Discrete Dynamics in Nature and Society

Volume 2018, Article ID 8545841, 15 pages

https://doi.org/10.1155/2018/8545841

## Pricing Vulnerable Options with Market Prices of Common Jump Risks under Regime-Switching Models

Correspondence should be addressed to Miao Han; moc.621@oaimnahkcul

Received 2 September 2017; Accepted 14 December 2017; Published 21 January 2018

Academic Editor: Paolo Renna

Copyright © 2018 Miao Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the valuation of vulnerable European options considering the market prices of common systematic jump risks under regime-switching jump-diffusion models. The way of regime-switching Esscher transform is adopted to identify an equivalent martingale measure for pricing vulnerable European options. Explicit analytical pricing formulae for vulnerable European options are derived by risk-neutral pricing theory. For comparison, the other two cases are also considered separately. The first case considers all jump risks as unsystematic risks while the second one assumes all jumps risks to be systematic risks. Numerical examples for the valuation of vulnerable European options are provided to illustrate our results and indicate the influence of the market prices of jump risks on the valuation of vulnerable European options.

#### 1. Introduction

Along with the development of the OTC market, people have recognized the influence of credit risk on financial derivatives pricing and attempted to establish all kinds of credit risk models. Johnson and Stulz [1] first incorporate credit risk with option pricing models and call it vulnerable options, which are vulnerable to counterparty default. They assume that if the counterparty writing an option is unable to make a promised payment, the holder of a derivative security would receive all assets of the counterparty. Klein [2] extends the work of Johnson and Stulz [1] by allowing the option writer to have other liabilities, which rank equally with payments under the option. It also extends the work of Hull and White [3] by relaxing the assumption of independence between the assets of the counterparty and the asset underlying the option. Jarrow and Turnbull [4] provide a new methodology for pricing and hedging derivative securities involving two types of credit risks. Klein and Inglis [5] also expand on the results of pricing vulnerable European options when the payoff of the option can increase the risk of financial distress from Johnson and Stulz [1] and Klein [2]. Hung and Liu [6] extend the framework of Klein [2] to price vulnerable option pricing when the market is incomplete. Chang and Huang [7] and Klein and Yang [8] study the valuation of vulnerable American options with correlated credit risk.

However, the above literatures on vulnerable options all assume that the dynamics of the assets are modelled by geometric Brownian motions with constant drift and volatility. Indeed, over the past decade or two, dozens of empirical evidences have revealed that risky asset prices not only present sudden shocks due to the arrival of important new information in financial markets but also exhibit different behaviors in different time periods due to the time-inhomogeneity generated by the financial market. For the former case, Merton [9] introduces the jump-diffusion models with compound Poisson processes into option pricing (see Kou [10], Xu et al. [11], Tian et al. [12], etc.). For the latter case, Markov regime-switching models have provided us with a natural and convenient way to describe structural changes in market interest rate, exchange rate, stock returns, and so forth, since Hamilton [13] introduced this class of models into economics and finance.

To better describe both the time-inhomogeneity and sudden shocks in the processes of asset prices, there have been lots of papers studying options pricing through incorporating jump-diffusion models with Markov regime-switching models. Elliott and Osakwe [14] consider option pricing for pure jump processes with Markov switching compensators. Bo et al. [15] study the pricing of some currency options based on the Markov-modulated jump-diffusion models for the spot foreign exchange rate. Wang and Wang [16] and Niu and Wang [17] study the pricing problem of vulnerable European options under the Markov regime switching jump-diffusion models. However, they regard all jump risks as systematic risks. Indeed, the jumps in an actual process may be caused by the market or may occur on its own. Tian et al. [12] divide the jumps into individual jumps for each asset price and common jumps that affect the prices of all assets. In this paper, considering the market prices of common systematic jump risks regardless of individual jump risks, we develop an equivalent martingale measure for two regime-switching jump-diffusion processes with correlated jumps via regime-switching Esscher transform and consider the differences between the physical jump-diffusion processes and the risk-neutral jump-diffusion processes. The dynamics of the risk assets prices in this paper are different from those obtained by considering the market prices of all jump risks in Niu and Wang [17] under the risk-neutral measure. In order to consider the influence of the market prices of jump risks on vulnerable option values, we also study vulnerable options pricing formulae with or without market prices of all jump risks. Finally, numerical results are also presented to illustrate our results by the Monte Carlo simulations.

The rest of the paper is organized as follows. Section 2 presents the basic setting and asset price model. In Section 3, we employ the regime-switching Esscher transform to determine an equivalent martingale measure considering the market prices of common systematic jump risks and derive the closed-form pricing formulae for vulnerable European options under a Markov regime switching jump-diffusion model. In Section 4, for comparison, the pricing formulae for vulnerable European options are obtained taking the market prices of all jump risks into consideration. Section 5 presents some numerical results to illustrate the valuation of vulnerable options for different parameters and different cases. Section 6 concludes the paper.

#### 2. The Model Description

Consider a continuous-time financial market with a finite time horizon , where . We consider a complete probability space , where is a physical probability measure, under which all stochastic processes are defined. We equip the probability space with a filtration and assume that there are three primary securities, namely, a risk-free bond , underlying asset , and option writer’s asset , which are traded continuously over the time horizon . is a continuous-time finite-state Markov chain on with a finite-state space . We use the states of to indicate the states of the economy. We adopt the assumptions of Elliott et al. [18] that the state space of is a limited collection of vectors , where with “1” in the* i*th component. Suppose that the time-invariant generator matrix of is with . Then, following Elliott et al. [18], the semimartingale decomposition of is given bywhere is an -valued martingale with respect to the filtration generated by under .

Let denote the value of the bank currency account at time . If one initially saves , then he can gain at time . We assume that the instantaneous market interest rates depend on the economic states of . Then, is given by where with , for each ; denotes the inner product in . For , , else, .

Let the dynamics of the underlying asset value of the option follow a regime-switching jump-diffusion process under the physical measure ,where is a standard Brownian motion on . The expected return rate and the volatility of the underlying asset depend on . They are defined as where and with for each . is a Poisson process with intensity . Following Tian et al. [12], both the jump term and the intensity consist of two parts, where, specifically, shocks to the underlying asset price are also composed of two parts: individual shocks for each asset price and common shocks affecting the prices of all assets. Here, we take the common shocks as market factors, which are considered as systematic risks and consider the individual shocks as unsystematic risks. We assume and are independent Poisson processes with intensities and , which depend on the states of and are described by where , , and . If the jump occurs at time , the jump amplitude of the underlying asset is controlled by . For any time , we assume that and are independently and identically distributed and The mean percentage jump size of the price is given by

We assume the dynamics of the counterparty’s asset value are also driven by the following regime-switching jump-diffusion process.where is a standard Brownian motion on . We assume the expected return rate and the volatility of the counterparty’s asset value depend on and are defined by where and with for each . is also a Poisson process with corresponding intensity . In addition, shocks to also include the individual component and the common component. Both the jump term and the intensity consist of two parts, where is a Poisson process with intensity , which is independent of and . The intensities , , and also depend on the states of ; they are given by where and . If the jump occurs at time , the jump amplitude of is controlled by . For , we assume that and are independently and identically distributed with distribution The mean percentage jump size is given by and have the correlation coefficient . Moreover, we assume that , , , , , and are mutually independent.

#### 3. Vulnerable European Option Pricing Considering the Market Prices of Common Jump Risks

##### 3.1. Equivalent Martingale Measure via Esscher Transform

The financial market described by a regime-switching model and the jump components is incomplete; hence the risk-neutral measure is not uniquely determined. In order to obtain the valuation of vulnerable European options, we need to choose a special martingale measure. Esscher transform, which was first introduced by Gerber and Shiu [19], is a method of selecting the equivalent martingale measure and has extensive applications in the fields of finance and insurance. In this subsection, we adopt the random Esscher transform of Elliott et al. [20] to identify an equivalent martingale measure.

Here, we divide jump risks into two parts and price common jump risks caused by market factors in the model while neglecting the jump risks caused by the assets themselves.

First, we define two processes and as follows:For simplicity, let where and are the continuous diffusive part, common jump part, and individual jump part, respectively, of They have the following forms:

Let and denote the -augmentations of natural filtrations generated by and , respectively. For each , set . For any , let . Then, the random Esscher transform on with regime-switching parameters and is given as follows:where denote the expectation under measure . For , , and , and for , and . Since then

By the fundamental theorem of asset pricing, the absence of arbitrage opportunities is “essentially” equivalent to the existence of an equivalent martingale measure under which the discounted stock price process is a martingale. Following Elliott et al. [20], due to the presence of the uncertainty generated by the Markov chain process, the martingale condition is obtained by considering an enlarged filtration . The following theorem presents the result.

Theorem 1. *The martingale conditions are satisfied if and only ifwhere*

*Proof. *LetThen, by Bayes’ rule, we have The martingale condition is satisfied if and only if whereSimilarly, since it follows that where and

Proposition 2. *Conditional on and under the equivalent martingale measure , and follow stochastic differential equations:where , , , and are given by (21). Moreover,are two standard Brownian motions. is still a Poisson process under the measure with the intensity rate . is given as follows:and the mean percentage jump sizes of and are*

*Proof. *Under the conditions of , we use Girsanov’s theorem and (20) to obtain which are two standard Brownian motions with the correlation coefficient . Note that the parameters describing the common jumps in the two risky assets values have been changed. Now, we prove the formulae of (32) and (33). Hence, under the condition and the probability measure , the intensity rate of is . The density function of is Similarly, we can also prove that the density function of is Note that the risk-neutral probability measure is selected by the Esscher transform, and the probability law of the Markov chain remains the same after the measure change; that is, under , still has the same semimartingale dynamics. For each , we find that the solutions of (21) are not unique. Hence, a special case needs to be considered to obtain an equivalent martingale measure. We suppose Then the following corollary gives a pair of solutions for the martingale condition.

Corollary 3. *The martingale conditions (21) are satisfied if the parameters are given as follows:The explicit Esscher parameters in Corollary 3 will be used to the Monte Carlo simulations in Section 5.*

##### 3.2. Vulnerable European Option Pricing

As in Klein [2], we assume that default will only occur if the counterparty’s asset value at maturity is less than some amount . Additionally, this amount is not set to the value of the option but corresponds to the amount of the counterparty’s claim at exercise time . Once a credit loss occurs at exercise time , only the proportion of the value of the option at is paid out by the counterparty. Then, the payoff of a vulnerable European call option is given by where is the strike price of the option and is the deadweight cost associated with bankruptcy, expressed as a percentage of the value of the counterparty’s assets. By the risk-neutral arguments, under the risk-neutral measure , the price of a vulnerable European call option at time 0 is given as is the risk-free rate. depends on for the Markov regime-switching in our pricing model.

Under the conditions and , for , the value of a vulnerable European call option at time 0 is given asClearly, represents the value of the vulnerable option given , conditional on and jumps on the underlying asset and the assets of the counterparty, respectively.

Therefore, the conditional vulnerable European call option price given isLet denote the occupation time of in state over the time duration , where . Then From Buffington and Elliott [21], the price of a vulnerable European call option is given bywhere is the joint distribution of the occupation time under the condition and the probability measure . Write for the vector of occupation times. Let denote a diagonal matrix with the elements in the vector as its diagonal. Then, for any , the characteristic function of under the condition and the martingale measure is given by where and . Note that can be completely determined by the characteristic function. As for the proof, refer to Buffington and Elliott [21].

At this point, the key problem that needs to be solved is to determine . According to the* Itô* formula, the following equalities hold: where , is bivariate normally distributed under the conditions and with the following properties:Let where and are standard normal random variables and the correlation coefficient is Now we need to calculate , and the derivation process can be referred to Tian et al. [12].whereThen, we can get , , , and in closed form: whereThe parameters are expressed as follows:Therefore, the conditional vulnerable European call option price given is

Note the valuation of vulnerable European call options depends on the parameters , , , and . For the numerical analysis, we use the special parameters that are given by Corollary 3:

*Note.* If we adopt the hypothesis of Merton [9] and regard the jump risks as unsystematic risks which should not be priced, thus, the parameters describing the jumps in the two risky asset price processes will not be changed after application of the measure transform from the physical measure to risk-neutral measure. That is, the parameters and in (18) are equal to 0. From Theorem 1 and Proposition 2, we can obtain the results of the following two propositions specifically.

Proposition 4. *The martingale conditions are satisfied if and only if*

Proposition 5. *Conditional on and under the equivalent martingale measure, the dynamics of and are given bywhere and are two standard Brownian motions which is defined by (31).*

Under the conditions and , for , is also bivariate normally distributed with the following properties:Similar to the previous results, we can obtain