match the “critical locations,” locations where nonlinearity of the derivative's value function occurs. Moreover, our lattice has the ability to model the jumps in the market variables such as regular withdraws from an investment account, which is hard to deal with analytically. Numerical results for vulnerable options, insurance contracts guaranteed minimum withdrawal benefit (GMWB), and defaultable bonds show that our methodology can be applied to the pricing of a wide range of complex financial contracts.     

ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES

We provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.     

Robust Hedging of Barrier Options

This article considers the pricing and hedging of barrier options in a market in which call options are liquidly traded and can be used as hedging instruments. This use of call options means that market preferences and beliefs about the future behavior of the underlying assets are in some sense incorporated into the hedge and do not need to be specified exogenously. Thus we are able to find prices for exotic derivatives which are independent of any model for the underlying asset. For example we do not need to assume that the underlying assets follow an exponential Brownian motion.
We find model-independent upper and lower bounds on the prices of knock-in and knock-out puts and calls. If the market prices the barrier options outside these limits then we give simple strategies for generating profits at zero risk. Examples illustrate that the bounds we give can be fairly tight.     

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.     

ARBITRAGE BOUNDS FOR PRICES OF WEIGHTED VARIANCE SWAPS

We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co‐maturing put options. We assume the put option prices do not admit arbitrage and deduce no‐arbitrage bounds on the weighted variance swap along with super‐ and sub‐replicating strategies that enforce them. We find that market quotes for variance swaps are surprisingly close to the model‐free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi‐infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model‐independent and probability‐free setup. In particular, we use and extend Föllmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the “log contract” and similar connections for weighted variance swaps. Our results take the form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk‐neutral expectations of discounted payoffs.