Valuation and Hedging of Differential Swaps

This paper derives a general‐form formula for pricing and hedging differential swaps with the principal denominated either in a domestic, foreign, or third‐country currency. We first derive the formula for differential swaps with the principal in a domestic currency and identify an error in the formula of Wei (1994). We then show the pricing duality between differential swaps with the principal in a domestic currency and differential swaps with the principal in a foreign currency. Finally, we complete the pricing and hedging analysis on differential swaps by deriving a formula for differential swaps with the principal denominated in a third‐country currency. Simulation results indicate that constant margin rates are generally smaller than interest rate differentials and decline with the tenor of swaps. Correlation parameters associated with the exchange rate play a more important role than correlation parameters among interest rates in pricing differential swaps. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:73–94, 2002     

A term structure model for dividends and interest rates

Over the last decade, dividends have become a standalone asset class instead of a mere side product of an equity investment. We introduce a framework based on polynomial jump‐diffusions to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. We present an efficient moment based approximation method for option pricing. In a calibration exercise we show that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 Index dividend futures and dividend options, and Euro Stoxx 50 Index options.     

Interest Rate,Currency and Equity Derivatives Valuation Using the Potential Approach

Based on the potential approach to interest rate modelling, we introduce a simple tractable model for the unified valuation of interest rate, currency and equity derivatives. Our model is able to accommodate the initial term structure of zero‐coupon bond prices, generate positive and bounded interest rates, and handle cross products such as differential swaps, quanto options and equity swaps. As our model is specified under the actual probability measure, it can be directly used for portfolio risk management and the computation of value at risk. Furthermore, our model yields simple analytical formulas that are easy to calibrate and implement.     

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.     

Pricing models of equity swaps

This article provides a generalized formula for pricing equity swaps with constant notional principal when the underlying equity markets and settlement currency can be set arbitrarily. To derive swap values using the risk‐neutral valuation method, the swap payment is replicated at each settlement date by constructing a self‐financing portfolio. To obtain the foreign equity index return denominated in the domestic or in a third currency, equity‐linked foreign exchange options are used to hedge the exchange rate risk. It is found that if the swap involves international equity markets, then the swap value contains an extra term which reflects the currency hedging costs. This methodology can easily be applied to price various types of equity swaps simply by modifying the specifications of the model presented here as required. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:751–772, 2003     

Capped equity swaps under the double‐jump stochastic volatility model with stochastic interest rates

This study proposes a double‐jump stochastic volatility model with stochastic interest rates to price capped equity swaps and other multi‐period derivative securities. Closed‐form solutions for capped equity swaps with a fixed or variable notional principle are derived. In addition, numerical examples are employed to analyze comparative statics properties, counterparty risks, and the dynamics of the forward smile. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:340–370, 2011     

A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid, or over‐the‐counter derivatives. We observe that our model matches the equity option implied volatility surface well since we properly account for the default risk in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque et al., which only accounts for stochastic volatility.     

Currency‐Protected Swaps and Swaptions with Nonzero Spreads in a Multicurrency LMM

Despite the fact that currency‐protected swaps and swaptions are widely traded in the marketplace, pricing models for zero‐spread swaps, and swaptions have rarely been examined in the extant literature. This study presents a multicurrency LIBOR market model and uses it to derive pricing formulas for currency‐protected swaps and swaptions with nonzero spreads. The resulting pricing formulas are shown to be feasible and tractable for practical implementation and their hedging strategies are also provided. Our pricing formulas provide prices close to those computed from Monte Carlo simulation, but involve far less computation time, and thereby offering almost instant price quotes to clients and daily marking‐to‐market trading books, and facilitating efficient risk management of trading positions.     

Asymmetric information and credit quality: Evidence from synthetic fixed‐rate financing

In this article the usage of synthetic fixed‐rate financing (SFRF) with interest rate swaps (i.e., borrowing short‐term and using swaps to hedge interest rate risk, instead of selecting conventional fixed‐rate financing) by Fortune 500 and S&P 500 nonfinancial firms is examined over the period 1991 through 1995. Credit ratings, debt issuance, and debt maturities of these firms are monitored through 1999. Strong evidence is found supporting the asymmetric information theory of swap usage as described by S. Titman (1992), even after controlling for industry, credit quality, size effects, and the simultaneity of the capital structure and the interest rate swap usage decision. Consistent with theoretical predictions, SFRF firms are more likely to undergo credit quality upgrades. When limiting the sample to firms where asymmetric information costs are potentially the greatest, the results are even stronger. These findings are important because they document that swaps serve a highly valuable service for firms subject to information asymmetries. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:595–626, 2006     

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 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     

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.     

Pricing and Hedging the Smile with SABR: Evidence from the Interest Rate Caps Market

This is the first comprehensive study of the SABR (stochastic alpha‐beta‐rho) model (Hagan, Kumar, Lesniewski, & Woodward, 2002) on the pricing and hedging of interest rate caps. I implement several versions of the SABR interest rate model and analyze their respective pricing and hedging performance using two years of daily data with seven different strikes and ten different tenors on each trading day. In‐sample and out‐of‐sample tests show that the fully stochastic version of the SABR model exhibits excellent pricing accuracy and, more importantly, captures the dynamics of the volatility smile over time very well. This is further demonstrated through examining delta‐hedging performance based on the SABR model. My hedging result indicates that the SABR model produces accurate hedge ratios that outperform those implied by the Black model. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:773‐791, 2012     

Forecasting performance of extreme‐value volatility estimators

This study evaluates the forecasting performance of extreme‐value volatility estimators for the equity‐based Nifty Index using two‐scale realized volatility. This benchmark mitigates the effect of microstructure noise in the realized volatility. Extreme‐value estimates with relatively simple forecasting methods provide substantially better short‐term and long‐term forecasts, compared to historical volatility. The higher efficiency of extreme‐value estimators is primarily responsible for this improvement. The extent of possible improvement in forecasts is likely to be economically significant for applications like options pricing. By including extremevalue estimators, the forecasting performance of generalized autoregressive conditional heteroscedasticity (GARCH) can also be improved. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27: 1085–1105, 2007     

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.     

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.     

Dissecting interbank risk using basis swap spreads

This paper analyses interbank risk using the information content of basis swap (BS) spreads, floating-to-floating interest rate swaps whose payments are associated with euro deposit rates for alternative tenors. To identify the impact of shocks affecting interbank risk, we propose an empirical model that decomposes BS quotes into their expected and unexpected components. These unobservable constituents of BS spreads are estimated by solving a signal extraction problem using a particle filter. We find that expected components covariate with aggregate liquidity and risk aversion while systemic risk arises as the main driver behind unexpected fluctuations. Our empirical findings suggest that macroprudential analysis emerges as a key device to ease asset pricing in a new multi-curve scenario.     

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS IN LÉVY MODELS WITH STOCHASTIC INTEREST RATE

Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.     

The Valuation Model for a Risky Asset When Its Risky Factors Follow Gamma Distributions

This paper constructed a pricing model for the asset with multi‐risks by specifying the risky factors (i.e., interest rate and termination hazard rates) to follow gamma distributions. The model not only avoids the possibility of the termination hazard rate taking an irrational (i.e., negative) value, but it also makes it easier to derive a valuation formula for a risky asset. Our model can also effortless apply because the parameters of the gamma distribution can easily be estimated from market data. An example using Taiwanese bond data illustrates how the model can be utilized for practical applications. To facilitate understanding of how accurately the different models price risky bonds, we compare their out‐of‐sample pricing errors for different hazard rate specifications assuming normal and gamma distributions. The results show that our pricing formula is realistic and accurate in its applications. Therefore, it should help market participants to accurately price risky assets and to effectively manage complicated portfolios.     

Effects of interest rate swaps

In this paper we examine the effect of interest rate swaps on the firm, and identify characteristics of firms that use interest rate swaps, reporting findings consistent with interest rate swaps being used as a risk-reducing instrument. Relative to nonswappers, firms using swaps are more likely to experience decreased cash flow variance in the five-year period subsequent to swap initiation. In addition, firms that engage in swaps are found to be larger and more highly levered than a control sample of nonswappers. Dividing our sample based upon type of swap, we find different characteristics explain different types of swap. In particular we find evidence consistent with swaps from variable to fixed interest rates being engaged in for risk reduction, i.e., hedging purposes.