The purpose of this paper is to test the time-invariance of the beta coefficients estimated by th... more The purpose of this paper is to test the time-invariance of the beta coefficients estimated by the Adaptive Multi-Factor (AMF) model. The AMF model is implied by the generalized arbitrage pricing theory (GAPT), which implies constant beta coefficients. The AMF model utilizes a Groupwise Interpretable Basis Selection (GIBS) algorithm to identify the relevant factors from among all traded ETFs. We compare the AMF model with the Fama-French 5-factor (FF5) model. We show that for nearly all time periods with length less than 6 years, the beta coefficients are time-invariant for the AMF model, but not for the FF5 model. This implies that the AMF model with a rolling window (such as 5 years) is more consistent with realized asset returns than is the FF5 model.
This paper studies contingent claim valuation of risky assets in a stochastic interest rate econo... more This paper studies contingent claim valuation of risky assets in a stochastic interest rate economy. The model employed generalizes the approach utilized by Heath, Jarrow, and Morton (1992) by imbedding their stochastic interest rate economy into one containing an arbitrary number of additional risky assets. We derive closed form formulae for certain types of European options in this context, notably call and put options on risky assets, forward contracts, and futures contracts. We also value American contingent claims whose payoffs are permitted to be general functions of both the term structure and asset prices generalizing Bensoussan (1984) and in this regard. Here, we provide an example where an American call's value is well defined, yet there does not exist an optimal trading strategy which attains this value. Furthermore, this example is not pathological as it is a generalization of formula for a call option on a stock that pays discrete dividends.
A constrained informationally efficient market is defined to be one whose price process arises as... more A constrained informationally efficient market is defined to be one whose price process arises as the outcome of some equilibrium where agents face restrictions on trade. This paper investigates the case of short sale constraints, a setting which despite its simplicity, generates new insights. In particular, it is shown that short sale constrained informationally efficient markets always admit equivalent supermartingale measures and local martingale deflators, but not necessarily local martingale measures. And if in addition some local martingale deflator turns the price process into a true martingale, then the market is constrained informationally efficient. Examples are given to illustrate the subtle phenomena that can arise in the presence of short sale constraints, with particular attention to representative agent equilibria and the different notions of no arbitrage.
It is commonly believed that the trading of futures on a commodity enables the market to overcome... more It is commonly believed that the trading of futures on a commodity enables the market to overcome short selling constraints on the spot commodity itself. This belief is embedded in the notion that trading strategies involving futures contracts enable traders to replicate the payoffs as if they were short the spot commodity. The purpose of this paper is to investigate this common belief in a general arbitrage-free semimartingale financial model with trading in futures and a short selling prohibition on the spot commodity. We show via various examples that, in general, this common belief is incorrect. Furthermore, we provide a set of sufficient conditions, albeit very restrictive, under which the common belief is true. KEY WORDS: Short sale constraints, futures contracts, futures prices, complete markets, martingale representation, supermartingale measures, overpricing hypothesis, bubbles * We would like to thank Professor Walter Schachermayer for providing the main ideas that led to Example 6.9. We would also like to thank the anonymous referees for their careful reviews of the manuscript.
The paper proposes a new algorithm for the high-dimensional financial data-the Groupwise Interpre... more The paper proposes a new algorithm for the high-dimensional financial data-the Groupwise Interpretable Basis Selection (GIBS) algorithm, to estimate a new Adaptive Multi-Factor (AMF) asset pricing model, implied by the recently developed Generalized Arbitrage Pricing Theory, which relaxes the convention that the number of risk-factors is small. We first obtain an adaptive collection of basis assets and then simultaneously test which basis assets correspond to which securities, using high-dimensional methods. The AMF model, along with the GIBS algorithm, is shown to have a significantly better fitting and prediction power than the Fama-French 5-factor model.
Incomplete information is at the heart of information-based credit risk models. In this paper, we... more Incomplete information is at the heart of information-based credit risk models. In this paper, we rigorously define incomplete information with the notion of "delayed filtrations". We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando ( ) and the partial information in Collin-Dufresne et al. ( ), under which structural models are translated into reduced-form intensity-based models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an information-based model. * The authors thank the Associate Editor and the two anonymous referees for their careful reading, constructive suggestions, and enlightening remarks.
In spite of the popularity of model calibration in …nance, empirical researchers have put more em... more In spite of the popularity of model calibration in …nance, empirical researchers have put more emphasis on model estimation than on the equally important goodness-of-…t problem. This is due partly to the ignorance of modelers, and more to the ability of existing statistical tests to detect speci…cation errors. In practice, models are often calibrated by minimizing a loss function of the di¤erences between the modelled and actual observations. Under this approach, it is challenging to disentangle model error from estimation error in the residual series. To circumvent the di¢ culty, we study an alternative way of estimating the model by exact calibration. Unlike the error minimization approach, all information about dynamic misspeci…cations is channeled to the parameter estimation residuals under exact calibration. In the context of option pricing, we illustrate that standard time series tests are powerful in detecting various kinds of dynamic misspeci…cations. Compared to the error minimization approach, exact calibration yields more reasonable model comparison result, and delivers more accurate hedging performance that is robust to both gradual and abrupt structural shifts of state variables. The authors would like to thank the Guest Co-editors, Michael McAleer, Shiqing Ling and Howell Tong, and the referees for their invaluable comments and suggestions, which lead to signi…cant improvements to the paper. We also thank the conference organizers and participants at the Frontiers of Time Series Analysis and Related Fields at HKU, the 2014 Asian Meeting of the Econometric Society at the Academia Sinica, and the 24th New Zealand Econometrics Study Group Meeting at the University of Waikato Management School, as well as all the seminar participants at the
William and Mary for helpful comments. We also thank Jens Christensen for excellent research assi... more William and Mary for helpful comments. We also thank Jens Christensen for excellent research assistance. David Lando acknowledges the partial support of the Danish Social Science Foundation.
Journal of International Money and Finance, Sep 1, 1991
In this paper, we build a general framework to price contingent claims on foreign currencies usin... more In this paper, we build a general framework to price contingent claims on foreign currencies using the Heath et al. (1987) model of the term structure. Closed form solutions are obtained for European options on currencies and currency futures assuming that the volatility functions determining the term structure are deterministic. As such, this paper provides an example of a bond price process (for both the domestic and foreign economies) consistent with Grabbe's (1983) formulation of the same problem.
This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] ... more This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager's information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the manager's information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azéma martingale, and we use excursion theory of Brownian motions to price risky debt.
This paper develops a structural credit risk model to characterize the difference between the eco... more This paper develops a structural credit risk model to characterize the difference between the economic and recorded default times for a firm. Recorded default occurs when default is recorded in the legal system. The economic default time is the last time when the firm is able to pay off its debt prior to the legal default time. It has been empirically documented that these two times are distinct (see ). In our model, the probability distribution for the time span between economic and recorded defaults follows a mixture of Arcsine Laws, which is consistent with the results contained in Guo, Jarrow, and Lin. In addition, we show that the classical structural model is a limiting case of our model as the time period between debt repayment dates goes to zero. As a corollary, we show how the firm value process's parameters can be estimated using the tail index and correlation structure of the firm's return.
Recent academic work has developed a method to determine, in real time, if a given stock is exhib... more Recent academic work has developed a method to determine, in real time, if a given stock is exhibiting a price bubble. Currently there is speculation in the financial press concerning the existence of a price bubble in the aftermath of the recent IPO of LinkedIn. We analyze stock price tick data from the short lifetime of this stock through May 24, 2011, and we find that LinkedIn has a price bubble.
We show that with suitable restrictions on allowable trading strategies, one has no arbitrage in ... more We show that with suitable restrictions on allowable trading strategies, one has no arbitrage in settings where the traditional theory would admit arbitrage possibilities. In particular, price processes that are not semimartingales are possible in our setting, for example, fractional Brownian motion.
Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these sw... more Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.
Classical theories of financial markets assume an infinitely liquid market and that all traders a... more Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a security's price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.
Economic theory tells us that market and credit risks are intrinsically related to each other and... more Economic theory tells us that market and credit risks are intrinsically related to each other and not separable. We describe the two main approaches to pricing credit risky instruments: the structural approach and the reduced form approach. It is argued that the standard approaches to credit risk management ± CreditMetrics, CreditRisk+ and KMV ± are of limited value when applied to portfolios of interest rate sensitive instruments and in measuring market and credit risk. Empirically returns on high yield bonds have a higher correlation with equity index returns and a lower correlation with Treasury bond index returns than do low yield bonds. Also, macro economic variables appear to in¯uence the aggregate rate of business failures. The CreditMetrics, CreditRisk+ and KMV methodologies cannot reproduce these empirical observations given their constant interest rate assumption. However, we can incorporate these empirical observations into the reduced form of Jarrow and Turnbull (1995b). Drawing the analogy. Risk 5, 63±70 model. Here default probabilities are correlated due to their dependence on common economic factors. Default risk and recovery rate uncertainty may not be the sole determinants of the credit spread. We show how to incorporate a convenience yield as one of the determinants of the credit spread. For credit risk management, the time horizon is typically one year or longer. This has two important implications, since the standard approximations do not apply over a one
Financial bubbles arise when the underlying asset's market price deviates from its fundamental va... more Financial bubbles arise when the underlying asset's market price deviates from its fundamental value. Unlike other bubble tests that use time series data and assume a reduced-form price process, we infer the existence of bubbles nonparametrically using option price data. Under no-arbitrage and acknowledging data constraints, we can partially identify asset price bubbles using a cross section of European option prices. In the empirical analysis, we obtain interval estimates of price bubbles embedded in the S&P 500 Index. The estimated index bubbles are then used to construct pro…table momentum trading strategies that consistently outperform a buy-and-hold trading strategy.
This paper provides a simple analytic formula for valuing default swaps with correlated market an... more This paper provides a simple analytic formula for valuing default swaps with correlated market and credit risk. We illustrate the numerical implementation of this model by inferring the default probability parameters implicit in default swap quotes for twenty two companies over the time period 8/21/00 to 10/31/00. For comparison, with also provide implicit estimates for the standard model (a special case of our approach) where market and credit risk are statistically independent.
Testing for the existence of asset price bubbles is a difficult task. This paper shows that the l... more Testing for the existence of asset price bubbles is a difficult task. This paper shows that the local martingale theory of bubbles provides three new approaches that can be used to test for the existence of asset price bubbles, two of which are currently unexplored in the literature. The first is based on specifying a stochastic process for the asset's price process. The second exploits option price data, if options trade on the asset suspected to exhibit a price bubble. The third uses a multiple-factor model for the return process of the underlying asset.
The purpose of this paper is to test the time-invariance of the beta coefficients estimated by th... more The purpose of this paper is to test the time-invariance of the beta coefficients estimated by the Adaptive Multi-Factor (AMF) model. The AMF model is implied by the generalized arbitrage pricing theory (GAPT), which implies constant beta coefficients. The AMF model utilizes a Groupwise Interpretable Basis Selection (GIBS) algorithm to identify the relevant factors from among all traded ETFs. We compare the AMF model with the Fama-French 5-factor (FF5) model. We show that for nearly all time periods with length less than 6 years, the beta coefficients are time-invariant for the AMF model, but not for the FF5 model. This implies that the AMF model with a rolling window (such as 5 years) is more consistent with realized asset returns than is the FF5 model.
This paper studies contingent claim valuation of risky assets in a stochastic interest rate econo... more This paper studies contingent claim valuation of risky assets in a stochastic interest rate economy. The model employed generalizes the approach utilized by Heath, Jarrow, and Morton (1992) by imbedding their stochastic interest rate economy into one containing an arbitrary number of additional risky assets. We derive closed form formulae for certain types of European options in this context, notably call and put options on risky assets, forward contracts, and futures contracts. We also value American contingent claims whose payoffs are permitted to be general functions of both the term structure and asset prices generalizing Bensoussan (1984) and in this regard. Here, we provide an example where an American call's value is well defined, yet there does not exist an optimal trading strategy which attains this value. Furthermore, this example is not pathological as it is a generalization of formula for a call option on a stock that pays discrete dividends.
A constrained informationally efficient market is defined to be one whose price process arises as... more A constrained informationally efficient market is defined to be one whose price process arises as the outcome of some equilibrium where agents face restrictions on trade. This paper investigates the case of short sale constraints, a setting which despite its simplicity, generates new insights. In particular, it is shown that short sale constrained informationally efficient markets always admit equivalent supermartingale measures and local martingale deflators, but not necessarily local martingale measures. And if in addition some local martingale deflator turns the price process into a true martingale, then the market is constrained informationally efficient. Examples are given to illustrate the subtle phenomena that can arise in the presence of short sale constraints, with particular attention to representative agent equilibria and the different notions of no arbitrage.
It is commonly believed that the trading of futures on a commodity enables the market to overcome... more It is commonly believed that the trading of futures on a commodity enables the market to overcome short selling constraints on the spot commodity itself. This belief is embedded in the notion that trading strategies involving futures contracts enable traders to replicate the payoffs as if they were short the spot commodity. The purpose of this paper is to investigate this common belief in a general arbitrage-free semimartingale financial model with trading in futures and a short selling prohibition on the spot commodity. We show via various examples that, in general, this common belief is incorrect. Furthermore, we provide a set of sufficient conditions, albeit very restrictive, under which the common belief is true. KEY WORDS: Short sale constraints, futures contracts, futures prices, complete markets, martingale representation, supermartingale measures, overpricing hypothesis, bubbles * We would like to thank Professor Walter Schachermayer for providing the main ideas that led to Example 6.9. We would also like to thank the anonymous referees for their careful reviews of the manuscript.
The paper proposes a new algorithm for the high-dimensional financial data-the Groupwise Interpre... more The paper proposes a new algorithm for the high-dimensional financial data-the Groupwise Interpretable Basis Selection (GIBS) algorithm, to estimate a new Adaptive Multi-Factor (AMF) asset pricing model, implied by the recently developed Generalized Arbitrage Pricing Theory, which relaxes the convention that the number of risk-factors is small. We first obtain an adaptive collection of basis assets and then simultaneously test which basis assets correspond to which securities, using high-dimensional methods. The AMF model, along with the GIBS algorithm, is shown to have a significantly better fitting and prediction power than the Fama-French 5-factor model.
Incomplete information is at the heart of information-based credit risk models. In this paper, we... more Incomplete information is at the heart of information-based credit risk models. In this paper, we rigorously define incomplete information with the notion of "delayed filtrations". We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando ( ) and the partial information in Collin-Dufresne et al. ( ), under which structural models are translated into reduced-form intensity-based models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an information-based model. * The authors thank the Associate Editor and the two anonymous referees for their careful reading, constructive suggestions, and enlightening remarks.
In spite of the popularity of model calibration in …nance, empirical researchers have put more em... more In spite of the popularity of model calibration in …nance, empirical researchers have put more emphasis on model estimation than on the equally important goodness-of-…t problem. This is due partly to the ignorance of modelers, and more to the ability of existing statistical tests to detect speci…cation errors. In practice, models are often calibrated by minimizing a loss function of the di¤erences between the modelled and actual observations. Under this approach, it is challenging to disentangle model error from estimation error in the residual series. To circumvent the di¢ culty, we study an alternative way of estimating the model by exact calibration. Unlike the error minimization approach, all information about dynamic misspeci…cations is channeled to the parameter estimation residuals under exact calibration. In the context of option pricing, we illustrate that standard time series tests are powerful in detecting various kinds of dynamic misspeci…cations. Compared to the error minimization approach, exact calibration yields more reasonable model comparison result, and delivers more accurate hedging performance that is robust to both gradual and abrupt structural shifts of state variables. The authors would like to thank the Guest Co-editors, Michael McAleer, Shiqing Ling and Howell Tong, and the referees for their invaluable comments and suggestions, which lead to signi…cant improvements to the paper. We also thank the conference organizers and participants at the Frontiers of Time Series Analysis and Related Fields at HKU, the 2014 Asian Meeting of the Econometric Society at the Academia Sinica, and the 24th New Zealand Econometrics Study Group Meeting at the University of Waikato Management School, as well as all the seminar participants at the
William and Mary for helpful comments. We also thank Jens Christensen for excellent research assi... more William and Mary for helpful comments. We also thank Jens Christensen for excellent research assistance. David Lando acknowledges the partial support of the Danish Social Science Foundation.
Journal of International Money and Finance, Sep 1, 1991
In this paper, we build a general framework to price contingent claims on foreign currencies usin... more In this paper, we build a general framework to price contingent claims on foreign currencies using the Heath et al. (1987) model of the term structure. Closed form solutions are obtained for European options on currencies and currency futures assuming that the volatility functions determining the term structure are deterministic. As such, this paper provides an example of a bond price process (for both the domestic and foreign economies) consistent with Grabbe's (1983) formulation of the same problem.
This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] ... more This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager's information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the manager's information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azéma martingale, and we use excursion theory of Brownian motions to price risky debt.
This paper develops a structural credit risk model to characterize the difference between the eco... more This paper develops a structural credit risk model to characterize the difference between the economic and recorded default times for a firm. Recorded default occurs when default is recorded in the legal system. The economic default time is the last time when the firm is able to pay off its debt prior to the legal default time. It has been empirically documented that these two times are distinct (see ). In our model, the probability distribution for the time span between economic and recorded defaults follows a mixture of Arcsine Laws, which is consistent with the results contained in Guo, Jarrow, and Lin. In addition, we show that the classical structural model is a limiting case of our model as the time period between debt repayment dates goes to zero. As a corollary, we show how the firm value process's parameters can be estimated using the tail index and correlation structure of the firm's return.
Recent academic work has developed a method to determine, in real time, if a given stock is exhib... more Recent academic work has developed a method to determine, in real time, if a given stock is exhibiting a price bubble. Currently there is speculation in the financial press concerning the existence of a price bubble in the aftermath of the recent IPO of LinkedIn. We analyze stock price tick data from the short lifetime of this stock through May 24, 2011, and we find that LinkedIn has a price bubble.
We show that with suitable restrictions on allowable trading strategies, one has no arbitrage in ... more We show that with suitable restrictions on allowable trading strategies, one has no arbitrage in settings where the traditional theory would admit arbitrage possibilities. In particular, price processes that are not semimartingales are possible in our setting, for example, fractional Brownian motion.
Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these sw... more Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.
Classical theories of financial markets assume an infinitely liquid market and that all traders a... more Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a security's price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.
Economic theory tells us that market and credit risks are intrinsically related to each other and... more Economic theory tells us that market and credit risks are intrinsically related to each other and not separable. We describe the two main approaches to pricing credit risky instruments: the structural approach and the reduced form approach. It is argued that the standard approaches to credit risk management ± CreditMetrics, CreditRisk+ and KMV ± are of limited value when applied to portfolios of interest rate sensitive instruments and in measuring market and credit risk. Empirically returns on high yield bonds have a higher correlation with equity index returns and a lower correlation with Treasury bond index returns than do low yield bonds. Also, macro economic variables appear to in¯uence the aggregate rate of business failures. The CreditMetrics, CreditRisk+ and KMV methodologies cannot reproduce these empirical observations given their constant interest rate assumption. However, we can incorporate these empirical observations into the reduced form of Jarrow and Turnbull (1995b). Drawing the analogy. Risk 5, 63±70 model. Here default probabilities are correlated due to their dependence on common economic factors. Default risk and recovery rate uncertainty may not be the sole determinants of the credit spread. We show how to incorporate a convenience yield as one of the determinants of the credit spread. For credit risk management, the time horizon is typically one year or longer. This has two important implications, since the standard approximations do not apply over a one
Financial bubbles arise when the underlying asset's market price deviates from its fundamental va... more Financial bubbles arise when the underlying asset's market price deviates from its fundamental value. Unlike other bubble tests that use time series data and assume a reduced-form price process, we infer the existence of bubbles nonparametrically using option price data. Under no-arbitrage and acknowledging data constraints, we can partially identify asset price bubbles using a cross section of European option prices. In the empirical analysis, we obtain interval estimates of price bubbles embedded in the S&P 500 Index. The estimated index bubbles are then used to construct pro…table momentum trading strategies that consistently outperform a buy-and-hold trading strategy.
This paper provides a simple analytic formula for valuing default swaps with correlated market an... more This paper provides a simple analytic formula for valuing default swaps with correlated market and credit risk. We illustrate the numerical implementation of this model by inferring the default probability parameters implicit in default swap quotes for twenty two companies over the time period 8/21/00 to 10/31/00. For comparison, with also provide implicit estimates for the standard model (a special case of our approach) where market and credit risk are statistically independent.
Testing for the existence of asset price bubbles is a difficult task. This paper shows that the l... more Testing for the existence of asset price bubbles is a difficult task. This paper shows that the local martingale theory of bubbles provides three new approaches that can be used to test for the existence of asset price bubbles, two of which are currently unexplored in the literature. The first is based on specifying a stochastic process for the asset's price process. The second exploits option price data, if options trade on the asset suspected to exhibit a price bubble. The third uses a multiple-factor model for the return process of the underlying asset.
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Papers by Robert Jarrow