CONTINUOUSLY MONITORED BARRIER OPTIONS UNDER MARKOV PROCESSES

In this paper, we present an algorithm for pricing barrier options in one‐dimensional Markov models. The approach rests on the construction of an approximating continuous‐time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Lévy process and a local volatility jump‐diffusion. We also provide a convergence proof and error estimates for this algorithm.     

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Ka? representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.     

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.     

LOCAL VARIANCE GAMMA AND EXPLICIT CALIBRATION TO OPTION PRICES

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 analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such models to match the market prices of European options with multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several one‐dimensional root‐search problems and applying elementary functions. 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.     

LEVERAGED ETF IMPLIED VOLATILITIES FROM ETF DYNAMICS

The growth of the exchange‐traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local‐stochastic volatility models. A closed‐form approximation for prices is derived for European‐style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed‐form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well‐known CEV and SABR models.     

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.     

Accuracy and Reliability Considerations of Option Pricing Algorithms

A wide variety of computational schemes have been proposed for the numerical valuation of various classes of options. Experiences in numerical computation have revealed that the details of the implementation of the auxiliary conditions in the numerical algorithms may have profound effects on numerical accuracy. Difficulties in designing algorithms that deal with the path‐dependent payoffs, monitoring features, etc., have been well reported in the literature. In this article, the theoretical issues on the assessment of numerical schemes with regard to accuracy of approximation of auxiliary conditions, rate of convergence, and oscillation phenomena are reviewed. In particular, the oscillation phenomena in bond‐price calculations and the intricacies in implementing the auxiliary conditions in barrier options, proportional step options, and lookback options are discussed. With different types of options and modes of monitoring (continuous or discrete), the optimal method of placing the lattice nodes with reference to the boundary (absorbing or reflecting) are examined in order to achieve linear temporal rate of convergence. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:875–903, 2001     

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.     

Toward A Convergence Theory For Continuous Stochastic Securities Market Models1

This article reviews some recently developed approximation schemes for financial markets with continuous trading. Two methods for approximating continuous-time stochastic securities market models whose exogenously given prices have continuous sample paths are described and compared One method approximates both the paths and the information structure; the other is an approximation in distribution with a Markovian structure. In both cases, the approximating models have a finite state space, discrete time, and possess the same “structural” properties (e.g., “no arbitrage” and “completeness”) as the continuous model. the latter characteristic is an important criterion for judging the merits of the approximations. Taking advantage of the “structure-preserving” characteristic, one can formulate a convergence theory for frictionless markets with continuous trading. the theory provides convergence results for objects such as contingent claim prices, replicating portfolio strategies (hedging policies), optimal consumption policies, and cumulative financial gains (i.e., stochastic integrals), which are constructed along the approximation. the convergence theory enables one to combine the intuitive appeal of discrete models and the analytic tractability of continuous models to provide new insight into the theory of modern financial markets. We survey the current state of such a convergence theory and illustrate the results with some examples of well-known continuous securities market models.     

Option pricing under Markov‐switching GARCH processes

This study proposes an N ‐state Markov‐switching general autoregressive conditionally heteroskedastic (MS‐GARCH) option model and develops a new lattice algorithm to price derivatives under this framework. The MS‐GARCH option model allows volatility dynamics switching between different GARCH processes with a hidden Markov chain, thus exhibiting high flexibility in capturing the dynamics of financial variables. To measure the pricing performance of the MS‐GARCH lattice algorithm, we investigate the convergence of European option prices produced on the new lattice to their true values as conducted by the simulation. These results are very satisfactory. The empirical evidence also suggests that the MS‐GARCH model performs well in fitting the data in‐sample and one‐week‐ahead out‐of‐sample prediction. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:444–464, 2010     

A direct solution method for pricing options in regime‐switching models

Pricing financial or real options with arbitrary payoffs in regime‐switching models is an important problem in finance. Mathematically, it is to solve, under certain standard assumptions, a general form of optimal stopping problems in regime‐switching models. In this article, we reduce an optimal stopping problem with an arbitrary value function in a two‐regime environment to a pair of optimal stopping problems without regime switching. We then propose a method for finding optimal stopping rules using the techniques available for nonswitching problems. In contrast to other methods, our systematic solution procedure is more direct as we first obtain the explicit form of the value functions. In the end, we discuss an option pricing problem, which may not be dealt with by the conventional methods, demonstrating the simplicity of our approach.     

VALUATION OF CONTINUOUSLY MONITORED DOUBLE BARRIER OPTIONS AND RELATED SECURITIES

In this paper, we apply Carr's randomization approximation and the operator form of the Wiener‐Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain perpetual contingent claims with first‐touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options. Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Lévy‐driven models including Variance Gamma processes, Normal Inverse Gaussian processes, KoBoL processes, CGMY model, and Kuznetsov's β ‐class. Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock‐out double barrier put/call options as well as double‐no‐touch options.     

The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

We consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.     

A new scheme for static hedging of European derivatives under stochastic volatility models

This study proposes a new scheme for static hedging of European path‐independent derivatives under stochastic volatility models. First, we show that pricing European path‐independent derivatives under stochastic volatility models is transformed to pricing those under one‐factor local volatility models. Next, applying an efficient static replication method for one‐dimensional price processes developed by Takahashi and Yamazaki (2008), we present a static hedging scheme for European path‐independent derivatives. Finally, a numerical example comparing our method with a dynamic hedging method under Heston's (1993) stochastic volatility model is used to demonstrate that our hedging scheme is effective in practice. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:397–413, 2009     

Pricing credit spread options under a Markov chain model with stochastic default rate

This paper studies a Markov chain model that, unlike the existing models, has a stochastic default rate model so as to reflect real world phenomena. We extend the existing Markov chain models as follows: First, our model includes both the economy‐wide and the rating‐specific factors, which affect credit ratings. Second, our model allows both continuous and discrete movements in credit spreads, even when there exist no changes in credit ratings. Under these assumptions, we provide a valuation formula for a credit spread option, and examine its effects. This paper suggests a parsimonious model. As in J. Wei ( 2003 ), we find that rating‐specific factors are important. Also, discrete movements seem to play a larger role depending on the firm's credit rating. Finally, we show that a model, like the Kodera model, that uses only a common factor without allowing for discrete movements, may overprice credit spread put options. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:631–648, 2004     

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.     

The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applied to a wide range of valuation problems including complicated contingent claims associated with the term structure of interest rates. We illustrate our method by giving two examples: the valuation problems of swaptions and average (Asian) options for interest rates. Our method gives some explicit formulas for solutions, which are sufficiently numerically accurate for practical purposes in most cases. The continuous stochastic processes for spot interest rates and forward interest rates are not necessarily Markovian nor diffusion processes in the usual sense; nevertheless our approach can be rigorously justified by the Malliavin–Watanabe Calculus in stochastic analysis.     

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.     

Pricing and hedging of quanto range accrual notes under Gaussian HJM with cross‐currency Levy processes

This study analyzes the pricing and hedging problems for quanto range accrual notes (RANs) under the Heath‐Jarrow‐Morton (HJM) framework with Levy processes for instantaneous domestic and foreign forward interest rates. We consider the effects of jump risk on both interest rates and exchange rates in the pricing of the notes. We first derive the pricing formula for quanto double interest rate digital options and quanto contingent payoff options; then we apply the method proposed by Turnbull (Journal of Derivatives, 1995, 3, 92–101) to replicate the quanto RAN by a combination of the quanto double interest rate digital options and the quanto contingent payoff options. Using the pricing formulas derived in this study, we obtain the hedging position for each issue of quanto RANs. In addition, by simulation and assuming the jump risk to follow a compound Poisson process, we further analyze the effects of jump risk and exchange rate risk on the coupons receivable in holding a RAN. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:973–998, 2009     

Existence,uniqueness, and stability of optimal payoffs of eligible assets

In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of prespecified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to be able to do so at the lowest possible cost and to identify the corresponding portfolios, or, equivalently, their payoffs. We study the existence and uniqueness of such optimal payoffs as well as their behavior under a perturbation or an approximation of the underlying capital position. This behavior is naturally linked to the continuity properties of the set‐valued map that associates to each capital position the corresponding set of optimal eligible payoffs. Upper continuity can be ensured under fairly natural assumptions. Lower continuity is typically less easy to establish. While it is always satisfied in a polyhedral setting, it generally fails otherwise, even when the reference risk measure is convex. However, lower continuity can often be established for eligible payoffs that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.