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46
A JumpDiffusion Model for Option Pricing
 Management Science
, 2002
"... Brownian motion and normal distribution have been widely used in the Black–Scholes optionpricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (as ..."
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Cited by 115 (3 self)
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Brownian motion and normal distribution have been widely used in the Black–Scholes optionpricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called “volatility smile ” in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jumpdiffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of optionpricing problems, including call and put options, interest rate derivatives, and pathdependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.
Spanning and DerivativeSecurity Valuation
, 1999
"... This paper proposes a methodology for the valuation of contingent securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payoff universe of most, if not all, derivative assets. In one specific application, from the char ..."
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Cited by 59 (5 self)
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This paper proposes a methodology for the valuation of contingent securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payoff universe of most, if not all, derivative assets. In one specific application, from the characteristic function of the stateprice density, it is possible to analytically price options on any arbitrary transformation of the underlying uncertainty. By differentiating (or translating) the characteristic function, limitless pricing and/or spanning opportunities can be designed. As made lucid via example contingent claims, by exploiting the unifying spanning concept, the valuation approach affords substantial analytical tractability. The strength and versatility of the methodology is inherent when valuing (1) Averageinterest options; (2) Correlation options; and (3) Discretelymonitored knockout options. For each optionlike security, the characteristic function is strikingly simple (although the corresponding density is unmanageable/indeterminate). This article provides the economic foundations for valuing derivative securities.
Do Bonds Span the Fixed Income Markets? Theory and Evidence for ‘Unspanned’ Stochastic Volatility
 Journal of Finance
, 2002
"... Most term structure models assume bond markets are complete, i.e., that all fixed income derivatives can be perfectly replicated using solely bonds. However, we find that, in practice, swap rates have limited explanatory power for returns on atthemoney straddles – portfolios mainly exposed to vola ..."
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Cited by 39 (0 self)
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Most term structure models assume bond markets are complete, i.e., that all fixed income derivatives can be perfectly replicated using solely bonds. However, we find that, in practice, swap rates have limited explanatory power for returns on atthemoney straddles – portfolios mainly exposed to volatility risk. We term this empirical feature “unspanned stochastic volatility ” (USV). While USV can be captured within an HJM framework, we demonstrate that bivariate models cannot exhibit USV. We determine necessary and sufficient conditions for trivariate Markov affine systems to exhibit USV. For such USVmodels, bonds alone may not be sufficient to identify all parameters. Rather, derivatives are needed.
Option Pricing by Transform Methods: Extensions, Unification, and Error Control
 Journal of Computational Finance
"... We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorith ..."
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Cited by 38 (3 self)
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We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorithms to select efficient quadrature parameters and to price with guaranteed numerical accuracy.
An Econometric Model of the Yield Curve with Macroeconomic Jump Effects
, 2000
"... This paper develops an arbitragefree timeseries model of yields in continuous time that incorporates central bank policy. Policyrelated events, such as FOMC meetings and releases of macroeconomic news the Fed cares about, are modeled as jumps. The model introduces a class of linearquadratic jump ..."
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Cited by 37 (2 self)
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This paper develops an arbitragefree timeseries model of yields in continuous time that incorporates central bank policy. Policyrelated events, such as FOMC meetings and releases of macroeconomic news the Fed cares about, are modeled as jumps. The model introduces a class of linearquadratic jumpdiffusions as state variables, which allows for a wide variety of jump types but still leads to tractable solutions for bond prices. I estimate a version of this model with U.S. interest rates, the Federal Reserve’s target rate, and key macroeconomic aggregates. The estimated model improves bond pricing, especially at short maturities. The “snakeshape ” of the volatility curve is linked to monetary policy inertia. A new monetary policy shock series is obtained by assuming that the Fed reacts to information available right before the FOMC meeting. According to the estimated policy rule, the Fed is mainly reacting to information contained in the yieldcurve. Surprises in analyst forecasts turn out to be merely temporary components of macro variables, so that the “humpshaped” yield response to these surprises is not consistent with a Taylortype policy rule.
Multiscale stochastic volatility asymptotics
 SIAM J. MULTISCALE MODELING AND SIMULATION
, 2003
"... In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the pri ..."
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Cited by 28 (11 self)
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In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the socalled term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the
The Term Structure of Simple Forward Rates with Jump Risk
 Jump Risk.” Mathematical Finance
, 2002
"... This paper characterizes the arbitragefree dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instan ..."
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Cited by 24 (5 self)
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This paper characterizes the arbitragefree dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instantaneous continuously compounded rates form the basis of most traditional interest rate models, simply compounded rates and their parameters are more directly observable in practice and are the basis of recent research on "market models." We consider very general types of jump processes, modeled through marked point processes, allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates. We make explicit how jump and diffusion risk premia enter into the dynamics of simple forward rates.
MeanReverting Stochastic Volatility
, 2000
"... We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of highfrequency S&P 500 index data confirms that volatility reverts slowly to its me ..."
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Cited by 22 (7 self)
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We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of highfrequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tickbytick fluctuations of the index value, but it is fast meanreverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for Europeanstyle securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the logmoneynesstomaturityratio. The results considerably simplify the estimation procedure, and the data produces estimates
A general formula for valuing defaultable securities
 Econometrica
, 2004
"... Previous research has shown that under a suitable nojump condition, the price of a defaultable security is equal to its riskneutral expected discounted cash flows if a modified discount rate is introduced to account for the possibility of default. Below, we generalize this result by demonstrating ..."
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Cited by 17 (0 self)
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Previous research has shown that under a suitable nojump condition, the price of a defaultable security is equal to its riskneutral expected discounted cash flows if a modified discount rate is introduced to account for the possibility of default. Below, we generalize this result by demonstrating that one can always value defaultable claims using expected riskadjusted discounting provided that the expectation is taken under a slightly modified probability measure. This new probability measure puts zero probability on paths where default occurs prior to the maturity, and is thus only absolutely continuous with respect to the riskneutral probability measure. After establishing the general result and discussing its relation with the existing literature, we investigate several examples for which the nojump condition fails. Each example illustrates the power of our general formula by providing simple analytic solutions for the prices of defaultable securities.