Equity swaps in a LIBOR market model

This study extends the BGM (A. Brace, D. Gatarek, & M. Musiela, 1997) interest rate model (the London Interbank Offered Rate [LIBOR] market model) by incorporating the stock price dynamics under the martingale measure. As compared with traditional interest rate models, the extended BGM model is both appropriate for pricing equity swaps and easy to calibrate. The general framework for pricing equity swaps is proposed and applied to the pricing of floating‐for‐equity swaps with either constant or variable notional principals. The calibration procedure and the practical implementation are also discussed. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:893–920, 2007     

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     

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.     

MULTI‐ASSET STOCHASTIC LOCAL VARIANCE CONTRACTS

Variance swaps now trade actively over‐the‐counter (OTC) on both stocks and stock indices. Also trading OTC are variations on variance swaps which localize the payoff in time, in the underlying asset price, or both. Given that the price of the underlying asset evolves continuously over time, it is well known that there exists a semirobust hedge for these localized variance contracts. Remarkably, the hedge succeeds even though the stochastic process describing the instantaneous variance is never specified. In this paper, we present a generalization of these results to the case of two or more underlying assets.     

A CLOSED‐FORM EXACT SOLUTION FOR PRICING VARIANCE SWAPS WITH STOCHASTIC VOLATILITY

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed‐form exact solution for the partial differential equation (PDE) system based on the Heston's two‐factor stochastic volatility model embedded in the framework proposed by Little and Pant. In comparison with the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed‐form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous‐sampling‐time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include the following: (1) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (2) numerical values can be very efficiently computed from the newly found analytic formula.     

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.     

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     

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.     

TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.     

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.     

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.     

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.     

Debt-equity swaps and the heavily indebted countries

In search of solutions to the international debt crisis, attention has recently been focused on a new financing technique, so-called debt-equity swaps. An essential difference between these and the usual swapping of debt into equity is that the former allow a wider range of applications. The following article seeks to elucidate the possible contribution of debt-equity swaps towards easing the debt burden and to estimate the potential for a reduction in external debt and its effect on the balance of payments of the debtor nation.     

Compositional Effects of Capital Controls – Theory and Evidence

This paper examines the effects of capital controls on the composition of inter‐national capital flows, paying particular attention to debt inflows versus equity inflows. A two‐period small open economy model with stochastic second‐period output and asymmetric information between domestic agents and international financiers is utilised to generate predictions regarding the effects of capital controls on the relative use of debt versus equity for financing first‐period investment. These capital control implications are then investigated with quarterly frequency panel data for Latin America. Capital controls are found to significantly affect the composition of the capital account.     

Pricing variance swaps under the Hawkes jump-diffusion process

This paper presents an analytical approach for pricing variance swaps with discrete sampling times when the underlying asset follows a Hawkes jump-diffusion process characterized with both stochastic volatility and clustered jumps. A significantly simplified method, with which there is no need to solve partial differential equations, is used to derive a closed-form pricing formula. A distinguished feature is that many recently published formulas can be shown to be special cases of the one presented here. Some numerical examples are provided with results demonstrating that jump clustering indeed has a significant impact on the price of variance swaps.     

Pricing American options with stochastic volatility: Evidence from S&P 500 futures options

This article is the first attempt to test empirically a numerical solution to price American options under stochastic volatility. The model allows for a mean‐reverting stochastic‐volatility process with non‐zero risk premium for the volatility risk and correlation with the underlying process. A general solution of risk‐neutral probabilities and price movements is derived, which avoids the common negative‐probability problem in numerical‐option pricing with stochastic volatility. The empirical test shows clear evidence supporting the occurrence of stochastic volatility. The stochastic‐volatility model outperforms the constant‐volatility model by producing smaller bias and better goodness of fit in both the in‐sample and out‐of‐sample test. It not only eliminates systematic moneyness bias produced by the constant‐volatility model, but also has better prediction power. In addition, both models perform well in the dynamic intraday hedging test. However, the constant‐volatility model seems to have a slightly better hedging effectiveness. The profitability test shows that the stochastic volatility is able to capture statistically significant profits while the constant volatility model produces losses. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:625–659, 2000     

ARBITRAGE‐FREE BILATERAL COUNTERPARTY RISK VALUATION UNDER COLLATERALIZATION AND APPLICATION TO CREDIT DEFAULT SWAPS

We develop an arbitrage‐free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on‐default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation‐induced contagion is illustrated with numerical examples.     

Black‐Scholes‐Merton revisited under stochastic dividend yields

European options are priced in a framework à la Black‐Scholes‐Merton, which is extended to incorporate stochastic dividend yield under a stochastic mean–reverting market price of risk. Explicit formulas are obtained for call and put prices and their Greek parameters. Some well‐known properties of the Black‐Scholes‐Merton formula fail to hold in this setting. For example, the delta of the call can be negative and even greater than one in absolute terms. Moreover, call prices can be a decreasing function of the underlying volatility although the latter is constant. Finally, and most importantly, option prices highly depend on the features of the market price of risk, which does not need to be specified at all in the standard Black‐Scholes‐Merton setting. The results are simulated in order to assess the economic impact of assuming that the dividend yield is deterministic when it is actually stochastic, as well as to assess the economic importance of the features of the market price of risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:703–732, 2006