Derivatives pricing with liquidity risk

This paper develops a novel, general derivative pricing model which introduces a liquidity risk factor. The model variants we outline offer a sufficient degree of flexibility so as to enable the valuation of various types of derivative classes including futures, American options, and mortgage backed security options, whereas existing derivative models can only price liquidity risk in European derivatives. We validate the model with oil and gold futures data and compare it to a classical benchmark model void of any liquidity risk. We find that our model is significantly more accurate than the classical model for pricing both oil and gold contracts.     

Valuation Bounds on Barrier Options Under Model Uncertainty

This article investigates valuation bounds on barrier options under model uncertainty. This investigation enriches the literature on the model‐free valuation of these exotic options. It is found that with weak assumptions on underlying price processes, tight valuation bounds on barrier options can be sought from a set of European options. As a result, the numerical routine developed in this article can be reviewed as a new method for the evaluation of barrier options, which is independent of model assumptions. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:199–234, 2013     

Valuation of futures and commodity options with information costs

In this article, futures and commodity options are analyzed in the context of Merton's (1987) model of capital market equilibrium with incomplete information. First, following Dusak (1973) and Black (1976), the conditions under which Merton's model can be applied to the valuation of forward and futures contracts are proposed. Then an application to futures markets is given. We provide a partial differential equation and the formulas for European commodity options, futures contracts, and American options in the same context. The models are simulated and compared to standard models with no information costs. We find that model prices are not significantly different from standard model prices. However, our models correct for some pricing biases in standard models. In particular, they reduce the overvaluation bias for European and American commodity options. It seems that the costs of gathering and processing information regarding the option and its underlying asset play a role in explaining the biases observed in standard models. This work can be applied to other futures markets. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 645–664, 1999     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

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.     

VALUATION OF BARRIER OPTIONS VIA A GENERAL SELF‐DUALITY

Classical put–call symmetry relates the price of puts and calls under a suitable dual market transform. One well‐known application is the semistatic hedging of path‐dependent barrier options with European options. This, however, in its classical form requires the price process to observe rather stringent and unrealistic symmetry properties. In this paper, we develop a general self‐duality theorem to develop valuation schemes for barrier options in stochastic volatility models with correlation.     

MODELING SOVEREIGN RISKS: FROM A HYBRID MODEL TO THE GENERALIZED DENSITY APPROACH

Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model that combines an accessible part taking into account the evolution of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed‐form formulas for the probability that the default occurs at critical political dates in a Markovian setting. Moreover, we introduce a generalized density framework for the hybrid default time and deduce the compensator process of default. Finally, we apply the hybrid model and the generalized density to the valuation of sovereign bonds and explain the significant jumps in long‐term government bond yields during the sovereign crisis.     

THE BLACK-SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS

In this paper, novel singular perturbation techniques are applied to price European, American, and barrier options. Employment of these methods leads to a significant simplification of the problem in all cases, by reducing the number of parameters. For American options, the valuation problem is reduced to a procedure that may be performed on a rudimentary handheld calculator. The method also sheds light on the evolution of option prices for all of the cases considered, the results being particularly illuminating for American and barrier options.     

Pricing VIX derivatives with infinite-activity jumps

In this paper, we investigate a two-factor VIX model with infinite-activity jumps, which is a more realistic way to reduce errors in pricing VIX derivatives, compared with Mencía and Sentana (2013), J Financ Econ, 108, 367–391. Our two-factor model features central tendency, stochastic volatility and infinite-activity pure jump Lévy processes which include the variance gamma (VG) and the normal inverse Gaussian (NIG) processes as special cases. We find empirical evidence that the model with infinite-activity jumps is superior to the models with finite-activity jumps, particularly in pricing VIX options. As a result, infinite-activity jumps should not be ignored in pricing VIX derivatives.     

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     

Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps

We investigate analytical solvability of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula for the conditional moment generating function of the log-asset price and providing the condition under which this new formula is explicit. The results lay a foundation for a range of valuation, calibration, and econometric problems. We then combine our theoretical results, the Hilbert transform method, various interpolation techniques, with the dimension reduction technique to propose unified simulation schemes for solvable models with affine SV and Lévy jumps. In contrast to traditional exact simulation methods, our approach is applicable to a broad class of models, maintains good accuracy, and enables efficient pricing of discretely monitored path-dependent derivatives. We analyze various sources of errors arising from the simulation approach and present error bounds. Finally, extensive numerical results demonstrate that our method is highly accurate, efficient, simple to implement, and widely applicable.     

Pricing American options on foreign currency with stochastic volatility,jumps, and stochastic interest rates

By applying the Heath–Jarrow–Morton (HJM) framework, an analytical approximation for pricing American options on foreign currency under stochastic volatility and double jump is derived. This approximation is also applied to other existing models for the purpose of comparison. There is evidence that such types of jumps can have a critical impact on earlyexercise premiums that will be significant for deep out‐of‐the‐money options with short maturities. Moreover, the importance of the term structure of interest rates to early‐exercise premiums is demonstrated as is the sensitivity of these premiums to correlation‐related parameters. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:867–891, 2007     

Vulnerable options,risky corporate bond,and credit spread

In the current literature, the focus of credit‐risk analysis has been either on the valuation of risky corporate bond and credit spread or on the valuation of vulnerable options, but never both in the same context. There are two main concerns with existing studies. First, corporate bonds and credit spreads are generally analyzed in a context where corporate debt is the only liability of the firm and a firm’s value follows a continuous stochastic process. This setup implies a zero short‐term spread, which is strongly rejected by empirical observations. The failure of generating non‐zero short‐term credit spreads may be attributed to the simplified assumption on corporate liabilities. Because a corporation generally has more than one type of liability, modeling multiple liabilities may help to incorporate discontinuity in a firm’s value and thereby lead to realistic credit term structures. Second, vulnerable options are generally valued under the assumption that a firm can fully pay off the option if the firm’s value is above the default barrier at the option’s maturity. Such an assumption is not realistic because a corporation can find itself in a solvent position at option’s maturity but with assets insufficient to pay off the option. The main contribution of this study is to address these concerns. The proposed framework extends the existing equity‐bond capital structure to an equity‐bond‐derivative setting and encompasses many existing models as special cases. The firm under study has two types of liabilities: a corporate bond and a short position in a call option. The risky corporate bond, credit spreads, and vulnerable options are analyzed and compared with their counterparts from previous models. Numerical results show that adding a derivative type of liability can lead to positive short‐term credit spreads and various shapes of credit‐spread term structures that were not possible in previous models. In addition, we found that vulnerable options need not always be worth less than their default‐free counterparts. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:301–327, 2001     

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.     

Is it important to consider the jump component for pricing and hedging short‐term options?

The usefulness of the jump component for pricing and hedging short‐term options is studied for the KOSPI (Korean Composite Stock Price Index) 200 Index options. It is found that jumps have only a marginal effect and stochastic volatility is of the most importance. There is evidence of jumps in the underlying index but no evidence of jumps in the corresponding index options. However, these results may not be valid for individual equity options. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:989–1009, 2005     

AN EQUILIBRIUM GUIDE TO DESIGNING AFFINE PRICING MODELS

The paper examines equilibrium models based on Epstein–Zin preferences in a framework in which exogenous state variables follow affine jump diffusion processes. A main insight is that the equilibrium asset prices can be computed using a standard machinery of affine asset pricing theory by imposing parametric restrictions on market prices of risk, determined inside the model by preference and model parameters. An appealing characteristic of the general equilibrium setup is that the state variables have an intuitive and testable interpretation as driving the consumption and dividend dynamics. We present a detailed example where large shocks (jumps) in consumption volatility translate into negative jumps in equilibrium prices of the assets as agents demand a higher premium to compensate for higher risks. This endogenous “leverage effect,” which is purely an equilibrium outcome in the economy, leads to significant premiums for out‐of‐the‐money put options. Our model is thus able to produce an equilibrium “volatility smirk,” which realistically mimics that observed for index options.     

The Valuation of American Options on Multiple Assets

In this paper we provide valuation formulas for several types of American options on two or more assets. Our contribution is twofold. First, we characterize the optimal exercise regions and provide valuation formulas for a number of American option contracts on multiple underlying assets with convex payoff functions. Examples include options on the maximum of two assets, dual strike options, spread options, exchange options, options on the product and powers of the product, and options on the arithmetic average of two assets. Second, we derive results for American option contracts with nonconvex payoffs, such as American capped exchange options. For this option we explicitly identify the optimal exercise boundary and provide a decomposition of the price in terms of a capped exchange option with automatic exercise at the cap and an early exercise premium involving the benefits of exercising prior to reaching the cap. Besides generalizing the current literature on American option valuation our analysis has implications for the theory of investment under uncertainty. A specialization of one of our models also provides a new representation formula for an American capped option on a single underlying asset.     

Currency option pricing: Mean reversion and multi‐scale stochastic volatility

This paper investigates the valuation of currency options when the underlying currency follows a mean‐reverting lognormal process with multi‐scale stochastic volatility. A closed‐form solution is derived for the characteristic function of the log‐asset price. European options are then valued by means of the Fourier inversion formula. The proposed model enables us to calibrate simultaneously to the observed currency futures and the implied volatility surface of the currency options within a unified framework. The fractional fast Fourier transform (FFT) is adopted to implement the Fourier inversion, thus ensuring that the grid spacing restriction of the standard FFT can be relaxed, which results in a more efficient computation. Using Monte Carlo simulation as a benchmark, our numerical examples show that the derived option pricing formula is accurate and efficient for practical use. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:938–956, 2010     

CREDIT SPREADS, OPTIMAL CAPITAL STRUCTURE, AND IMPLIED VOLATILITY WITH ENDOGENOUS DEFAULT AND JUMP RISK

We propose a two-sided jump model for credit risk by extending the Leland–Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) Jumps and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the "smooth fitting" principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.     

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