Financial Derivatives Pricing

The following sections are included:

• Acknowledgments

• Preface

• Foreword

• Contents

The following sections are included:

• References

We show how a given probability distribution can be approximated by an arbitrary distribution in terms of a series expansion involving second and higher moments. This theoretical development is specialized to the problem of option valuation where the underlying security distribution, if not lognormal, can be approximated by a lognormally distributed random variable. The resulting option price is expressed as the sum of a Black–Scholes price plus adjustment terms which depend on the second and higher moments of the underlying security stochastic process. This approach permits the impact on the option price of skewness and kurtosis of the underlying stock's distribution to be evaluated.

This paper studies the impact that margin requirements have on both the existence of arbitrage opportunities and the valuation of call options. In the context of the Black-Scholes economy, margin restrictions are shown to exclude continuous-trading arbitrage opportunities and, with two additional hypotheses, still to allow the Black-Scholes call model to apply. The Black-Scholes economy consists of a continuously traded stock with a price process that follows a geometric Brownian motion and a continuously traded bond with a price process that is deterministic.

This paper investigates the relation between ex-dividend stock price behavior and arbitrage opportunities. In a continuous trading, frictionless economy, we demonstrate that it is possible for the ex-dividend stock price drop to differ from the dividend, and still short-term traders cannot generate arbitrage profits. Our argument is independent of transaction costs. The relevance of this insight to estimating the marginal tax bracket based on ex-dividend stock price drops is explored. Furthermore, this insight is also applied to the area of option pricing in which the special class of escrowed dividend stock price processes is studied. We show that most elements from this class of stock price processes generate invalid option pricing formulas.

The following sections are included:

• The Black–Scholes Model, Terminology, and the Stop-Loss Start-Gain Strategy

• Resolution of the Paradox

• Valuation Results

• Generalizing the Stock-Price Process

• Conclusions

• Appendix
  • Continuous semimartingales
  • Geometric Brownian motion
  • Time-homogeneous diffusion processes

• References

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.

This paper investigates market manipulation trading strategies by large traders in a securities market. A large trader is defined as any investor whose trades change prices. A market manipulation trading strategy is one that generates positive real wealth with no risk. Market manipulation trading strategies are shown to exist under reasonable hypotheses on the equilibrium price process. Sufficient conditions for their nonexistence are also provided.

This paper studies a new theory for pricing options in a large trader economy. This theory necessitates studying the impact that derivative security markets have on market manipulation. In an economy with a stock, money market account, and a derivative security, it is shown, by example, that the introduction of the derivative security generates market manipulation trading strategies that would otherwise not exist. A sufficient condition is provided on the price process such that no additional market manipulation trading strategies are introduced by a derivative security. Options are priced under this condition, where it is shown that the standard binomial option model still applies but with random volatilities.

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.

This article studies the pricing of options in an extended Black Scholes economy in which the underlying asset is not perfectly liquid. The resulting liquidity risk is modeled as a stochastic supply curve, with the transaction price being a function of the trade size. Consistent with the market microstructure literature, the supply curve is upward sloping with purchases executed at higher prices and sales at lower prices. Optimal discrete time hedging strategies are then derived. Empirical evidence reveals a significant liquidity cost intrinsic to every option.

The following sections are included:

• References

Within the term structure of interest rate literature, three different quantifications of the expectations hypothesis are commonly employed. This paper demonstrates under very general conditions that the three quantifications are inconsistent. Each quantification implies a different price for the same bond. The paper concludes with a brief discussion of both the theoretical and empirical implications of these results.

This paper provides a detailed discussion of the similarities and differences between forward contracts and futures contracts. Under frictionless markets and continuous trading, simple arbitrage arguments are invoked to value forward contracts, to relate forward prices and spot prices, and to relate forward prices and futures prices. We also argue that forward prices need not equal futures prices unless default free interest rates are deterministic.

The following sections are included:

• INTRODUCTION

• THE ECONOMY

• THE ARBITRAGE PRICING METHODOLOGY

• FORWARD CONTRACTS

• FUTURES CONTRACTS

• COMMODITY OPTIONS ON THE SPOT COMMODITY

• COMMODITY OPTIONS ON FUTURES CONTRACTS

• SUMMARY

• APPENDIX

• NOTES

• REFERENCES

This paper presents a unifying theory for valuing contingent claims under a stochastic term structure of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processes for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results.

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 studies contingent claim valuation of risky assets in a stochastic interest rate economy. The model employed generalizes the approach utilized by Heath, larrow, 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 Karatzas (1988) 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 Roll's (1977) formula for a call option on a stock that pays discrete dividends.

This paper uses an HJM model to price TIPS and related derivative securities. First, using the market prices of TIPS and ordinary U.S. Treasury securities, both the real and nominal zero-coupon bond price curves are obtained using standard coupon bond price stripping procedures. Next, a three-factor arbitrage-free term structure model is fit to the time-series evolutions of the CPI-U and the real and nominal zero-coupon bond price curves. Then, using these estimated term structure parameters, the validity of the HJM model for pricing TIPS is confirmed via its hedging performance. Lastly the usefulness of the pricing model is illustrated by valuing call options on the inflation index.

The following sections are included:

• References

This article provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a "spot exchange rate." Arbitrage-free valuation techniques are then employed. This methodology can be applied to corporate debt and over the counter derivatives, such as swaps and caps.

The following sections are included:

• The Jarrow–Turnbull Model

• Credit Ratings and Default-Probabilities: The Discrete Time Case
  • Valuation
  • Options and hedging
  • Fitting the credit class zero-curves
  • Discussion

• Credit Ratings and Default Probabilities: The Continuous Time Case
  • Valuation
  • Options and hedging
  • Examples

• Parameter estimation
  • Estimation of default-free parameters
  • Estimation of the bankruptcy process parameters
  • Estimating the recovery rate
  • Estimating the generator matrix Λ
  • Estimation of the empirical generator matrix Λ
  • Estimation of the risk premium
  • Survival probabilities and spreads under risk neutrality
  • An illustrative estimation of the risk premia

• Conclusion

• Appendix A

• Appendix B

• References

Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity's diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of "diversifiable default risk." The equivalence between the empirical and martingale default intensities that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. We emphasize that this is not an equivalence in distribution, and illustrate its importance using credit spread dynamics estimated in Duffee (1999). We also argue that the assumption of diversifiability is implicitly used in certain existing models of mortgage-backed securities.

Motivated by recent financial crises in East Asia and the United States where the downfall of a small number of firms had an economy-wide impact, this paper generalizes existing reduced-form models to include default intensities dependent on the default of a counterparty. In this model, firms have correlated defaults due not only to an exposure to common risk factors, but also to firm-specific risks that are termed "counterparty risks." Numerical examples illustrate the effect of counterparty risk on the pricing of defaultable bonds and credit derivatives such as default swaps.

This paper investigates the forecasting accuracy of bankruptcy hazard rate models for U.S. companies over the time period 1962–1999 using both yearly and monthly observation intervals. The contribution of this paper is multiple-fold. One, using an expanded bankruptcy database we validate the superior forecasting performance of Shumway's (2001) model as opposed to Altman (1968) and Zmijewski (1984). Two, we demonstrate the importance of including industry effects in hazard rate estimation. Industry groupings are shown to significantly affect both the intercept and slope coefficients in the forecasting equations. Three, we extend the hazard rate model to apply to financial firms and monthly observation intervals. Due to data limitations, most of the existing literature employs only yearly observations. We show that bankruptcy prediction is markedly improved using monthly observation intervals. Fourth, consistent with the notion of market efficiency with respect to publicly available information, we demonstrate that accounting variables add little predictive power when market variables are already included in the bankruptcy model.

We provide an approach to the market valuation of deposit insurance that is based on reduced-form methods for the pricing of fixed-income securities under default risk. By reference to bank debt prices as well as qualitative-response models of the probability of bank failure, we suggest how a risk-neutral valuation model for deposit insurance can be applied both to the calculation of fair-market deposit insurance premia and to the valuation of long-term claims against the insurer.

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.