Valuing power options under a regime-switching model

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1 ( ) Journal of East China Normal University (Natural Science) No. 6 Nov. 13 Article ID: (13)6-3-8 Valuing power options under a regime-switching model SU Xiao-nan 1, WANG Wei, WANG Wen-sheng 3 (1. School of Mathematics and Statistics, Nanjing Audit University, Nanjing 11815, China;. Department of Mathematics, Ningbo University, Ningbo Zhejiang 3151, China; 3. Department of Hangzhou Normal University, Hangzhou 3136, China) Abstract: The pricing of European style power call options was considered when the dynamics of the underlying risky asset are driven by the Markov-modulated Geometric Brownian Motion. In particular, the market interest rate, the appreciation rate and the volatility of the underlying risky asset switched over time according to the sates of the continuous time Markov chain process. Since the market is incomplete, the regime switching Esscher transform was employed to determine an equivalent martingale measure and derive the valuation of the options. Then, the numerical analysis of our result was given. Key words: regime switching; power options; regime switching Esscher transform; option pricing; numerical analysis CLC number: O11 Document code: A DOI: /j.issn Regime switching 1,, 3 (1., 11815;., 31511; 3., 3136) :.,,., regime switching Esscher.,. : regime switching; ; regime switching Esscher ; ; : 13-1 : (117176); (1YJC919); (LQ1A16); :,,,. suxiaonanecnu@163.com.

2 6, Regime switching ( ) 33 Introduction Option valuation is one of the major concerns in financial economics. As the financial markets develop, more exotic options are created, such as Bermudan options, Asian options, Lookback options, Barrier options, etc. At first, the pricing and hedging of these options were based on the pioneering work by Black and Scholes [1]. In the Black-Scholes model, the appreciation rate, volatility rate of the underlying assets are assumed to be constants. It has been well documented using empirical data that these assumptions are not consistent with the reality. Over the past three decades, various extensions of the standard model with more realistic price dynamics have been proposed and tested. We can refer to Merton [], Kou and Wang [3]. Chan [4], Elliott and Osakwe [5]. Harrison and Kreps [6], and Harrison and Pliska [7] established that the absence of arbitrage is equivalent to the existence of an equivalent martingale measure under which all discounted price processes are martingales. When the market is incomplete, there are infinitely many equivalent martingale measures and perfect hedging is impossible. How to choose a consistent pricing measure from the set of equivalent martingale measures becomes an important problem. Gerber and Shiu [8] provided an elegant way to choose a pricing measure using the Esscher transform for option valuation. This pricing result can be justified by maximizing the expected power utility of an economic agent. Their work highlights the interplay between financial and insurance pricing problems in an incomplete market. Here, we also adopt the Esscher transform to determine an equivalent martingale measure to price options. Recently, there has been considerable attention paid to many areas which are described by regime-switching models. A key feature of regime switching models is that model parameters are functions of a Markov chain whose states represent states of an economy, or different states of business cycles. Hamilton [9] pioneered applications of regime-switching models in economics and econometrics. Empirical studies revealed that regime-switching models fit economic and financial time series well and explained some important stylized facts of these series. Several researchers investigated the option valuation under regime-switching. For the literature on the regime-switching model, we can refer to Elliott et al. [1], Elliott et al. [11], Siu et al. [1], Gopal et al. [13], Yoon et al. [14]. However, none of the studies handle the valuation of power options under regime-switching. In this paper, we investigate the valuation of power options under a regime switching model. By the risk neutral pricing theorem, we can obtain the formula of the option price. Then, we assume that the state of the Markov chain is two. From an economic perspective, the two states may stand for bull market and bear market. Surprise events or changes in macroeconomic variables can cause asset price processes to follow completely different dynamics. That is, the market interest rate, the appreciation rate and the volatility switch between the two states over time. Employing the regime-switching Esscher transform, we obtain an equivalent martingale measure. Then we get the valuation of the European power call option. At last, numerical analysis is also provided.

3 34 ( ) 13 1 Underlying market Fix a complete probability space (Ω, F, P), where P is a real-world probability measure. Write T > for the finite time horizon. Let {X t, t [, T]} be a continuous-time Markov chain with a finite state space χ := {1,,..., N}, whose transition probabilities satisfy P(X t+δt = j X t = i) = q ij δt + o(δt), i j; P(X t+δt = i X t = i) = 1 + q ii δt + o(δt), when δ, where q ij, i j; q ii = N q ij. Let Q = [q ij ] denote the generating Q-matrix j=1 of the Markov chain. In contrast to the classical Black-Scholes model, we suppose that the interest rate r t, the appreciation rate µ t and the volatility rate σ t switch over time according to the economy represented by the state of X t. Let r i, σ i >, and µ i be given real numbers and let r t = r(x t ), µ t = µ(x t ) and σ t = σ(x t ) be functions of X t, that is, r t = r(i) = r i, µ t = µ(i) = µ i and σ t = σ(i) = σ i when the state of X t is i, i = 1,,...N. Throughout this paper, we consider a financial model consisting of a locally risk-free money market account and only one stock as a risky asset. The value at time t of the locally risk-free asset is described by db t = r t B t dt, B = 1. We assume that under P, S t follows a geometric Brownian motion with Markov-modulated parameters ds t = µ t S t dt + σ t S t dw t, S > where W t is a standard Brownian motion on (Ω, F, P). Moreover, we assume that X is independent of W under P. Let S t denote the discounted stock price at time t, that is S t = e t rsds S t. The dynamic of S t is governed by d S t = (µ t r t ) S t dt + σ t St dw t. In this paper, we want to value the European power call options whose payoff at the maturity T is the power function of the stock price S T and the strike price K: (S α T Kα ) +, where α R is the parameter. Equivalent martingale measure Since the market described by the Markov modulated model is incomplete, there are more than one equivalent martingale measures. Following from Elliott et al. [1], we employ the regime swithcing Esscher transform to determine an equivalent martingale measure. Let Y t = ln St S, for t [, T]. Suppose (F Y ) t [,T], (F X ) t [,T] denote the P-augments of the natural filtrations generated by {Y t } t [,T], {X t } t [,T] respectively. For each t [, T], set H t = F Y t F X T. Then we can define a regime switching Esscher transform P θ P on H T with respect to a family of parameters θ t by L θ T = d P θ dp = e θsdys E[e (1) θsdys FT X ],

4 6, Regime switching ( ) 35 where E[ ] denotes expectation under P. Note that given FT X, θ sdy s follows a normal distribution with mean θ s(µ s σ )ds and variance θ sσsds under P. So, the Radon- Nikodým derivative L θ T can be written as Lθ T = e θsσsdws 1 θ s σ s ds. Moreover, we assume that E[exp( 1 T θ sσsds)] <. By the fundamental theorem of asset pricing, the absence of arbitrage is essentially equivalent to the existence of an equivalent martingale measure under which the discounted stock price process is a martingale. As in Elliott et al. [1], due to the presence of the uncertainty generated by X, the martingale conditon in our case is given by S t = Ê[ S T H t ] which is defined with respect to the enlarged filtration H under the probability P θ. Here Ê[ ] denotes the expectation under P θ. Then, by applying Bayes rule, we can obtain that the martingale condition holds if and only if θ t = rt µt σt obtain the following remark. Remark.1 Given F X T, is a standard P θ - Brownian motion. t Ŵ t = W t 3 Valuation of power options for all t T. By Girsanov s theorem, we can r s µ s ds σ s In this section, we shall investigate the valuation of the European-style power call option. For the sake of simplifying the notation, we suppose that the current time is. By risk neutral pricing theorem, the value of the European power call option with initial regime i {1,,,, N} and maturity T is given by C i (S, K, T, α) = Ê[e rsds (S α T Kα ) + ]. In order to get the price of power call option C i (S, K, T, α), we may first compute the conditional power call option price Ê[e T rsds (ST α Kα ) + FT X]. Theorem 3.1 The conditional power call option price given F X T Ê[e rsds (ST α K α ) + FT X ] = S α e (α 1) rsds+ 1 α(α 1)σ s ds N(d 1 ) K α e rsds N(d ), where d = ln S K + (r s σ s )ds T, d 1 = d + α σ sds T N( ) denotes the cumulative distribution of normal random variable. Proof Note that is σ sds. Ê[e rsds (S α T Kα ) + F X T ] = e rsds Ê[S α T I {S α T Kα } F X T ] K α e rsds Ê[I {S α T K α } F X T ] I 1 I.

5 36 ( ) 13 First, I = K α e rsds E[I {S α T K α } F X T ]. We set A = {Sα T Kα }. By Itô s lemma, the set A is equivalent to { σ s dŵ s ln K S ( r s σ s ) } ds. Since under P θ, σ sdŵs is a normal distribution N(, σ s ds) given FX T, we immediately obtain that I = K α e rsds N(d ). Then, for the caculation of I 1, we introduce a measure Q equivalent to P θ by the Radon Nikodým derivative dq d P = ST α θ Ê[ST α FX T ] = e T ασsd Ws c 1 α σs ds. Girsanov s theorem implies that W Q t = Ŵt t ασ sds is a standard Brownian motion under measure Q. By Bayes rule, we can obtain I 1 = e rsds Ê[S α TI {S α T K α } F X T ] = e rsds EQ [ d P bθ dq Sα T I {S α T Kα } F X T ] E Q [ d b P θ dq FX T ] = e rsds EQ [I {S α T K α } FT X] E Q [ 1 S F X T α T ] = e rsds E Q [I {S α T K α } F X T ]Ê[Sα T FX T ] = e rsds S α e α (rs σ s )ds+ 1 α σ s ds E Q [I {S α T } F X Kα T ] Under the measure Q, the set A is equivalent to { σ s dŵ s Q ln K S Analogously to the caculation of I, we can get ( r s σ s ) ds α σs }. ds I 1 = S α e(α 1) rsds+ 1 T α(α 1)σ s ds N(d 1 ) By the previous results, we complete the proof. 4 Pricing power options under -state regime switching model In this section, we investigate the valuation of power options under -state regime switching model. We suppose that the state space of X = {X t } t [,T] is χ = {1, }. Let ( ) λ 1 λ 1 Q = λ λ

6 6, Regime switching ( ) 37 denote the generator or Q-matrix of X t where λ 1, λ >. For s, t, the transition probabilities satisfy P(X t+s = 1 X s = 1) = 1 λ 1 t P(X t+s = X t = 1) = λ 1 t P(X t+s = 1 X s = ) = λ t; P(X t+s = X s = ) = 1 λ t. By solving the forward Kolmogorov s equation, we obtain the transition probability over an interval of length t: i. Define So, P ij (t) = λ i λ i + λ j (1 e (λi+λj)t ), i j i, j {1, }. Let J i (t) denote the occupation time of X t in state 1 between and T starting from regime P,T = r s ds = r s, X s ds, U,T = σ sds = P,T = r 1 J i (t) + r (T J i (t)) = (r 1 r )J i (t) + r T, σ s, X s ds. U,T = σ1 J i(t) + σ (T J i(t)) = (σ1 σ )J i(t) + σ T. () By Theorem 3.1 and the definition of P,T, U,T, we can denote the conditional power call price by C i (S, K, T, α, J i (t)). In order to evaluate the power call price C i (S, K, T, α), we need to know the distribution of P,T and U,T. We can see from formula () that the distribution of P,T and U,T can be determined completely by the distribution of the occupation time J i (t). Let f i (t, u) be a probability density of J i (t) for a fixed time < t < and regime i {1, }. As in Yoon et al. [14], f i (t, u) satisfies, for < u < t f 1 (t, u) = e λ(t u) λ1u(( λ 1 λ u t u f (t, u) = e λ(t u) λ1u(( λ 1 λ (t u) u ) 1 I 1 ((λ 1 λ u(t u)) 1 ) + λ1 I ((λ 1 λ u(t u)) 1 ) ) ) 1 I 1 ((λ 1 λ u(t u)) 1 ) + λ I ((λ 1 λ u(t u)) 1 ) ) and f 1 (t, ) =, f 1 (t, t) = e λ1t, f (t, ) = e λt, f (t, t) = where I a (z) is the modified Bessel function defined by I a (z) = ( z )a n= ( z )n n!γ(a+n+1). Then, by Theorem 3.1 and the probability density of J i (t), we can obtain the valuation of the European power call option. Theorem 4.1 C i (S, K, T, α) = C i (S, K, T, α, u)f i (T, u)du + δ (i)e λt C i (S, K, T, α, ) + δ 1 (i)e λ1t C i (S, K, T, α, T), (3) where δ i (k) = { 1 if i = k, if i k.

7 38 ( ) 13 Corollary 4.1 When α = 1, (3) reduces to the formula for the price of a vanilla European call option under regime switching. 5 Numerical analysis In this section, we shall perform a numerical experiment for pricing the power call option under the -state Markov-modulated model. By Matlab language, we can give a numerical solution of formula (3). We suppose that the interest rate is r 1 =.4 when the economy is good; r =.1 when the economy is bad; the volatility is σ 1 =.1, σ =.3, respectively and λ 1 = 1, λ =.5. The initial value S = 1. For each α =.,.5,.8, 1, we consider the impact of the strike price on the option price. We also consider the change of the option price against the time to maturity for each of the fixed K = 8, 1, 1 when α =.5. Fig. 1 The stock price process Fig. Option price vs strike price Fig. 1 displays the stock price process starting from regime 1. Fig. illustrates the power call option prices against the strike prices for each α =.5,.8, 1, 1.5 with initial regime 1, respectively. As the figure shows, the option price is decreasing as the strike price increases. We can see that the greater the α, the greater the option price. When α = 1, the power call option reduces to the vanilla European call option. Fig. 3 depicts the power call price initially at state 1 and state against the time to maturity for each of the K = 8, 1, 1 when α =.5. As is revealed in Fig. 3 that the option price can be a increasing function of the time to maturity.

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