American Options in the Heston Model with Stochastic Interest Rate and Its Generalizations

Abstract

We consider the Heston model with the stochastic interest rate of Cox–Ingersoll–Ross (CIR) type and more general models with stochastic volatility and interest rates depending on two CIR-factors; the price, volatility and interest rate may correlate. Time-derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options arising in the time discretization of a Markov-modulated Lévy model. Options in this sequence are solved using an iteration method based on the Wiener–Hopf factorization. Typical shapes of the early exercise boundary are shown, and good agreement of option prices with prices calculated with the Longstaff–Schwartz method and Medvedev–Scaillet asymptotic method is demonstrated.     

Option pricing under joint dynamics of interest rates, dividends, and stock prices

This paper proposes a unified framework for option pricing, which integrates the stochastic dynamics of interest rates, dividends, and stock prices under the transversality condition. Using the Vasicek model for the spot rate dynamics, I compare the framework with two existing option pricing models. The main implication is that the stochastic spot rate affects options not only directly but also via an endogenously determined dividend yield and return volatility; consequently, call prices can be decreasing with respect to interest rates.     

Hybrid Lévy Models: Design and Computational Aspects

ABSTRACT

A hybrid model is a model, where two markets are studied jointly such that stochastic dependence can be taken into account. Such a dependence is well known for equity and interest rate markets on which we focus here. Other pairs can be considered in a similar way. Two different versions of a hybrid approach are developed. Independent time-inhomogeneous Lévy processes are used as the drivers of the dynamics of interest rates and equity. In both versions, the dynamics of the interest rate side is described by an equation for the instantaneous forward rate. Dependence between the markets is generated by introducing the driver of the interest rate market as an additional term into the dynamics of equity in the first version. The second version starts with the equity dynamics and uses a corresponding construction for the interest rate side. Dependence can be quantified in both cases by a single parameter. Numerically efficient valuation formulas for interest rate and equity derivatives are developed. Using market quotes for liquidly traded assets we show that the hybrid approach can be successfully calibrated.     

No Arbitrage and the Growth Optimal Portfolio

Abstract

Recently, several papers have expressed an interest in applying the Growth Optimal Portfolio (GOP) for pricing derivatives. We show that the existence of a GOP is equivalent to the existence of a strictly positive martingale density. Our approach circumvents two assumptions usually set forth in the literature: 1) infinite expected growth rates are permitted and 2) the market does not need to admit an equivalent martingale measure. In particular, our approach shows that models featuring credit constrained arbitrage may still allow a GOP to exist because this type of arbitrage can be removed by a change of numéraire. However, if the GOP exists the market admits an equivalent martingale measure under some numéraire and hence derivatives can be priced. The structure of martingale densities is used to provide a new characterization of the GOP which emphasizes the relation to other methods of pricing in incomplete markets. The case where GOP denominated asset prices are strict supermartingales is analyzed in the case of pure jump driven uncertainty.     

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract

The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.     

Using Utility Functions to Model Risky Bonds

This paper prices defaultable bonds by incorporating inherent risks with the use of utility functions. By allowing risk preferences into the valuation of bonds, nonlinearity is introduced in their pricing. The utility‐function approach affords the advantage of yielding exact solutions to the risky bond pricing equation when familiar stochastic models are used for interest rates. This can be achieved even when the default probability parameter is itself a stochastic variable. Valuations are found for the power‐law and log utility functions under the interest‐rate dynamics of the extended Vasicek and CIR models.     

Modelling Credit Risk in the Jump Threshold Framework

ABSTRACT

The jump threshold framework for credit risk modelling developed by Garreau and Kercheval enjoys the advantages of both structural- and reduced-form models. In their article, the focus is on multidimensional default dependence, under the assumptions that stock prices follow an exponential Lévy process (i.i.d. log returns) and that interest rates and stock volatility are constant. Explicit formulas for default time distributions and basket credit default swap (CDS) prices are obtained when the default threshold is deterministic, but only in terms of expectations when the default threshold is stochastic. In this article, we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds and CDS premia. The main idea is to make use of the Duffie–Pan–Singleton method of evaluating expectations of exponential integrals of affine diffusions.     

Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates

We study indifference pricing of mortality contingent claims in a fully stochastic model. We assume both stochastic interest rates and stochastic hazard rates governing the population mortality. In this setting we compute the indifference price charged by an insurer that uses exponential utility and sells k contingent claims to k independent but homogeneous individuals. Throughout we focus on the examples of pure endowments and temporary life annuities. We begin with a continuous-time model where we derive the linear pdes satisfied by the indifference prices and carry out extensive comparative statics. In particular, we show that the price-per-risk grows as more contracts are sold. We then also provide a more flexible discrete-time analog that permits general hazard rate dynamics. In the latter case we construct a simulation-based algorithm for pricing general mortality-contingent claims and illustrate with a numerical example.     

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.     

Valuation of guaranteed annuity options using a stochastic volatility model for equity prices

Guaranteed annuity options are options providing the right to convert a policyholder’s accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970’s and 1980’s when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990’s. Currently, these options are frequently sold in the US and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper explicit expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.     

Carr's randomization for American options in regime-switching models

In the paper, we solve the pricing problem for American put-like options in Markov-modulated Lévy models. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization for regime-switching models. An efficient iteration pricing procedure is developed. The computational time is of order m², where m is the number of states, and of order m, if the parallel computations are allowed. The payoffs, riskless rates and class of Lévy processes may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modelled as finite-state Markov chains. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Heston model with stochastic elasticity of variance

Empirical evidence suggests that single factor models would not capture the full dynamics of stochastic volatility such that a marked discrepancy between their predicted prices and market prices exists for certain ranges (deep in‐the‐money and out‐of‐the‐money) of time‐to‐maturities of options. On the other hand, there is an empirical reason to believe that volatility skew fluctuates randomly. Based upon the idea of combining stochastic volatility and stochastic skew, this paper incorporates stochastic elasticity of variance running on a fast timescale into the Heston stochastic volatility model. This multiscale and multifactor hybrid model keeps analytic tractability of the Heston model as much as possible, while it enhances capturing the complex nature of volatility and skew dynamics. Asymptotic analysis based on ergodic theory yields a closed form analytic formula for the approximate price of European vanilla options. Subsequently, the effect of adding the stochastic elasticity factor on top of the Heston model is demonstrated in terms of implied volatility surface. Copyright © 2016 John Wiley & Sons, Ltd.     

Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching

In this paper, we consider the problem of pricing discretely-sampled variance swaps based on a hybrid model of stochastic volatility and stochastic interest rate with regime-switching. Our modeling framework extends the Heston stochastic volatility model by including the Cox-Ingersoll-Ross (CIR) stochastic interest rate model. In addition, certain model parameters in our model switch according to a continuous-time observable Markov chain process. This enables our model to capture several macroeconomic issues such as alternating business cycles. A semi-closed form pricing formula for variance swaps is derived. The pricing formula is assessed through numerical implementation, where we validate our pricing formula against the Monte Carlo simulation. The impact of incorporating regime-switching for pricing variance swaps is also discussed, where variance swaps prices with and without regime-switching effects are examined in our model. We also explore the economic consequence for the prices of variance swaps by allowing the Heston-CIR model to switch across three different regimes.     

Valuation of life insurance products under stochastic interest rates

In this paper, we introduce a consistent pricing method for life insurance products whose benefits are contingent on the level of interest rates. Since these products involve mortality as well as financial risks, we present an approach that introduces stochastic models for insurance products through stochastic interest rate models. Similar to Black et al. [Black, Fisher, Derman, Emanuel, Toy, William, 1990. A one-factor model of interest rates and its application to treasury bond options. Financ. Anal. J. 46 (January-February), 33-39], we assume that the premiums and volatilities of standard insurance products are given exogenously. We then project insurance prices to extract underlying martingale probability structures. Numerical examples on variable annuities are provided to illustrate the implementation of this method.     

Rare Shock,Two-Factor Stochastic Volatility and Currency Option Pricing

Abstract

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.     

Pricing and hedging of guaranteed minimum benefits under regime-switching and stochastic mortality

This paper presents a novel framework for pricing and hedging of the Guaranteed Minimum Benefits (GMBs) embedded in variable annuity (VA) contracts whose underlying mutual fund dynamics evolve under the influence of the regime-switching model. Semi-closed form solutions for prices and Greeks (i.e. sensitivities of prices with respect to model parameters) of various GMBs under stochastic mortality are derived. Pricing and hedging is performed using an accurate, fast and efficient Fourier Space Time-stepping (FST) algorithm. The mortality component of the model is calibrated to the Australian male population. Sensitivity analysis is performed with respect to various parameters including guarantee levels, time to maturity, interest rates and volatilities. The hedge effectiveness is assessed by comparing profit-and-loss distributions for an unhedged, statically and semi-statically hedged portfolios. The results provide a comprehensive analysis on pricing and hedging the longevity risk, interest rate risk and equity risk for the GMBs embedded in VAs, and highlight the benefits to insurance providers who offer those products.     

Stochastic models for bond prices,function space integrals and immunization theory

This article presents several diffusion models for bond prices. By considering the spot interest rate as a state variable and invoking the no-arbitrage principle, the price of a default-free and non-callable pure discount bond is represented as a conditional expectation. The Ornstein—Uhlenbeck (O.U.) stochastic process is described, and used to model the spot rate. The O.U. process is then modified to exclude negative interest rates and the resulting bond-price partial differential equation is solved. By considering the yield rate as a state variable and using the Brownian bridge process, a simpler bond price model is obtained. Applications to immunization theory are presented.     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

The pricing of Asian options under stochastic interest rates

The purpose of this paper is to analyse the effect of stochastic interest rates on the pricing of Asian options. It is shown that a stochastic, in contrast to a deterministic, development of the term structure of interest rates has a significant influence. The price of the underlying asset, e.g. a stock or oil, and the prices of bonds are assumed to follow correlated two-dimensional Itô processes. The averages considered in the Asian options are calculated on a discrete time grid, e.g. all closing prices on Wednesdays during the lifetime of the contract. The value of an Asian option will be obtained through the application of Monte Carlo simulation, and for this purpose the stochastic processes for the basic assets need not be severely restricted. However, to make comparison with published results originating from models with deterministic interest rates, we will stay within the setting of a Gaussian framework.     

Testing robustness in calibration of stochastic volatility models

Many numerical aspects are involved in parameter estimation of stochastic volatility models. We investigate a model for stochastic volatility suggested by Hobson and Rogers [Complete models with stochastic volatility, Mathematical Finance 8 (1998) 27] and we focus on its calibration performance with respect to numerical methodology.In recent financial literature there are many papers dealing with stochastic volatility models and their capability in capturing European option prices; in Figà-Talamanca and Guerra [Towards a coherent volatility pricing model: An empirical comparison, Financial Modelling, Phisyca-Verlag, 2000] a comparison between some of the most significant models is done. The model proposed by Hobson and Rogers seems to describe quite well the dynamics of volatility.In Figà-Talamanca and Guerra [Fitting the smile by a complete model, submitted] a deep investigation of the Hobson and Rogers model was put forward, introducing different ways of parameters' estimation. In this paper we test the robustness of the numerical procedures involved in calibration: the quadrature formula to compute the integral in the definition of some state variables, called offsets, that represent the weight of the historical log-returns, the discretization schemes adopted to solve the stochastic differential equation for volatility and the number of simulations in the Monte Carlo procedure introduced to obtain the option price.The main results can be summarized as follows. The choice of a high order of convergence scheme is not fully justified because the option prices computed via calibration method are not sensitive to the use of a scheme with 2.0 order of convergence or greater. The refining of the approximation rule for the integral, on the contrary, allows to compute option prices that are often closer to market prices. In conclusion, a number of 10 000 simulations seems to be sufficient to compute the option price and a higher number can only slow down the numerical procedure.