PRICING OF POWER OPTIONS UNDER THE REGIME-SWITCHING MODEL

Abstract

Power options have payoffs that are determined by the price of the underlying asset raised to some power. In this paper, power options are considered under a regime-switching model which can capture complex asset dynamics by permitting switching between different regimes. The pricing formulas for the Laplace transforms of power options are obtained. The prices of power options are calculated using the formulas and compared with the results of the Monte Carlo simulation.

1. Introduction

Power options are one of exotic options and have payoffs that are determined by the stock price raised to some power, whereas plain vanilla options have linear payoffs. Because of the non-linear characteristics of these power options, the power option is appropriate for hedging non-linear price risks [11]. When the exponent of power option is greater than 1, the power option provides a greater leverage than a plain vanilla option. Therefore, for an investor with a strong view of the market, a power call option brings more benefits than a plain vanilla option [7]. On the other hand, it can lead an option seller to high losses. Thus the power call options are usually capped at some predefined level [4].

Power options, also known as polynomial options and leveraged options, are widely traded in financial markets. For example, an option whose payoff is a polynomial function of the Nikkei level at the maturity was issued in Tokyo [7]. Bankers Trust in Germany has issued capped foreign-exchange power options with power exponent two [12]. For more examples, see Tompkins [11], and Macovschi and Quittard-Pinon [9].

Under the Black-Scholes model, Heynen and Kat [7] obtained the pricing formula for power options. Esser [3] and Macovschi and Quittard-Pinon [9] derived a quasi-closed form pricing equation for power options in a general framework. Under the Heston model, pricing formulas for power options were derived analytically in Kim et al. [8].

In this paper, power option prices are considered in a regime-switching model. The regime-switching model, first introduced by Hamilton [6], is one of the most popular nonlinear time series models in the literature. This model can capture complex asset dynamics by permitting switching between different regimes [5]. Su et al. [10] studied this model and obtained a conditional power option value, which can be computed only for a two-state regime-switching model. In this paper, explicit formulas for the Laplace transforms of power option prices are derived under the general regime-switching model.

The rest of the paper is organized as follows. In Section 2, the regimeswitching model and power options are described. Then explicit formulas for the Laplace transforms of power option prices are derived in Section 3. Section 4 calculates prices of power options using the formulas in Section 3, and compares them with the results of the Monte Carlo simulation.

 

2. The model and power options

Under the risk-neutral measure ℙ, the underlying asset price St is given by

where {Z(t) : t ≥ 0} represents the market regime as an irreducible continuoustime Markov process with a finite state space E = {1, . . . ,m} and an infinitesimal generator Q. Here, {W(t) : t ≥ 0} is standard Brownian motion, and {Z(t) : t ≥ 0} and {W(t) : t ≥ 0} are independent under ℙ. We consider a filtration {Ft : t ≥ 0}, where Ft is the smallest σ-algebra generated by {W(u),Z(u) : u ≤ t}. For each i ∈ E, ri and σi represent the expected rate of return and the volatility of the stock price at regime i, respectively.

There are three kinds of power options: standard power option, capped power option and powered option. The payoffs of all power options depend on the price of the underlying asset raised to the power α > 0. For a standard power call, the payoff is and for a standard power put, it is with strike K and maturity T. Standard power call options can be lead to very high risk for a option seller and thus the call options are usually capped at some predefined level L. The payoff at maturity for a capped power call is A powered call has the payoff (max{ST − K, 0})α, and a powered put has the payoff (max{K − ST, 0})α.

It is not necessary to obtain all of these price. The payoff of the capped power call option can be represented as the difference between the payoffs of two standard power call options. In addition, a price of a standard power call option can be obtained from a price of a standard power put option by the putcall parity with the conditional α-th moment of the underlying asset. Hence, in the remainder of this paper, standard power put, powered call and powered put options will be considered. The values of the power options are given as follows:

 

3. Laplace transforms of power option prices

For w ∈ ℂ and t > 0, define ψi(w, t) as

with the conditional expectation given Z(0) = i.

Lemma 3.1. For w ∈ ℂ, ψi(w, t) is given by

where ei is the m-dimensional column vector with all 0s except for a 1 in the ith component, 1 is the m-dimensional column vector with all its components equal to one, and

where diag(ai)i∈E is an m × m diagonal matrix whose diagonal entries are ai, i ∈ E.

Proof. Define Then a bivariate Markov process {(Y(t),Z(t)) : t ≥ 0} is a continuous time Markov addictive process. By Proposition XI.2.2 in Asmussen [2], the m × m matrix Φ(t) with (i, j)th entry is given by

Since (1) is obtained.

Now analytic formulas for the Laplace transform of the standard power options and the powered options are presented.

Standard power option. Recall that the Powp(α, S0, K, T, i) is the price of the standard power put option with strike K and maturity T given that the price of the underlying asset is S0 and the current regime is i. Let

The Laplace transform of f with respect to k is defined as

for complex number w such that the right hand side is well-defined. The following theorem provides an explicit formula for the transform

Theorem 3.2. Suppose that w is a complex number such that Re(w) > α. Then the Laplace transform is given by

where ei, 1 and Λ(w) are given in Lemma 3.1.

Proof. Let X(t) = log S(t). For can be written as

The integration in (2) is

Substituting (3) into (2) yields

By (1) in Lemma 3.1, the proof is completed.

Powered option. Recall that Powdc(α, S0, K, T, i) and Powdp(α, S0, K, T, i) are the prices of the powered call and put options, respectively, with strike K and maturity T given that the price of the underlying asset is S0 and the current regime is i. Let

Laplace transforms with respect to k are defined, respectively, as

for complex number w such that the right-hand side of each equation is welldefined.

If α is a positive integer, the payoff of a powered option is the sum of payoffs for standard power options by the binomial theorem. For a general α > 0, the Newton’s generalized binomial theorem is used. The theorem is stated as follows: For any complex numbers w, x and y such that |x| > |y|,

where with falling factorial (w)n = w(w − 1) · · · (w − n + 1).

The complete beta function is used in Theorem 3.3. It is defined as

for Re(x) > 0 and Re(y) > 0. Note that B(x, y) is symmetric by its definition, and has another form as follows:

Theorem 3.3. (a) Suppose that w is a complex number such that Re(w) < 0. Then the Laplace transform is given by

where ei, 1 and Λ(w) are given in Lemma 3.1.

(b) Suppose that w is a complex number such that Re(w) > α. Then the Laplace transform is given by

Proof. Let X(t) = log S(t). For can be written as

Using (4), for k < XT

Combine (7) and (8),

Since with ⌊x⌋ the largest integer less than or equal to x,

for k ≤ XT. By the dominated convergence theorem, (9) follows

For Re(w) < 0 and n > 0, the integration in the above equation is Then

Since and (5), the summation in (10) can be written as

Substituting the above equation into (10) yields (6). The proof of (b) is omitted since it is similar to that of (a).

 

4. Numerical examples

In this section, the numerical examples of power options are provided. The Laplace transforms for the prices of the standard power put and powered put options are given by Theorem 3.2 and Theorem 3.3, respectively. To invert the Laplace transforms, the Euler inversion method in Abate and Whitt [1] is used. The model in the case of two regimes are considered. Suppose that Z is a Markov process with state space {1, 2} and infinitesimal generator

For the calculation of the data in Tables 1 and 2, the parameters used for underlying asset prices are: S0 = 1, r1 = 0.05, r2 = 0.03, σ1 = 0.2 and σ2 = 0.1.

TABLE 1.Comparison of Theorem 3.2 and the Monte Carlo results for the values of standard power put options with S0 = 1.

TABLE 2.Comparison of Theorem 3.3 and the Monte Carlo results for the values of powered put options with S0 = 1.

In Tables 1 and 2, the transform method and the Monte Carlo simulation results are presented for the prices of standard power put options Powp(α, S0, K, T, i) and powered put options Powdp(α, S0, K, T, i) with power α = 2, 3, 5, maturity T = 1, 2, and strike price K = 0.9, 1, 1.1. For each simulation result in Tables 1 and 2, 107 replications are generated. Tables 1 and 2 show that the values of transform method are included in 95% confidence intervals. It is observed that the price of each standard power put and powered put option for Z(0) = 2 is lower than that for Z(0) = 1, and the difference between these values decreases as T increases. As α increases, the standard power put option value increases in Table 1, and the powered put option value decreases in Table 2.