### Peter Carr Research

Peter’s research interests are primarily in the field of derivative securities, especially American-style and exotic derivatives. In collaboration with others, he has developed several new financial contracts (e.g. variance and corridor variance swaps), hedging strategies (e.g. static and semi-static hedging), and valuation models (e.g. VG, CGMY, and FMLS). He also has previously consulted for several firms and have given numerous talks at both practitioner and academic conferences.

Journal Articles

 A Simple Robust Link Between American Puts and Credit Protection by Peter Carr and Liuren Wu Review of Financial Studies, forthcoming   Abstract: We develop a simple robust link between deep out-of-the-money American put options on a company’s stock and a credit insurance contract on the company’s bond. We assume that the stock price stays above a barrier B before default but drops below a lower barrier A after default, thus generating a default corridor [A,B] that the stock price can never enter. Given the presence of this default corridor, a spread between two co-terminal American put options struck within the corridor replicates a pure credit contract paying off when and only when default occurs prior to the option expiry Article (pdf) Variance Swaps on Time-Changed Levy Processes by Peter Carr, Roger Lee, and Liuren Wu Finance and Stochastics , Forthcoming   Abstract: We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Levy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Levy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Levy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula as a special case of our framework. In the presence of negatively- skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Levy processes to equity options data. Finally we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly-quoted 2 multiple undervalues the continuously-sampled variance, then it undervalues furthermore the discretely-sampled variance. Article (pdf) Volatility Derivatives by Peter Carr and Roger Lee Annual Review of Financial Economics, Vol 1.   Abstract: Volatility derivatives are a class of derivative securities where the payoff explicitly depends on some measure of the volatility of an underlying asset. Prominent examples of these derivatives include variance swaps and VIX futures and options. We provide an overview of the current market for these derivatives. We also survey the early literature on the subject. Finally, we provide relatively simple proofs of some fundamental results related to variance swaps and volatility swaps. Article (pdf) Variance Risk Premiums by Peter Carr and Liuren Wu Review of Financial Studies, Vol 22, 3, pp 1311-1341   Abstract: We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the risk-neutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premia on five stock indexes and 35 individual stocks. Article (pdf) Options on Realized Variance and Convex Orders by Peter Carr, Helyette Geman, Dilip Madan, and Marc Yor Quantitative Finance, April, 2010   Abstract: Realized variance options and options on quadratic variation normalized to unit expectation are analyzed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds, the risk neutral densities are said to be increasing in the convex order. For Levy processes such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Levy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order. Article (pdf) Pricing swaps and options on quadratic variation under stochastic time change models - discrete observations case by Andrey Itkin and Peter Carr Review of Derivatives Research , July 2010   Abstract: We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of L\'evy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closed-form expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar log-contract approach is provided. Article (pdf) A Class of Levy Process Models with almost exact calibration of both barrier and vanilla FX options by Peter Carr and John Crosby Quantitative Finance, May 2010   Abstract: Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange. The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important of the FX Options market is that barrier options, especially Double-no-touch (DNT) options, are now so activley traded that they are o longer considered, in ay way, exotic options. Instead, traders would, in principle, like ot use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we indtoruce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The seocnd stage fits to vanilla options. The model allows for jumps (ether finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the overshoot''. Article (pdf) Online Supplement (pdf) Local Volatility Enhanced by a Jump to Default by Peter Carr and Dilip Madan SIAM Journal on Financial Mathematics, January 2010   Abstract: A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a pre-specified negative power and a deterministic function of time. The empirical work uses a power of -1.5. It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for GM and FORD over the period October 2004 to September 2007. Article (pdf) Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation by Peter Carr and Liuren Wu Journal of Financial Econometrics, July 2009   Abstract: We propose a dynamically consistent framework that allows joint valuation and estimation of stock options and credit default swaps written on the same reference company. We model default as controlled by a Poisson process with a stochastic default arrival rate. When default occurs, the stock price drops to zero. Prior to default, the stock price follows a jump-diffusion process with stochastic volatility. The instantaneous default rate and variance rate follow a bivariate continuous Markov process, with its dynamics specified to capture the empirical evidence on stock option prices and credit default swap spreads. Under this joint specification, we derive tractable pricing solutions for stock options and credit default swaps. We estimate the joint dynamics using data from both markets for eight companies that span five sectors and six major credit rating classes from B to AAA. The estimation highlights the interaction between market risk (return variance) and credit risk (default arrival) in pricing stock options and credit default swaps. Article (pdf) Multi-asset Stochastic Local Variance by Peter Carr and Peter Laurence Mathematical Finance , forthcoming   Abstract: 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 semi-robust 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. Working Paper (pdf) Saddlepoint Methods for Option Pricing by Peter Carr and Dilip Madan Journal of Computational Finance , forthcoming   Abstract: We show a new way in which saddlepoint methods can be used to determine the arbitrage-free price of a European-style option. Working Paper (pdf) Time Changed Markov Processes in Unified Credit-Equity Modelling by Rafael Mendoza, Peter Carr, and Vadim Linetsky Mathematical Finance , forthcoming   Abstract: 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 CEV model (JDCEV) 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 and hedge equity and credit derivatives. Working Paper (pdf) Hedging Variance Options on Continuous Semimartingales by Peter Carr and Roger Lee Finance and Stochastics , February 2010   Abstract: We find robust model-free hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forward-starting variance options, by dynamically trading the underlying asset, and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans. Article (pdf) On the qualitative effect of volatility and duration on prices of Asian options by Peter Carr, Christian-Oliver Ewald, and Yajun Xiao Finance Research Letters, Sept. 2008   Abstract: We show that under the Black Scholes assumption the price of an arithmetic average Asian call option with fixed strike increases with the level of volatility . This statement is not trivial to prove and for other models in general wrong. In fact we demonstrate that in a simple binomial model no such relationship holds. Under the Black-Scholes assumption however, we give a proof based on the maximum principle for parabolic partial differential equations. Furthermore we show that an increase in the length of duration over which the average is sampled also increases the price of an arithmetic average Asian call option, if the discounting effect is taken out. To show this, we use the result on volatility and the fact that a reparametrization in time corresponds to a change in volatility in the Black-Scholes model. Both results are extremely important for the risk management and risk assessment of portfolios that include Asian options. Article (pdf) Put-Call Symmetry: Extensions and Applications by Peter Carr and Roger Lee Mathematical Finance, October 2009   Abstract:Classic put-call symmetry relates the prices of puts and calls at strikes on opposite sides of the forward price. We extend put-call symmetry in several directions. Relaxing the assumptions, we generalize to unified local/stochastic volatility models and time-changed \levy processes, under a symmetry condition. Further relaxing the assumptions, we generalize to various \emph{asymmetric} dynamics. Extending the conclusions, we take an arbitrarily given payoff of European style or single/double/sequential-barrier style, and we construct a conjugate European-style claim of equal value, and thereby a semi-static hedge of the given payoff. Article (pdf) Hedging under the Heston Model with Jump-to-Default by Peter Carr and Wim Schoutens International Journal of Theoretical and Applied Finance, June, 2008   Abstract: We explain how to perfectly hedge under Heston's stochastic volatility model with jump to default. Besides the usual stock and bond, we use variance swaps and Credit Default Swaps (CDS) as hedge instruments. We explain how to perfectly hedge theoretical payoffs such as power payoffs, gamma payoffs and Dirac payoffs, before turning to the hedge for the vanillas. Article (pdf) Stochastic Risk Premiums: Stochastic Skewness in Currency Options, and Stochastic Discount Factors in International Economics by Gurdip Bakshi, Peter Carr, and Liuren Wu Journal of Financial Economics, Jan., 2008   Abstract: We develop models of stochastic discount factors in international economies that produce stochastic risk premiums and stochastic skewness in currency options. We estimate the model using time-series returns and options prices on three currency pairs that form a triangular relation. Estimation shows that the average risk premium in Japan is larger than that in the US or the UK, the global risk premium is more persistent and volatile then the country specific risk premiums, and investors respond differently to different shocks. We also identify high-frequency jumps in each economy, but find that only downside jumps are priced. Finally, our analysis shows that the risk premiums are economically compatible with movements in stock and bond market fundamentals. Article (pdf) Stochastic Skew for Currency Options by Peter Carr and Liuren Wu Journal of Financial Economics, October 2007   Abstract: We document the behavior of over-the-counter currency option prices across moneyness, maturity, and calendar time on two of the most active exchange rates over the past eight years. We find that the risk-neutral distribution of currency returns is relatively symmetric on average. However, on any given date, the conditional currency return distribution can show strong asymmetry. Furthermore, all of the standard skewness measures exhibit substantial time variation, switching signs several times over our sample. We design and estimate a class of models that captures this stochastic skewness. We show that our class of models outperform traditional jump-diffusion stochastic volatility models both in-sample and out-of-sample. Article (pdf) Overheads (pdf) On the Numerical Valuation of Option Prices in Jump Diffusion Processes by Peter Carr and Anita Mayo The European Journal of Finance , June 2007   Abstract: The fair price of a financial option on an asset that follows a Poisson jump diffusion process satisfies a partial integro-differential equation. When numerical methods are used to solve such equations, the integrals are usually evaluated using either quadrature methods or fast Fourier methods. Quadrature methods are expensive since the integrals must be evaluated at every point of the mesh. Though less so, Fourier methods are also computationally intensive since in order to avoid wrap around effects, they require enlargement of the computational domain. They are also slow to converge when the the parameters of the jump process are not smooth and for efficiency require uniform meshes. We present a different and more efficient class of methods which are based on the fact that the integrals often satisy differential equations. Depending on the process the asset follows, the equations are either ordinary differential equations or parabolic partial differential equations. Both types of equations can be accurately solved very rapidly. We discuss the methods and present results of numerical experiments. Article (pdf) A New Approach for Option Pricing Under Stochastic Volatility by Peter Carr and Jian Sun Review of Derivatives Research , May 2007   Abstract: We develop a new approach for pricing European-style contingent claims written on the time $T$ spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time $T$ variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying's futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives. Article (pdf) Realized Volatility and Variance: Options via Swaps by Peter Carr and Roger Lee Risk, May 2007   Abstract: We present explicit and readily applicable formulas for pricing options on realized variance and volatility. We use variance and volatility swaps -- or alternatively vanilla options -- as pricing benchmarks and hedging instruments. We also cover the pricing of VIX options. Article (pdf) Theory and Evidence on the Dynamic Interactions Between Sovereign Credit Default Swaps and Currency Options by Peter Carr and Liuren Wu Journal of Banking and Finance, August 2007   Abstract: Using sovereign CDS spreads and currency option data for Mexico and Brazil, we document that CDS spreads covary with both the currency option implied volatility and the slope of the implied volatility curve in moneyness. We propose a joint valuation framework, in which currency return variance and sovereign default intensity follow a bivariate diffusion with contemporaneous correlation. Estimation shows that default intensity is much more persistent than currency return variance. The market price estimates on the two risk factors also explain the well-documented evidence that historical average default probabilities are lower than those implied from credit spreads. Article (pdf) Self Decomposability and Option Pricing by Peter Carr, Helyette Geman, Dilip Madan, and Marc Yor Mathematical Finance, January 2007   Abstract: The risk-neutral process is modeled by a four parameter self-similar process of independent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all 6 models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with six to ten free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices. Article (pdf) Generating integrable one dimensional driftless diffusions by Peter Carr, Peter Laurence, and Tai-Ho Wang Comptes Rendus de l'Academie des Sciences ,Sept. 2006   Abstract: A criterion on the diffusion coefficient is formulated that allows the classification of driftless time and state dependent diffusions that are integrable in closed form via point transformations. In the time dependent and state dependent case a remarkable intertwining with the inhomogeneous Burger's equation is exploited. The criterion is constructive. It allows us to devise families of driftless diffusions parametrized by a rich class of several arbitrary functions for which the solution of any initial value problem can be expressed in closed form. We also derive an elegant form for the masters equation for infinitesimal symmetries, previously considered only in the time homogeneous case. Article (pdf) A Jump to Default Extended CEV Model: An Application of Bessel Processes by Peter Carr and Vadim Linetsky Finance and Stochastics ,August 2006   Abstract: We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for risk-neutral survival probabilities, CDS spreads, corporate bond values, and European-style equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and at-the-money implied volatilities. Article (pdf) A Tale of Two Indices by Peter Carr and Liuren Wu Journal of Derivatives, Spring 2006 Abstract: In 1993, the Chicago Board of Options Exchange (CBOE) introduced the COBE Volatility Index (VIX). This index has become the de facto benchmark for stock market volatility. On September 22, 2003, the CBOE revamped the definition and calculation of the VIX, and back-calculated the new VIX up to 1990 based on historical option prices. The CBOE is also planning to launch futures and options on the new VIX. In this paper, we describe the major differences between the old and the new VIXs, derive the theoretical underpinnings for the two indices, and discuss the practical motivation for the recent switch. We also study the historical behaviors of the two indices. Article (pdf) Overheads (pdf) A Note on Sufficient Conditions for No Arbitrage by Peter Carr and Dilip Madan Finance Research Letters , Sept. 2005 Abstract: It is shown that the absence of call spread, butterfly spread, and calendar spread arbitrages is sufficient to exclude all static arbitrages from a set of option price quotes across strikes and maturities on a single underlier. Article (pdf) Pricing Options on Realized Variance by Peter Carr, Dilip Madan, Helyette Geman, and Marc Yor Finance and Stochastics, 2005, issue 4 Abstract: Models which hypothesize that returns are pure jump processes with independent increments have been shown to be capable of capturing the observed variation of market prices of vanilla stock options across strike and maturity. In this paper, these models are employed to derive in closed form the prices of derivatives written on future realized quadratic variation. Alternative work on pricing derivatives on quadratic variation has alternatively assumed that the underlying returns process is continuous over time. We compare the model values of derivatives on quadratic variation for the two types of models and find substantial differences. Article (pdf) The Forward PDE for European Options on Stocks with Fixed Fractional Jumps by Peter Carr and Alireza Javaheri IJTAF , March 2005 Abstract: We derive a partial integro differential equation (PIDE) which relates the price of a calendar spread to the prices of butterfly spreads and the functions describing the evolution of the process. These evolution functions are the forward local variance rate and a new concept called the {\it forward local default arrival rate}. We then specialize to the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-jump value. This is a standard assumption when valuing an option written on a stock which can default. We discuss novel strategies for calibrating to a term and strike structure of European options prices. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate. Article (pdf) From Local Volatility to Local Levy Models by Peter Carr, Dilip Madan, Helyette Geman, and Marc Yor Quantitative Finance, October 2004 Abstract: We define the class of local Levy processes. These are Levy processes time changed by an inhomogeneous local speed function. The local speed function is a deterministic function of time and the level of the process itself. We show how to reverse engineer the local speed function from traded option prices of all strikes and maturities. The local Levy processes generalize the class of local volatility models. Closed forms for local speed functions for a variety of cases are also presented. Numerical methods for recovery are also described. Article (pdf) Corridor Variance Swaps by Peter Carr and Keith Lewis Risk, February 2004 Abstract: This paper studies a recent variation of a variance swap called a corridor variance swap (CVS). For this swap, returns are not counted in the realized variance calculation if the reference index level is outside some specified corridor. CVS's allow speculators to bet on both the level of the index and its realized variance. The authors propose a robust hedge of CVS's which requires that entry and exit of the corridor be treated asymmetrically in the contract specification. The main conclusion is that the complications introduced by the asymmetric treatment of entry and exit are outweighed by the improved hedge performance. Article (pdf) What Type of Process Underlies Options? A Simple Robust Test by Peter Carr and Liuren Wu Journal of Finance, December, 2003 Article (pdf) Frequently Asked Questions in Option Pricing Theory by Peter Carr Journal of Derivatives, forthcoming Abstract: We consider several Frequently Asked Questions (FAQ’s) in option pricing theory. Article (pdf) Overheads (pdf) Bessel Processes, The Integral of Geometric Brownian motion and Asian options by Peter Carr and Michael Schröder Theory of Probability and its Applications, issue 3, 2004 Abstract: This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the so-called Asian options.  Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman-Watson theory of [Y80[.  Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance.  Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper.  One of them in particular is of a principal nature and originates with the Hartman-Watson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices.  The main mathematical contribution of this paper is the development of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion for their consequences for the valuation of Asian options proper. Article (pdf) Time-Changed Levy Processes and Option Pricing by Peter Carr and Liuren Wu Journal of Financial Economics, January 2004, Vol. 71 No. 1, pp. 113-141 Abstract: The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-changed Lévy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology. Article (pdf) Overheads (pdf) Stochastic Volatility for Levy Processes by Peter Carr, Hélyette Geman, Dilip Madan, and Marc Yor Mathematical Finance, July 2003, Vol. 13 No. 3, pp. 345-382. Abstract: Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general, mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date. Article (pdf) The Finite Moment Log Stable Process and Option Pricing by Peter Carr and Liuren Wu Journal of Finance, April 2003, Vol. 58 No. 2, pp. 753-778. Abstract: We document a surprising pattern in S&P 500 option prices. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives. Article (pdf) Overheads (pdf) Why Be Backward? Forward Equations for American Options by Peter Carr and Ali Hirsa Risk, January 2003, pp. 103-107 Abstract: Originally developed as a tool for calibrating smile models, so-called forward methods can also be used to price options and derive Greeks. Here, Peter Carr and Ali Hirsa apply the technique to the pricing of continuously exercisable American-style put options, developing a forward partial integro-differential equation within a jump diffusion framework Article (pdf) Overheads (pdf) The Fine Structure of Asset Returns: An Empirical Investigation by Peter Carr, Hélyette Geman, Dilip B. Madan, and Marc Yor Journal of Business, April 2002, Volume 75 Number 2, pp. 305-32. Abstract: We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation. Article (pdf) Black Scholes Goes HyperGeometric by Claudio Albanese, Giuseppe Campolieti, Peter Carr, and Alexander Lipton Risk, December 2001, Volume14 No 12,  pp. 99-103. Abstract: We introduce a general pricing formula that extends Black-Scholes and contains as particular cases most analytically solvable models in the literature, including the quadratic and the constant elasticity of variance models for European-style and barrier options. In addition, large families of new solutions are found, containing as many as seven free parameters Article (pdf) Pricing and Hedging in Incomplete Markets by Peter Carr, Hélyette Geman, and Dilip B. Madan Journal of Financial Economics, October 2001, Volume 62 Issue 1, pp. 131-167. Abstract: We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures and associated floors which expected payoffs must exceed in order for the investor to consider the hedged and financed investment to be acceptable. By assuming that the liquid assets are priced so that each portfolio of assets has negative expected return under at least one measure, we derive a counterpart to the first fundamental theorem of asset pricing. We also derive a counterpart to the second fundamental theorem, which leads to unique derivative security pricing and hedging even though markets are incomplete. For products that are not spanned by the liquid assets of the economy, we show how our methodology provides more realistic bid–ask spreads. Article (pdf) Overheads (pdf) Commodity Covariance Contracting by Peter Carr and Anthony Corso Energy Risk, April 2001 Abstract: ER presents a solution to the problems in developing the replication of variance and covariance swaps. Article (pdf) Overheads (pdf) Optimal Positioning in Derivative Securities by Peter Carr and Dilip Madan Quantitative Finance, Jan. 2001, Volume 1, Number 1 Abstract: We consider a simple single period economy in which agents invest so as to maximize expected utility of terminal wealth. We assume the existence of three asset classes, namely a riskless asset (the bond), a single risky asset (the stock), and European options of all strikes (derivatives). In this setting, the inability to trade continuously potentially induces investment in all three asset classes. We consider both a partial equilibrium where all asset prices are initially given, and a more general equilibrium where all asset prices are endogenously determined. By restricting investor beliefs and preferences in each case, we solve for the optimal position for each investor in the three asset classes. We find that in partial or general equilibrium, heterogeneity in preferences or beliefs induces investors to hold derivatives individually, even though derivatives are not held in aggregate. Article (pdf) Overheads (pdf) Optimal Investment in Derivative Securities by Peter Carr, Xing Jin, and Dilip Madan Finance and Stochastics, January 2001, volume 5, issue1, pp:33-59 Abstract: We consider the problem of optimal investment in a risky asset, and in derivatives written on the price process of this asset, when the underlying asset price process is a pure jump L´evy process. The duality approach of Karatzas and Shreve is used to derive the optimal consumption and investment plans. In our economy, the optimal derivative payoff can be constructed from dynamic trading in the risky asset and in European options of all strikes. Specific closed forms illustrate the optimal derivative contracts when the utility function is in the HARA class and when the statistical and risk-neutral price processes are in the variance gamma (VG) class. In this case, we observe that the optimal derivative contract pays a function of the price relatives continuously through time. Article (pdf) Deriving Derivatives of Derivative Securities by Peter Carr Journal of Computational Finance, Winter 2000, Volume 4, Number 2 Abstract: Various techniques are used to simplify the derivations of “greeks” of path-independent claims in the Black-Merton-Scholes model. First, delta, gamma, speed, and other higher-order spatial derivatives of these claims are interpreted as the values of certain quantoed contingent claims.  It is then shown that all partial derivatives of such claims can be represented in terms of these spatial derivatives.  These observations permit the rapid deployment of higher-order Taylor series expansions, and this is illustrated for the case of European options. Article (pdf) Overheads (pdf) Going with the Flow: Alternative Approach for Valuing Continuous Cash Flows by Peter Carr, Alex Lipton, and Dilip Madan Risk, August 2000, Volume 13 No 8, pp. 85-89 Abstract: We consider the problem of replicating the payoffs from variable annuities with a continuous cash flow given by a function of some traded asset's price. The standard approaches involve either dynamic trading in this underlying asset or a static position in a continuum of options of all strikes and maturities. We present an alternative approach which combines dynamic trading in the underlying asset with a static position in options of a single maturity. In many instances, our approach yields explicit valuation formulas and hedging strategies when the volatility of the underlying is an arbitrary function of its price. Article (pdf) Working Paper (pdf) The Valuation of Executive Stock Options in an Intensity Based Framework by Peter Carr and Vadim Linetsky   January 2000, Volume 4,  pp.211-230.   Abstract: This paper presents a general intensity-based framework to value executive stock options (ESOs). It builds upon the recent advances in the credit risk modeling arena. The early exercise or forfeiture due to voluntary or involuntary employment termination and the early exercise due to the executive’s desire for liquidity or diversification are modeled as an exogenous point process with random intensity dependent on the stock price. Two analytically tractable specifications are given where the ESO value, expected time of exercise or forfeiture, and the expected stock price at the time of exercise or forfeiture are calculated in closed-form. Article (pdf) Correction (pdf) Option Pricing and the Fast Fourier Transform by Peter Carr and Dilip Madan Summer 1999,  Volume 2 Number 4, pp. 61-73. Abstract: In this paper the authors show how the fast Fourier transform may be used to value options when the characteristic function of the return is known analytically. Article (pdf) Overheads (pdf) Static Hedging of Timing Risk by Peter Carr and Jean Picron April 1999 Abstract: Many exotic options involve a payoff which occurs at the first passage time to a constant barrier. Although the amount to be paid is known, the time at which it is paid is not. This paper shows how a static position in vanilla European options can be used to hedge against this timing risk. We also show how the results can be used to price any barrier options. Article (pdf) Currency Covariance Contracting by Peter Carr and Dilip Madan Risk, February 1999, pp. 47-51 Abstract: We show how contracts paying the realized covariance between two currencies can be constructed by combining static positions in a continuum of options with continuous trading in underlying futures or forward contracts. The construction is general in that the volatilities and correlations are arbitrary. Article (pdf) Static Hedging of Exotic Options by Peter Carr, Katrina Ellis, and Vishal Gupta June 1998, pp. 1165-90. Abstract: This paper develops static hedges for several exotic options using standard options. The method relies on a relationship between European puts and calls with different strike prices. The analysis allows for constant volatility or for volatility smiles and frowns. Article (pdf) The Variance Gamma Process and Option Pricing by Dilip Madan, Peter Carr, and Eric Chang June 1998, Vol. 2 No. 1 Abstract: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here. Article (pdf) Randomization and the American Put by Peter Carr 1998, pp. 597-626 Abstract: While American calls on non-dividend-paying stocks may be valued as European, there is no completely explicit exact solution for the values of American puts. We use a technique called randomization to value American puts and calls on dividend-paying stocks. This technique yields a new semi-explicit approximation for American option values in the Black-Scholes model. Numerical results indicate that the approximation is both accurate and computationally efficient. Article (pdf) Overheads (pdf) Breaking Barriers by Peter Carr and Andrew Chou Risk, September 1997, pp. 139-145 Article (pdf) Two Extensions to Barrier Option Valuation by Peter Carr May 1995 Article (pdf) Static Simplicity by Peter Carr and Jonathan Bowie Risk, August 1994, pp. 44-50 Article (pdf) Overheads (pdf) Alternative Characterizations of American Put Options by Peter Carr, Robert Jarrow, and Ravi Myneni Mathematical Finance, April 1992, Volume 2, Issue 4 Abstract: We derive alternative representations of the McKean equation for the value of the American put option. Our main result decomposes the value of an American put option into the corresponding European put price and the early exercise premium. We then represent the European put price in a new manner. This representation allows us to alternatively decompose the price of an American put option into its intrinsic value and time value, and to demonstrate the equivalence of our results to the McKean equation. Article (pdf) The Stop-Loss Start-Gain Strategy and Option Valuation by Peter Carr and Robert Jarrow Fall 1990, pp. 469-92 Abstract: The downside risk in a leveraged stock position can be eliminated by using stop-loss orders. The upside potential of such a position can be captured using contingent buy orders. The terminal payoff of this stop-loss start-gain strategy is identical to that of a call option, but the strategy costs less initially. This article resolves this paradox by showing that the strategy is not self-financing for continuous stock-price processes of unbounded variation  The resolution of the paradox leads to a new decomposition of an option’s price into its intrinsic and time value. When the stock price follows geometric Brownian motion, this decomposition is proven to be mathematically equivalent to the Black-Scholes (1973) formula. Article (pdf) The Valuation of Sequential Exchange Opportunities by Peter Carr December 1988, pp. 1235-56 Abstract: Sequential exchange opportunities are valued using the techniques of modern option-pricing theory. The vehicle for analysis is the concept of a compound exchange option. This security is shown to exist implicitly in several contractual settings. A valuation formula for this option is derived.  The formula is shown to generalize much previous work in option pricing.  Several applications of the formula are presented. Article (pdf) A Calculator Program for Option Values and Implied Standard Deviations by Peter Carr Journal of Financial Education, Fall 1988 Article (pdf) A Note on the Pricing of Commodity-Linked Bonds by Peter Carr September 1987, pp. 1071-76 Abstract: The owner of a commodity-linked bond can exchange the bond’s face value for the maturity value of a commodity.  In an interesting application of option-pricing theory, Schwartz provides a general framework for valuing commodity-linked bonds.  This very general valuation framework allows for commodity price risk, default risk, and interest rate risk, along with interest payments and dividends.  A partial differential equation and its associated boundary condition are derived but not solved. Schwartz states that the solution to the general problem is difficult even by numerical methods.  In order to find closed-form solutions, Schwartz considers commodity price risk in conjunction with default risk or interest rate risk but not both.  This note presents a closed-form solution for the value of a commodity-linked bond when all three types of risk are present. Article (pdf)