Option pricing under regime switching

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case. This paper develops a family of option pricingmodels obtained when the underlying stock dynamic is modelled by a Markov regime switching process. In such models, the stochastic process remains in one regime for a random amount of time before switching over into a new regime. In our case, the regimes are characterized by different volatility levels. Rather than permitting volatilities to follow a continuous time, continuous state process, as in most stochastic volatility models, our primary focus is on cases where volatilities can take on a finite set of values, and can only switch regimes at finite times. In this regard, our models can be viewed as special cases of the large family of stochastic volatility models, in which the number of distributions for the logarithmic return are constrained to a finite collection. While this may, at first glance, appear unnecessarily restrictive, our family of models includes as special limiting cases, many well known models, including the family of GARCH option pricing models discussed by Duan (1995). We also can obtain more general limiting models in which variance updating schemes depend not only on levels of variance and on asset innovations, but also on a second factor that is uncorrelated with asset returns. As a result variance levels are not completely determined by the path of prices. 1469-7688/02/000001+17$30.00 © 2002 IOP Publishing Ltd PII: S1469-7688(02)27838-0 1 J-C Duan et al QUANTITATIVE FINANCE This second factor allows for further flexibility in capturing the properties of stock return processes. Our family of models also include the regime switching models of Hamilton (1989) as a special case. Hismodels allow volatility regimes to impact returns, but they do not allow returns to impact future volatilities. Our models do permit this feedback effect. Thus our models allow us to capture the correlation between asset and volatility innovations, or equivalently, the asymmetric volatility response to good and bad news in asset returns. Our family of models fills the gap of models between the Black–Scholes (BS) model, which in our framework can be viewed as a single volatility regime model with no feedback effects, and the extended GARCH models, which have infinitely many volatility regimes with feedback effects. We demonstrate that it is possible to establish models with a relatively small number of volatility regimes that produce option prices indistinguishable from models with a continuum of volatility states. We present an algorithm that permits American derivatives to be priced for our most general bidirectional regime switching process. If one is only interested in the rich family of GARCH option models, then our algorithm provides an alternative to the numerical procedure of Ritchken and Trevor (1999) andDuan and Simonato (1999). The addition of the second orthogonal factor causes little complication for the algorithm, and provides meaningful extensions to processes beyond the GARCH family. Our study is certainly not the first to consider regime switching mechanisms nor option pricing under regime switching. The early regime switching models were primarily designed to capture changes in the underlying economic mechanism that generated the data. Examples include Hamilton (1989) andGray (1996), Bekaert andHodrick (1993) and Durland and McCurdy (1994). Recently, attention has been placed on volatility regime switching models, solely for the purpose of better understanding option price behaviour. Bollen et al (1999), for example, show that a very simple regime switching model with independent shifts in the mean and variance dominate a range of GARCH models in the foreign exchange market. Bollen (1998) presents a latticebased algorithm that permits American options to be priced for these regime switching models. The type of regime switching models that are typically considered assume that asset innovations have no feedback effects on volatilities. Further, the assumption that regime shift risk is not priced is made so as to allow option pricing to proceed in the usual risk neutral manner. Our models weaken these restrictions. The paper proceeds as follows. In section 1 we present the bi-directional regime switching model for asset returns and describe some of its properties. In section 2 we investigate how options can be priced when the underlying follows a regime switching process with feedback. In section 3 we investigate a special case of the model where there are only two volatility regimes and asset innovations have no feedback effects on volatilities. The resulting option model turns out to be a weighted sum of BS prices. In section 4 we investigate a second special case that leads to models which include GARCH and stochastic volatility models as limiting cases. In section 5 we provide an efficient numerical scheme for pricing European and American options under our most general bi-directional regime switching process. We illustrate how regime switching models with relatively few volatility states can serve as excellent proxies for GARCH models. We also demonstrate how the second orthogonal factor provides significant flexibility beyond GARCH models in the shapes that return distributions can take on over the lifetime of the option. While there are a huge number of models in the regime switching family, the specific models that we evaluate have up to six unknownparameters, with the simplestmodel containing just four. In section 6 we investigate the performance of a few specifications of our regime switching models on S&P 500 stock index option prices. The example provides sufficiently encouraging results to warrant ongoing empirical research in this area. 1. Regime switching model with feedback effects We assume that the asset price is governed by a regime switching with feedback dynamic. Let St be the asset price at date t , and letσ 2 t+1 be the conditional variance of the logarithmic return at date t that holds for the period [t, t + 1]. Given σt+1, the dynamics of the price over the next period is assumed to be ln St+1 St = r + λσt+1 − 1 2 σ 2 t+1 + σt+1εt+1 (1) where εt+1 is a standard normal random variable and λ can be interpreted as the risk premium per one standard deviation. We shall assume that there areK distinct volatility regimes, and in each period there is a chance that the volatility will move into a new regime. The volatility follows a Markov chain, which is fully determined by the K × K transition matrix between volatility states. The transition probabilities are determined by a threshold model. In particular, the volatility, σt+1, depends on its previous value, σt , on the most recent return innovation, εt , and on a variable, ξt , that is independent of εt . The random process ξt can be viewed as a state variable process that impacts the variance but is orthogonal to the asset return innovation process, εt . Let F(εt , ξt ) be a function, that determines the impact of the most recent return innovation and orthogonal volatility innovation, in the form of a non-negative real number. The new volatility state is completely determined by this functional value together with the existing level of volatility. Specifically, corresponding to each volatility level, δi, i = 1, 2, . . . , K , is a set of values {c0(δi), c1(δi), . . . , cK(δi)} such that c0(δi) = 0 and cK(δi) = ∞. At date t , we have, for i = 1, 2, . . . , K: σt+1 = δi if ci−1(σt ) F(εt , ξt ) < ci(σt ). (2) That is, conditional on the current level of volatility, the switch into a new regime, is completely determined by the magnitude of the functional value, F(·). While the updating function, F(·) could be fairly general, to make matters specific, we will assume that it has the structure F(εt , ξt ) = q1(εt −ω)+ +q2(εt −ω)− + (1−q1−q2)|ξt | (3)