We consider the role of options when markets in its underlying asset are frictionless and when th... more We consider the role of options when markets in its underlying asset are frictionless and when this underlying has a volatility process and jump arrival rates which are arbitrarily stochastic. By combining a static option position with a particular dynamic hedging strategy, we characterize the option’s time value as the (risk-neutral) expected benefit from being able to buy or sell one share of the underlying at the option’s strike whenever the strike price is crossed. The buy/sell decision can be based on the post jump price, so that a rational investor buys on rises and sells on drops. Thus, an option provides liquidity at its strike even when the market doesn’t. We next present two methods for extending this local liquidity to every price between the pre and post jump level. The first method involves holding a continuum of options of all strikes. The second method holds one option, but adjusts the dynamic hedging strategy. We discuss the advantages and disadvantages of each appro...
In this paper we formulate a regression problem to predict realized volatility by using option pr... more In this paper we formulate a regression problem to predict realized volatility by using option price data and enhance VIX-styled volatility indices' predictability and liquidity. We test algorithms including regularized regression and machine learning methods such as Feedforward Neural Networks (FNN) on S&P 500 Index and its option data. By conducting a time series validation we find that both Ridge regression and FNN can improve volatility indexing with higher prediction performance and fewer options required. The best approach found is to predict the difference between the realized volatility and the VIX-styled index's prediction rather than to predict the realized volatility directly, representing a successful combination of human learning and machine learning. We also discuss suitability of different regression algorithms for volatility indexing and applications of our findings.
We develop a new, decentralized theory that determines the fair value of the yield to maturity on... more We develop a new, decentralized theory that determines the fair value of the yield to maturity on a bond or bond portfolio based purely on the near-term dynamics of its own yield, without the need to make assumptions on the instantaneous interest rate dynamics, nor the need to know whether and how the yield dynamics will change in the future. The new theory decomposes the yield into three components: near-term expectation, risk premium, and convexity effects. We propose to estimate the convexity effect with its recent time series and determine the expectation from either statistical models or economists' forecasts, leaving the remaining component of the yield as a risk premium estimate. Empirical analysis on US and UK swap rates shows that this risk premium component can predict future bond excess returns.
We use simulation to develop a Markov chain approximation for the value of caplets and Bermudan i... more We use simulation to develop a Markov chain approximation for the value of caplets and Bermudan interest rate derivatives in the Market Model developed by Brace, Gatarek, and Musiela (1995) and Jamshidian (1996a,b). One and two factor versions of the Market Model were numerically studied. Our approach yields numerical values for caplets which are in close agreement with analytic solutions. We also provide numerical solutions for several Bermudan swaptions.
This paper synthesizes the existing discrete-time option pricing literature. We posit a binomial ... more This paper synthesizes the existing discrete-time option pricing literature. We posit a binomial process on a term structure of futures prices and derive existing discrete-time results as special cases of our general framework.
This paper describes another extension of the Local Variance Gamma model originally proposed by P... more This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded Local Variance Gamma model), here we provide two innovations. First, in all previous papers the model was constructed based on a Gamma timechanged arithmetic Brownian motion: with no drift in CI2017, and with drift in CI2018, and the local variance to be a function of the spot level only. In contrast, here we develop a geometric version of this model with drift. Second, in CN2017 the model was calibrated to option smiles assuming the local variance is a piecewise constant function of strike, while in CI2018 the local variance is a piecewise linear function of strike. In this paper we consider 3 piecewise linear models: the local variance as a function of strike, the local variance as function of log-strike, and the local volatility as a function of strike (so, the local variance is a piecewise quadratic function of strike). We show that for all these new constructions it is still possible to derive an ordinary differential equation for the option price, which plays a role of Dupire's equation for the standard local volatility model, and, moreover, it can be solved in closed form. Finally, similar to CI2018, we show that given multiple smiles the whole local variance/volatility surface can be recovered which does not require solving any optimization problem. Instead, it can be done term-by-term by solving a system of non-linear algebraic equations for each maturity which is fast.
In option pricing, it is customary to first specify a stochastic underlying model and then extrac... more In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an $\ell ^{p}$ ℓ p vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide par...
Diffusions are widely used in finance due to their tractability. Driftless diffusions are needed ... more Diffusions are widely used in finance due to their tractability. Driftless diffusions are needed to describe ratios of asset prices under a martingale measure. We provide a simple example of a tractable driftless diffusion which also has a bounded state space.
A single saddlepoint approximation for call prices seen as complementary probabilities that log p... more A single saddlepoint approximation for call prices seen as complementary probabilities that log price exceeds log strike by an independent exponential under the share measure is developed using a non-Gaussian base. The suggested base is that of a Gaussian random variable less an exponential with parameter λ. It is suggested that λ be chosen to match the volatility under the share measure. The method is implemented and observed to be exact for the Black-Scholes model. Six other models with closed forms for the cumulant generating function are also investigated.
In this paper the authors show how the fast Fourier transform may be used to value options when t... more 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.
We de¯ne the class of local L ¶ evy processes. These are L ¶ evy processes time changed by an inh... more We de¯ne the class of local L ¶ evy processes. These are L ¶ evy 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 L ¶ evy 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.
International Journal of Numerical Analysis and Modeling, 2011
The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness ... more The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness in the risk-neutral distribution governing currency returns. To capture this unique phenomenon Carr and Wu developed a model (SSM) with three dynamical state variables. They then used Fourier methods to value simple European-style options. However pricing exotic options requires numerical solution of 3D unsteady PIDE with mixed derivatives which is expensive. In this paper to achieve this goal we propose a new splitting technique. Being combined with another method of the authors, which uses pseudo-parabolic PDE instead of PIDE, this reduces the original 3D unsteady problem to a set of 1D unsteady PDEs, thus allowing a significant computational speedup. We demonstrate this technique for single and double barrier options priced using the SSM.
ABSTRACT this paper, the fixed payment is independent of the occupation time of the corridor. How... more ABSTRACT this paper, the fixed payment is independent of the occupation time of the corridor. However, variations exist in which the fixed payment accrues over time at a constant rate only while the underlying is in the corridor
We consider several Frequently Asked Questions (FAQ's) in option pricing theory. I thank Ajay Kha... more We consider several Frequently Asked Questions (FAQ's) in option pricing theory. I thank Ajay Khanna and Carol Marquardt for their comments.
Page 1. The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic... more Page 1. The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value Peter P. Carr Robert A. Jarrow Cornell University The downside risk in a leveraged stock position can be eliminated by using stop-loss orders. ...
We derive a simple relationship between the values and exercise boundaries of American puts andca... more We derive a simple relationship between the values and exercise boundaries of American puts andcalls. The relationship holds for options with the same "moneyness", although the absolute level ofthe strike price and underlying may differ. The result holds in both the Black Scholes model and ina more general diffusion setting.We thank Neil Chriss, the editor, and two anonymous referees for
We consider the role of options when markets in its underlying asset are frictionless and when th... more We consider the role of options when markets in its underlying asset are frictionless and when this underlying has a volatility process and jump arrival rates which are arbitrarily stochastic. By combining a static option position with a particular dynamic hedging strategy, we characterize the option’s time value as the (risk-neutral) expected benefit from being able to buy or sell one share of the underlying at the option’s strike whenever the strike price is crossed. The buy/sell decision can be based on the post jump price, so that a rational investor buys on rises and sells on drops. Thus, an option provides liquidity at its strike even when the market doesn’t. We next present two methods for extending this local liquidity to every price between the pre and post jump level. The first method involves holding a continuum of options of all strikes. The second method holds one option, but adjusts the dynamic hedging strategy. We discuss the advantages and disadvantages of each appro...
In this paper we formulate a regression problem to predict realized volatility by using option pr... more In this paper we formulate a regression problem to predict realized volatility by using option price data and enhance VIX-styled volatility indices' predictability and liquidity. We test algorithms including regularized regression and machine learning methods such as Feedforward Neural Networks (FNN) on S&P 500 Index and its option data. By conducting a time series validation we find that both Ridge regression and FNN can improve volatility indexing with higher prediction performance and fewer options required. The best approach found is to predict the difference between the realized volatility and the VIX-styled index's prediction rather than to predict the realized volatility directly, representing a successful combination of human learning and machine learning. We also discuss suitability of different regression algorithms for volatility indexing and applications of our findings.
We develop a new, decentralized theory that determines the fair value of the yield to maturity on... more We develop a new, decentralized theory that determines the fair value of the yield to maturity on a bond or bond portfolio based purely on the near-term dynamics of its own yield, without the need to make assumptions on the instantaneous interest rate dynamics, nor the need to know whether and how the yield dynamics will change in the future. The new theory decomposes the yield into three components: near-term expectation, risk premium, and convexity effects. We propose to estimate the convexity effect with its recent time series and determine the expectation from either statistical models or economists' forecasts, leaving the remaining component of the yield as a risk premium estimate. Empirical analysis on US and UK swap rates shows that this risk premium component can predict future bond excess returns.
We use simulation to develop a Markov chain approximation for the value of caplets and Bermudan i... more We use simulation to develop a Markov chain approximation for the value of caplets and Bermudan interest rate derivatives in the Market Model developed by Brace, Gatarek, and Musiela (1995) and Jamshidian (1996a,b). One and two factor versions of the Market Model were numerically studied. Our approach yields numerical values for caplets which are in close agreement with analytic solutions. We also provide numerical solutions for several Bermudan swaptions.
This paper synthesizes the existing discrete-time option pricing literature. We posit a binomial ... more This paper synthesizes the existing discrete-time option pricing literature. We posit a binomial process on a term structure of futures prices and derive existing discrete-time results as special cases of our general framework.
This paper describes another extension of the Local Variance Gamma model originally proposed by P... more This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded Local Variance Gamma model), here we provide two innovations. First, in all previous papers the model was constructed based on a Gamma timechanged arithmetic Brownian motion: with no drift in CI2017, and with drift in CI2018, and the local variance to be a function of the spot level only. In contrast, here we develop a geometric version of this model with drift. Second, in CN2017 the model was calibrated to option smiles assuming the local variance is a piecewise constant function of strike, while in CI2018 the local variance is a piecewise linear function of strike. In this paper we consider 3 piecewise linear models: the local variance as a function of strike, the local variance as function of log-strike, and the local volatility as a function of strike (so, the local variance is a piecewise quadratic function of strike). We show that for all these new constructions it is still possible to derive an ordinary differential equation for the option price, which plays a role of Dupire's equation for the standard local volatility model, and, moreover, it can be solved in closed form. Finally, similar to CI2018, we show that given multiple smiles the whole local variance/volatility surface can be recovered which does not require solving any optimization problem. Instead, it can be done term-by-term by solving a system of non-linear algebraic equations for each maturity which is fast.
In option pricing, it is customary to first specify a stochastic underlying model and then extrac... more In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an $\ell ^{p}$ ℓ p vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide par...
Diffusions are widely used in finance due to their tractability. Driftless diffusions are needed ... more Diffusions are widely used in finance due to their tractability. Driftless diffusions are needed to describe ratios of asset prices under a martingale measure. We provide a simple example of a tractable driftless diffusion which also has a bounded state space.
A single saddlepoint approximation for call prices seen as complementary probabilities that log p... more A single saddlepoint approximation for call prices seen as complementary probabilities that log price exceeds log strike by an independent exponential under the share measure is developed using a non-Gaussian base. The suggested base is that of a Gaussian random variable less an exponential with parameter λ. It is suggested that λ be chosen to match the volatility under the share measure. The method is implemented and observed to be exact for the Black-Scholes model. Six other models with closed forms for the cumulant generating function are also investigated.
In this paper the authors show how the fast Fourier transform may be used to value options when t... more 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.
We de¯ne the class of local L ¶ evy processes. These are L ¶ evy processes time changed by an inh... more We de¯ne the class of local L ¶ evy processes. These are L ¶ evy 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 L ¶ evy 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.
International Journal of Numerical Analysis and Modeling, 2011
The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness ... more The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness in the risk-neutral distribution governing currency returns. To capture this unique phenomenon Carr and Wu developed a model (SSM) with three dynamical state variables. They then used Fourier methods to value simple European-style options. However pricing exotic options requires numerical solution of 3D unsteady PIDE with mixed derivatives which is expensive. In this paper to achieve this goal we propose a new splitting technique. Being combined with another method of the authors, which uses pseudo-parabolic PDE instead of PIDE, this reduces the original 3D unsteady problem to a set of 1D unsteady PDEs, thus allowing a significant computational speedup. We demonstrate this technique for single and double barrier options priced using the SSM.
ABSTRACT this paper, the fixed payment is independent of the occupation time of the corridor. How... more ABSTRACT this paper, the fixed payment is independent of the occupation time of the corridor. However, variations exist in which the fixed payment accrues over time at a constant rate only while the underlying is in the corridor
We consider several Frequently Asked Questions (FAQ's) in option pricing theory. I thank Ajay Kha... more We consider several Frequently Asked Questions (FAQ's) in option pricing theory. I thank Ajay Khanna and Carol Marquardt for their comments.
Page 1. The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic... more Page 1. The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value Peter P. Carr Robert A. Jarrow Cornell University The downside risk in a leveraged stock position can be eliminated by using stop-loss orders. ...
We derive a simple relationship between the values and exercise boundaries of American puts andca... more We derive a simple relationship between the values and exercise boundaries of American puts andcalls. The relationship holds for options with the same "moneyness", although the absolute level ofthe strike price and underlying may differ. The result holds in both the Black Scholes model and ina more general diffusion setting.We thank Neil Chriss, the editor, and two anonymous referees for
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