Option Pricing under Stochastic Interest Rates: An Empirical Investigation

Using daily data of the Nikkei 225 index, call option prices and call money rates of the Japanese financial market,a comparison is made of the pricing performance of stock option pricing modelsunder several stochastic interest rate processes proposedby the existing term structure literature.The results show that (1) one option pricing modelunder a specific stochastic interest ratedoes not significantly outperformanother option pricing model under an alternative stochasticinterest rate, and (2) incorporating stochastic interest ratesinto stock option pricing does not contribute to the performanceimprovement of the original Black–Scholes pricing formula.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-coupon bond in a Heath–Jarrow–Morton (1992) framework when the value and/or functional form of forward interest rates volatility is unknown, but is assumed to lie between two fixed values. Due to the link existing between the drift and the diffusion coefficients of the forward rates in the Heath, Jarrow and Morton framework, this is equivalent to hedging and pricing the option when the underlying interest rate model is unknown. We show that a continuous rangeof option prices consistent with no arbitrage exist. This range is bounded by the smallest upper-hedging strategy and the largest lower-hedging strategy prices, which are characterized as the solutions of two non-linear partial differential equations. We also discuss several pricing and hedging illustrations.     

On the valuation of compositions in Lévy term structure models

We derive explicit valuation formulae for an exotic path-dependent interest rate derivative, namely an option on the composition of LIBOR rates. The formulae are based on Fourier transform methods for option pricing. We consider two models for the evolution of interest rates: an HJM-type forward rate model and a LIBOR-type forward price model. Both models are driven by a time-inhomogeneous Lévy process.     

Empirical Performance of Alternative Option Pricing Models

Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance.     

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

Regression-based algorithms for life insurance contracts with surrender guarantees

We present a general framework for pricing life insurance contracts embedding a surrender option. The model allows for several sources of risk, such as uncertainty in mortality, interest rates and other financial factors. We describe and compare two numerical schemes based on the Least Squares Monte Carlo method, emphasizing underlying modeling assumptions and computational issues.     

Parameter estimation risk in asset pricing and risk management: A Bayesian approach

Parameter estimation risk is non-trivial in both asset pricing and risk management. We adopt a Bayesian estimation paradigm supported by the Markov Chain Monte Carlo inferential techniques to incorporate parameter estimation risk in financial modelling. In option pricing activities, we find that the Merton's Jump-Diffusion (MJD) model outperforms the Black-Scholes (BS) model both in-sample and out-of-sample. In addition, the construction of Bayesian posterior option price distributions under the two well-known models offers a robust view to the influence of parameter estimation risk on option prices as well as other quantities of interest in finance such as probabilities of default. We derive a VaR-type parameter estimation risk measure for option pricing and we show that parameter estimation risk can bring significant impact to Greeks' hedging activities. Regarding the computation of default probabilities, we find that the impact of parameter estimation risk increases with gearing level, and could alter important risk management decisions.     

Futures Options and the Volatility of Futures Prices

Assuming nonstochastic interest rates, European futures options are shown to be European options written on a particular asset referred to as a futures bond. Consequently, standard option pricing results may be invoked and standard option pricing techniques may be employed in the case of European futures options. Additional arbitrage restrictions on American futures options are derived. The efficiency of a number of futures option markets is examined. Assuming that at-the-money American futures options are priced accurately by Black's European futures option pricing model, the relationship between market participants' ex ante assessment of futures price volatility and the term to maturity of the underlying futures contract is also investigated empirically.     

Pricing Options under Stochastic Interest Rates: A New Approach

We will generalize the Black-Scholes option pricing formula by incorporating stochastic interest rates. Although the existing literature has obtained some formulae for stock options under stochastic interest rates, the closed-form solutions have been known only under the Gaussian (Merton type) interest rate processes. We will show that an explicit solution, which is an extended Black-Scholes formula under stochastic interest rates in certain asymptotic sense, can be obtained by extending the asymptotic expansion approach when the interest rate volatility is small. This method, called the small-disturbance asymptotics for Itô processes, has recently been developed by Kunitomo and Takahashi (1995, 1998) and Takahashi (1997). We found that the extended Black-Scholes formula is decomposed into the original Black-Scholes formula under the deterministic interest rates and the adjustment term driven by the volatility of interest rates. We will illustrate the numerical accuracy of our new formula by using the Cox–Ingersoll–Ross model for the interest rates.     

Pricing and Hedging Discount Bond Options in the Presence of Model Risk

This paper focuses on pricing and hedging options on a zero-couponbond in a Heath—Jarrow—Morton (1992) framework whenthe value and/or functional form of forward interest rates volatilityis unknown, but is assumed to lie between two fixed values.Due to the link existing between the drift and the diffusioncoefficients of the forward rates in the Heath, Jarrow and Mortonframework, this is equivalent to hedging and pricing the optionwhen the underlying interest rate model is unknown. We showthat a continuous range of option prices consistent with noarbitrage exist. This range is bounded by the smallest upper-hedgingstrategy and the largest lower-hedging strategy prices, whichare characterized as the solutions of two non—linear partialdifferential equations. We also discuss several pricing andhedging illustrations.     

Option pricing and hedging in different cyclical structures: a two-dimensional Markov-modulated model

The critical role of interest rate risk and associated regime-switching risk in pricing and hedging options is examined using a closed-form valuation model. Equity call options are valued under the proposed 2-dimensional Markov-modulated model in which asset prices and interest rates exhibit Markov regime-switching features. In addition, the relationship between cyclical structures and option prices are analyzed using a time-varying transition probability matrix. The proposed model can enhance the forecast transition probabilities in an out-sample period. The cycle-stylized effect of an economy exhibits different impacts on option prices and hedging strategies in a short- and a long-cycle economy. Our closed-form formula based on more realistic specifications with respect to business-cyclical structures in various financial markets is more appropriate for pricing and hedging options.     

Implied Spot Rates as Predictors of Currency Returns: A Note

Currency call option transactions data and the Black-Scholes option pricing model, as modified by Merton for continuous dividends and as adapted to currency options by Biger and Hull and by Garman and Kohlhagen, are used to imply spot foreign exchange rates. The proportional deviation between implied and simultaneously observed spot rates is found to be a direct and statistically significant determinant of subsequent returns on foreign currency holdings after controlling for interest rate differentials. Further, an ex ante trading rule reveals that the additional information contained in implied rates often is sufficient to generate significant economic profits.     

Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model

The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. This paper investigates whether the use of models which allow for negative interest rates can improve option pricing and implied volatility forecasting. This is done with special attention to foreign exchange and index options. To this end, we carried out an empirical analysis on the prices of call and put options on the US S&P 500 index and Eurodollar futures using a generalization of the Heston model in the stochastic interest rate framework. Specifically, the dynamics of the option’s underlying asset is described by two factors: a stochastic variance and a stochastic interest rate. The volatility is not allowed to be negative, but the interest rate is. Explicit formulas for the transition probability density function and moments are derived. These formulas are used to estimate the model parameters efficiently. Three empirical analyses are illustrated. The first two show that the use of models which allow for negative interest rates can efficiently reproduce implied volatility and forecast option prices (i.e. S&P index and foreign exchange options). The last studies how the US three-month government bond yield affects the US S&P 500 index.     

An Option-Pricing Approach to the Costs of Export Credit Insurance

This article investigates the relationship between a debtor country's external financial indicators and the costs associated with the insurance of export credits to that country. For this purpose a stylized model of export credit insurance (ECI) is developed, the central idea being that ECI is similar to a contingent claim such as a European put option. Thus, tools from option pricing theory were used to calculate the price of ECI, implying that not only the current financial position but also the volatility of the changes in that position determine such costs. The empirical results of a statistical analysis of the premium rates for ECI, applied by a private export credit insurer to seventy-seven developing countries during 1993, provide some support for these hypotheses. In particular, the reserves-over-imports ratio of a debtor country and the volatility of the rates of change of this ratio appear to contribute significantly to the premium rates that apply to that country. Thus, the article provides evidence that option pricing parameters do play role in practical insurance pricing, even if this pricing is not explicitly based on these parameters. Premium rates are set as if an underlying option market operated. Thus, the trade of countries with volatile external financial positions is saddled with higher costs than that of countries with more stable positions.     

An equilibrium approach to pricing foreign currency options

The paper presents a modified version of the Garman-Kohlhagen formula for pricing European currency options. The equilibrium approach deviates from the no-arbitrage approach by allowing domestic and foreign interest rates and their dynamics to be determined endogenously in the model. By using the relations between exchange rate dynamics and the dynamics of interest rates, I provide a new characterisation of the relevant volatilities for European currency option pricing, which only depends on parameters describing the variability of the log-exchange rate. The implications of the model for the valuation of American currency options and optimal exercise strategies are examined by applying numerical methods.     

HOW CHANGES IN BOND CALL FEATURES AFFECT COUPON RATES

In April 1998, Level 3, a telecommunications company, sold $2 billion of 9.19%, ten-year bonds to finance the building of a fiber-optic network. Like most below-investment-grade issues, as well as many investment-grade issues, the Level 3 issue contained an embedded call option that gave the company the right to repurchase the bonds after five years at par value plus the (semi-annual) coupon rate, with the call price declining to par two years before maturity.
Because issuers must pay for the call provision in the form of a higher coupon rate, the choice of whether or not to include a call option can be a difficult one. And, once management decides to include a call option, it must then decide how to structure the call. The most important call structure decisions are how long to make the call protection period and how to set the call price—both of which can have a significant impact upon the coupon yields required to attract investors. Using a well-known option pricing model, the authors of this article summarize their recent research on how variations in bond call features can be expected to affect par coupon yields of new issues under different market circumstances—circumstances that include market conditions relevant to option valuation such as the shape of the term structure and the volatility of interest rates.     

Microfoundations of a mortgage prepayment function

This article focuses on the following question: how much of an interest rate decline is needed to justify refinancing a typical home mortgage? Modern option pricing theory is used to answer the question; this theory indicates that the answer depends upon several factors, which include the volatility of interest rates and the expected holding period of the borrower. The analysis suggests that the commonly espoused “rule of thumb” refinance if the interest rate declines by 200 basis points — is a fair approximation to the more precisely derived differential for many households. We also construct the prepayment behavior of a pool of mortgages in which the expected holding periods of the borrowers in the pool vary. The prepayment behavior of this simulated pool is used to generate a series of empirically testable hypotheses regarding the likely shape of an actual prepayment function and its determinants. Finally, actual prepayment data are used to estimate a hazard function that explains prepayment behavior. We find that the estimated model understates prepayment behavior relative to that predicted by the simulation model, which suggests that the simple option pricing model is not adequate to explain aggregate prepayment behavior.     

VALUING REAL OPTIONS: CAN RISK‐ADJUSTED DISCOUNTING BE MADE TO WORK?

This paper examines three alternative approaches to valuing real options: (1) the standard option pricing technique using "risk-neutral" probabilities; (2) the use of risk-adjusted discount rates; and (3) discounting certainty-equivalent values with a riskless discount rate. As suggested by the title, a question of particular interest is whether an approach based on risk-adjusted discount rates can be "made to work" for valuing options. The answer is yes. Indeed, the authors show that any of the three approaches will provide a correct valuation if properly employed.
Nevertheless, there are important differences in the information requirements associated with each of the three methods. Another important issue is the relative degree of difficulty in calculating the correct option value. When these two considerations are taken into account, the risk-neutral option pricing procedure generally proves to be the preferred method. It tends to be computationally more convenient—often much more convenient—and to require less information than either the risk-adjusted discounting or certainty-equivalent procedures.     

The value of embedded real options: Evidence from consumer automobile lease contracts—A note

Giaccotto et al. [2007. Journal of Finance 62, 411–445] provide a simple model for pricing the cancellation and the purchase options typically embedded in automobile lease contracts, assuming constant interest rates. They show that the cancellation option is worthless because of a penalty applied if the lease is terminated before maturity. We extend their results by developing a model with stochastic interest rates, and show that the cancellation option has a significant value also in presence of the penalty. We provide sufficient conditions to make the cancellation option worthless in our more general framework.     

Valuing an Early‐Stage Biotechnology Investment as a Rainbow Option

In an article published in this journal in 2003, Richard Shockley and three of his students presented a detailed valuation of an early‐stage biotechnology investment using a binomial lattice option pricing model. The article demonstrates how investments with multiple stages can be treated as “compound sequential options”—that is, as series of options in which investments in one option provide the opportunity to invest in the next in the series. In this article, the author uses the same business case analyzed by Shockley et al. to demonstrate how to value this early‐stage biotechnology investment by separately modeling the two types of risks: technology and product market. An option that has two distinct kinds of risk that develop differently over time is known as a “rainbow option.” The key adjustment to the option pricing model required to value such an option is that, instead of the standard binomial option pricing model with two outcomes at each point in time, the author uses a “quadranomial” option pricing model with four outcomes at each point in time. By distinguishing technology risks from product market risks and allowing them to develop differently over time, the author's analysis leads to a very different valuation and, indeed, a different decision about the initial investment than the one produced by Shockley's model.