American option valuation under stochastic interest rates

By applying Ho, Stapleton and Subrahmanyam's (1997, hereafter HSS) generalised Geske–Johnson (1984, hereafter GJ) method, this paper provides analytic solutions for the valuation and hedging of American options in a stochastic interest rate economy. The proposed method simplifies HSS's three-dimensional solution to a one-dimensional solution. The simulations verify that the proposed method is more efficient and accurate than the HSS (1997) method. We illustrate how the price, the delta, and the rho of an American option vary between the stochastic and non-stochastic interest rate models. The magnitude of this effect depends on the moneyness of the option, interest rates, volatilities of the underlying asset price and the bond price, as well as the correlation between them. This revised version was published online in June 2006 with corrections to the Cover Date.     

Pricing of foreign exchange options under the MPT stochastic volatility model and the CIR interest rates

We consider an extension of the model proposed by Moretto, Pasquali, and Trivellato [2010. “Derivative Evaluation Using Recombining Trees under Stochastic Volatility.” Advances and Applications in Statistical Sciences 1 (2): 453–480] (referred to as the MPT model) for pricing foreign exchange (FX) options to the case of stochastic domestic and foreign interest rates driven by the Cox, Ingersoll, and Ross dynamics introduced in Cox, Ingersoll, and Ross [1985. “A Theory of Term Structure of Interest Rates.” Econometrica 53(2): 385–408]. The advantage of the MPT model is that it retains some crucial features of Heston's stochastic volatility model but, as demonstrated in Moretto, Pasquali, and Trivellato [2010. “Derivative Evaluation Using Recombining Trees under Stochastic Volatility.” Advances and Applications in Statistical Sciences 1 (2): 453–480], it is better suited for discretization through recombining lattices, and thus it can also be used to value and hedge exotic FX products. In the model examined in this paper, the instantaneous volatility is correlated with the exchange rate dynamics, but the domestic and foreign short-term rates are assumed to be mutually independent and independent of the dynamics of the exchange rate. The main result furnishes a semi-analytical formula for the price of the FX European call option, which hinges on explicit expressions for conditional characteristic functions.     

A generalization of reset options pricing formulae with stochastic interest rates

A generalization of reset call options with predetermined dates is derived in the case of time-dependent volatility and time-dependent interest rate by applying martingale method and change of nume?aire or change of probability measure. An analytical pricing formula for the reset call option is also obtained when the interest rate follows an extended Vasicek’s model. Numerical results show that the correlated coefficient between the stock price and interest rate is almost unacted on the price of reset call option with short maturity and Monte Carlo method is inefficient. Monte Carlo method should be only used if there is no closed-formed solution for option pricing.     

On financial guarantee insurance under stochastic interest rates

We extend the financial guarantee insurance literature by modeling, under stochastic interest rates, private financial guarantees when the guarantor potentially defaults. By performing numerical simulations under plausible parameters values, we characterize the differential impact of the incorporation of stochasticity of interest rates on the valuation of both public and private guarantees.     

Savings bonds,retractable bonds and callable bonds

Savings bonds, retractable bonds and callable bonds are each equivalent to a straight bond with an option. Neglecting default risk the value of these contingent claims depends upon the riskless interest rate. This paper employs the option pricing framework to value these bonds, under the assumptions that the interest rate follows a Gauss-Wiener process and that the pure expectations hypothesis holds.     

Implied volatilities,stochastic interest rates,and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.     

Pricing American interest rate options under the jump-extended constant-elasticity-of-variance short rate models

This paper demonstrates how to value American interest rate options under the jump-extended constant-elasticity-of-variance (CEV) models. We consider both exponential jumps (see Duffie et al., 2000) and lognormal jumps (see Johannes, 2004) in the short rate process. We show how to superimpose recombining multinomial jump trees on the diffusion trees, creating mixed jump-diffusion trees for the CEV models of short rate extended with exponential and lognormal jumps. Our simulations for the special case of jump-extended Cox, Ingersoll, and Ross (CIR) square root model show a significant computational advantage over the Longstaff and Schwartz’s (2001) least-squares regression method (LSM) for pricing American options on zero-coupon bonds.     

A fast Fourier transform technique for pricing American options under stochastic volatility

This paper develops a non-finite-difference-based method of American option pricing under stochastic volatility by extending the Geske-Johnson compound option scheme. The characteristic function of the underlying state vector is inverted to obtain the vector’s density using a kernel-smoothed fast Fourier transform technique. The method produces option values that are closely in line with the values obtained by finite-difference schemes. It also performs well in an empirical application with traded S&P 100 index options. The method is especially well suited to price a set of options with different strikes on the same underlying asset, which is a task often encountered by practitioners.     

Dynamic asset allocation with stochastic income and interest rates

We solve for optimal portfolios when interest rates and labor income are stochastic with the expected income growth being affine in the short-term interest rate in order to encompass business cycle variations in wages. Our calibration based on the Panel Study of Income Dynamics (PSID) data supports this relation with substantial variation across individuals in the slope of this affine function. The slope is crucial for the valuation and riskiness of human capital and for the optimal stock/bond/cash allocation both in an unconstrained complete market and in an incomplete market with liquidity and short-sales constraints.     

Portfolio management with stochastic interest rates and inflation ambiguity

We solve, in closed form, a stock-bond-cash portfolio problem of a risk- and ambiguity-averse investor when interest rates and the inflation rate are stochastic. The expected inflation rate is unobservable, but the investor can learn about it from observing realized inflation and stock and bond prices. The investor is ambiguous about the inflation model and prefers a portfolio strategy which is robust to model misspecification. Ambiguity about the inflation dynamics is shown to affect the optimal portfolio fundamentally different than ambiguity about the price dynamics of traded assets, for example the optimal portfolio weights can be increasing in the degree of ambiguity aversion. In a numerical example, the optimal portfolio is significantly affected by the learning about expected inflation and somewhat affected by ambiguity aversion. The welfare loss from ignoring learning or ambiguity can be considerable.     

Term structure estimation and pricing of callable Treasury bonds

This article describes a methodology of term structure estimation incorporating callable Treasury bonds using a bond-option valuation model. This article also examines whether some simple approximation of the option value suffice for providing a useful estimation procedure. The authors find that the errors in estimating the option value can generate significant errors for estimating the discount function. A call provision on a Treasury bond is not negligible at least our framework. This procedure is consistent with two aspects of the Treasury market. First, it provides the discount function that best determines the prices of observed Treasury securities, and second, it obtains a discount function that explains callable Treasuries.     

Reduced-form valuation of callable corporate bonds: Theory and evidence

We develop a reduced-form approach for valuing callable corporate bonds by characterizing the call probability via an intensity process. Asymmetric information and market frictions justify the existence of a call-arrival intensity from the market's perspective. Our approach both extends the reduced-form model of Duffie and Singleton (1999) for defaultable bonds to callable bonds and captures some important differences between call and default decisions. A comprehensive empirical analysis of callable bonds using both our model and the more traditional American option approach for valuing callable bonds shows that the reduced-form model fits callable bond prices well and that it outperforms the traditional approach both in- and out-of-sample.     

American options under stochastic volatility: control variates,maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of an American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow-scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price an American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where the scaling parameter value is equal to unity, fast and slow-scale approximations are equally accurate.     

Valuation of FX barrier options under stochastic volatility

This paper describes European-style valuation and hedging procedures for a class of knockout barrier options under stochastic volatility. A pricing framework is established by applying mean self-financing arguments and the minimal equivalent martingale measure. Using appropriate combinations of stochastic numerical and variance reduction procedures we demonstrate that fast and accurate valuations can be obtained for down-and-out call options for the Heston model.     

The term structure of real interest rates: theory and evidence from UK index-linked bonds

This paper studies the behavior of the default-risk-free real term structure and term premia in two general equilibrium endowment economies with complete markets but without money. In the first economy there are no frictions as in Lucas (Econometrica 46 (1978) 1429) and in the second risk-sharing is limited by the risk of default as in Alvarez and Jermann (Econometrica 68 (2000) 775; Rev. Financial Studies 14 (2001) 1117). Both models are solved numerically, calibrated to UK aggregate and household data, and the predictions are compared to data on real interest rates constructed from the UK index-linked data. While both models produce time-varying risk or term premia, only the model with limited risk-sharing can generate enough variation in the term premia to account for the rejections of expectations hypothesis.     

Valuation of American options under the CGMY model

In the present work, we concentrate on the analytical study of American options under the CGMY process. The decomposition formula of the American option and the integral equation for the optimal-exercise boundary are established in explicit forms. Moreover, an analytical approximation formula is obtained for the American value. This approximation is valid when time to maturity is either very short or very long. Numerical simulations are provided for European options, optimal-exercise prices and approximate values for American options.