# Pricing of Power Exchange Option with Jumps under the Double Risk of Exchange and Default

Authors: Kaili Xiang; Peng Hu; Jie Shen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Power exchange option is an exotic option which combines power option and exchange option. In this paper, we consider the pricing of the power exchange option under exchange rate volatility risk and issuing company bankruptcy risk. Meanwhile, considering the major events between the two countries, we add the Poisson jump process to the option model in order to reflect the impact of sudden factors on the price of transnational derivatives in the international market. According to the no-arbitrage principle, a mathematical model for pricing such problems is established, and explicit solutions are obtained. The numerical examples show that the model established in this paper is effective.


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## Body

## 1. Introduction

As the influence of fierce market competition and homogenization of financial products, the traditional European option or American option can no longer meet the needs of investors. Various types of overseas derivatives and risk management financial products have been designed by many financial institutions. Some of them are customized products which can cater to the special risk hedging needs of institutional investors. Power exchange option is an exotic option which is derived from exchange option and power option; both of them are powerful financial instruments for hedging nonlinear risks or incentive design of the executive stock option [1, 2], which are mainly traded in the OTC market. Compared with the traditional options, the application scenarios of power exchange options are more diversified, with better flexibility, innovation, and practicability. As the option is trade in the OTC market, the power exchange option is typically nonstandardized contracts. The elements are designed and adjusted on the basis of the specific needs of both parties, and there is no unified and standardized standard. On the one hand, the OTC option meets the needs of both buyers and sellers, so the volume of trading and the volume of trading far exceed the OTC option. On the other hand, different security firms or future companies have different pricing standards for royalties, and only the contracting parties can understand the content of the contract, let alone obtain market data, which leads to the almost opaque phenomenon of the OTC option. Because the exchange could not monitor whether the parties to the contract have fulfilled their obligations, when the issuing company is in bankruptcy, liquidation, etc., the counterparty would not be able to complete the payment according to the contract when the option expires; thus, a breach of contract occurred.Many scholars have conducted in-depth research on the power exchange option and jump risk.Margrabe [3] introduced the definition of exchange option firstly, which was a new option whose underlying assets are two risky assets. It allowed option holders to exchange the two risky assets at the maturity date. At the same time, Margrabe proposed that the value of American exchange options with short-term maturity was consistent with that of European exchange options with standard conditions. That is to say, American exchange options with early exercise rights were not superior to the European exchange options. Fischer [4] studied the pricing of an exchange option and explained that the exercise price was the price of untranslated assets.Blenman and Clark [5] summarized the exchange option of Fischer-Margrabe [3, 4] and Tompkins [1] as the power exchange option, whose payoffs were λ1S1α1T−λ2S2α2T+. Under the risk-neutral probability, assumed that the values of underlying assets were controlled by geometric Brownian motion, they found the pricing formula of the European power exchange option:(1)PEt,S1,S2,ξ;T=γ1t,S1,ξ;TNd1−γ2t,S2,ξ;TNd2,where(2)d1=lnλ1S1α1/λ2S2α2+α1r−δ1−α2r−δ2−α11−α1σ12/2+α21−α2σ22/2+1/2υ2T−tυT−t,d2=lnλ1S1α1/λ2S2α2+α1r−δ1−α2r−δ2−α11−α1σ12/2+α21−α2σ22/2−1/2υ2T−tυT−t,ϒ1t,S1,ξ;T=λ1S1α1expα1−1r−α1δ1−α11−α1σ122T−t,ϒ1t,S1,ξ;T=λ1S1α1expα1−1r−α1δ1−α11−α1σ122T−t,where N⋅ denotes the cumulative normal density function.With this formula, they proved that perfect hedging could be achieved by holding multiple positions of assetS1 value α1Nd1ϒ1, short positions of asset S2 value α2Nd2ϒ2, and buying or selling portfolios of riskless assets.Blenman and Clark also proved the equivalence of American power exchange option and European power exchange option under certain conditions. For the American power exchange option with parametersξ=r,α1,α2,λ1,λ2,δ1,δ2, if α2≤1,α1r−δ1≥r and α2r−δ2≤r, it is not optimal to exercise option rights early, and its value is the same as that of European options.The appearance of new information in option pricing is likely to bring about discontinuous changes in asset prices. In order to capture discontinuous asset prices, Merton [6] based on Black–Scholes model by using underlying assets included a compound Poisson process to simulate discontinuous changes in stock prices. The pricing formulas of several kinds of exotic options in the jump process were given, which laid a foundation for further research.By using the jump-diffusion model, Tian et al. [7] assumed that the dynamics of asset prices are controlled by jump-diffusion and that the two assets were interrelated, and a closed valuation formula for the fragile European option was calculated. It was proved that the price of the fragile call option was fragile. If there was no consideration of jump risk, the value of fragile options and stocks were calculated. Price may run counter to reality.Wang [8] incorporated the discontinuous change of risky asset prices into the power-exchange option model. The jump-diffusion process was used to describe the dynamic change of asset prices. Not only the common jump time but also the jump strength and the difference of the impact of common jump components on asset prices were considered. The total jump risk was divided into special components and general jump risk, and the differences between special jump risk and general jump risk are considered. In addition, the correlation between two kinds of risky assets was included in both the continuous part and the discontinuous part, while the correlation between the discontinuous part was linked through a common jump process, which improved the pricing framework of the power exchange option of Blenman and Clark [5] and obtained a clear pricing formula of the power exchange option.Owing to the fact that the exchange does not assume the responsibility of the over-the-counter (OTC) parties to fulfill their obligations, the holders of over-the-counter (OTC) contracts are vulnerable to the risk of counterparties, that is, the risk that one counterparty in a financial contract fails to meet the agreed terms. Hull and White [9] calculated the pricing formulas of defaultable European option and compared the pricing analytic formulas of defaultable European option, American option, and ordinary European option by numerical methods. Klein [10] put forward a more realistic assumption that the final payment of default depended on the final market value of assets and other companies with the same level of liability. Brigo and Mercurio [11] showed how to consider the event that counterparties may default in a risk-neutral valuation of financial returns.Wang et al. [12] proposed a power-exchange option pricing model including default risk and jump risk. In their model, they not only included jump components in all asset price processes but also made these processes interrelated. In the calculation process, the pricing formulas of the power exchange option under counterparty risk and jump risk are obtained by using measure transformation technology.Xu et al. [13] studied the pricing of power exchange options with default risk. They considered the possibility of bankruptcy and liquidation of the company at any time before the expiration of the option. That is to say, the time when the counterparty actually defaults was uncertain. Furthermore, the pricing formula of the power exchange option with uncertain default risk was given.With the integration of the international financial environment, cross-trading of various types of asset securities is frequent, and various pricing problems between the two kinds of currency transactions have been considered. Reiner [14] linked foreign stock and currency risk on the premise that both underlying asset value and exchange rate fluctuations met the B-S equation. For the first time, the pricing model of dual-currency stock options in four cases was proposed, which progressively progressed according to the degree of complexity.At present, the research on foreign exchange options has formed a relatively complete theoretical structure, while the deeper level of dual-currency options is still under continuous research. Kwok and Wong [15] gave the pricing formula of the singular option with dual-currency assets. Li and Zhou [16] studied the pricing of the power exchange option under the influence of exchange rate fluctuations. Gao and Wang [17], under the dual-currency model, used the method of measure transformation to give the European option pricing formula of asset price with jump process. Huang and He [18] studied the pricing theory of dual-currency European option and reset option under jump risk related to floating exchange rate. Furthermore, underlying asset can be considered drived by Lévy process. Yu et al. [19] modeled the price of the found by an exponential Lévy process. Zhang et al. [20] linked to the performance of the underlying asset, which is modeled by an exponential Lévy process.According to the research path of the above literature studies, this paper comprehensively considers the power exchange option with the dual risk of exchange rate and default. Also, the Poisson jump process is added into the option to analyze the pricing of the power exchange options with dual currency with jump process.


## 2. Model Building

In recent years, financial market emergencies and important policies occur frequently, and the release of such major financial events and important policies has a seismic impact on the entire financial market. In order to reflect the unexpected and unpredictable major events in the market, we add the Poisson jump process to the basic assets, company value, and exchange rate, based on which we analyze the dual-currency power-exchange option model with default risk.Assume that the probability spaceΩ,F,Ρ represents an economic environment with uncertainty, where P is the risk-neutral probability measure. We define the market environment that includes two risky assets. At the expiration date T, the payoffs of European power exchange options are as follows:(3)VS1T,S2T,T=λ1S1α1T−λ2S2α2T+,where λii=1,2 and αii=1,2 are positive constants. Especially, for λ1=λ2=1 and α1=α2=1, it simplifies to the standard exchange option.According to Merton’s [6] assumption, the jump risks common to the underlying assets are diversified in the market, and the risk premium is 0. We use the method of partial differential equation to research the pricing of dual-currency power exchange options with jump process.With the risk-neutral probability measureP, according to the different combinations of stocks that investors hold, we divide the dual-currency power exchange options with jump process into two types for discussion.Type 1.
A domestic investor invests in a foreign stock and a domestic stock at the same time. Over the life of the investment, he will sell foreign stocks and buy domestic ones if they outperform the foreign ones. So, to reduce the transaction cost of switching the underlying asset, he bought a power exchange option that would exchange the two assets at maturity time.
Assume the underlying assetsSit,i=1,2 and the exchange rate Xt follow the following equations; each of these equations contains a jump representing a major event Nt.(4)dS1t=rf−ρ1Xσ1σX−k1λS1tdt+σ1S1tdW1t+eZ1t−−1S1tdNt,dS2t=rd−k2λS2tdt+σ2S2tdW2t+eZ2t−−1S2tdNt,dXt=rd−rfXtdt+σXXtdWXt+eZXt−−1XtdNt,where rf is the foreign risk-free interest rate; rd is the risk-free interest rate of the country; σ1 and σ2 are the instantaneous standard deviations of foreign stock return volatility and domestic stock return, respectively; σX is the instantaneous standard deviation of the exchange rate; ρ1X=CovdW1t,dWXt represents the instantaneous covariance of foreign stocks and exchange rates; and W1t, W2t, and WXt are standard Brownian motion under the measure, which meet dW1t⋅dW2t=ρ12dt, dW1t⋅dWXt=ρ1Xdt, and dW2t⋅dWXt=ρ2Xdt. In addition, as described by Merton [6], Nt is a Poisson process with an intensity of λ, and it is independent of other Brownian motions. It is used to simulate the discontinuous changes of asset prices and affect the general changes of asset prices. If the general jump occurs at time t, then the jump of the asset is Sit,i=1,2 controlled by Zit, where Zit is the normal distribution with mean μi and standard deviation γi>0.Type 2.
A domestic investor is interested in two foreign stocks, so he buys foreign power exchange options which use the two stocks as the underlying asset. Since he buys in his own currency, he should exchange the option at the exchange rate. In this case, assume that the price of the underlying assetSit,i=1,2 and the exchange rate Xt follow the following equations, where the parameters have the same meaning as those of Type 1.(5)dS1t=rf−ρ1Xσ1σX−k1λS1tdt+σ1S1tdW1t+eZ1t−−1S1tdNt,dS2t=rf−ρ2Xσ2σX−k2λS2tdt+σ2S2tdW2t+eZ2t−−1S2tdNt,dXt=rd−rf−kXλXtdt+σXXtdWXt+eZXt−−1XtdNt.


## 3. Explicit Solution of the Model

### 3.1. Solution of the Pricing Model ofType 1

According to the principle of no arbitrage and using ITO’s lemma, we get the following conclusions for the first kind of the dual-currency power exchange option model.Theorem 1.
Pricing formula of dual-currency power exchange options under the first kind of jump risk is(6)C∗=e−rdTEλ2S2α2Τ⋅∑n=0∞λQTne−λQTn!K1−K2+K3−K4=λ2e−rdT⋅S2α20⋅expα2rd−σ222−k2λ+α22σ222T⋅expλeα2μ2+1/2α22γ22−1T⋅∑n=0∞λQTne−λQTn!K1−K2+K3−K4,where(7)K1=λ1λ2eM1+1/2H1N2M1−lnλ2/λ1+H1H1,M2−lnL∗+ρH1H2H2,ρ,K2=N2M1−lnλ2/λ1H1,M2−lnL∗H2,ρ,K3=λ1λ2⋅1−αL⋅eM1+M2+1/2H1+1/2H2+ρH1H2⋅N2M1−lnλ2/λ1+H1+ρH1H2H1,−lnL∗−M2−ρH1H2−H2H2,−ρ,K4=1−αLeM2+1/2H2N2M1−lnλ2/λ1+ρH1H2H1,−M2−lnL∗+H2H2,−ρ.Proof.
LetGt=Xt⋅S1t, so Gt satisfy the following random process:(8)dGt=dXtS1t=XtdS1t+S1tdXt+dS1tdXt+XtS1t−Xt−S1t−dNt=XtS1trf−ρ1Xσ1σX−k1λdt+σ1dW1t+S1tXtrd−rf−kXλdt+σXdWXt+σ1S1tσXXtdW1tdWXt+eZXt−−1Xt−+Xt−⋅eZ1t−−1S1t−+S1t−−Xt−S1t−dNt=Gtrd−k1λ−kXλdt+Gtσ1dW1t+σXdWXt+eZXt−+Z1t−−1Gt−dNt=rd−k1λ−kXλGtdt+σGGtdWGt+eZGt−−1Gt−dNt,where WGt is the Brownian motion under risk-neutral measure and σG2=σ12+σX2−2σ1σXρ1X.
According toσGρ2G=σ1ρ12+σXρ2X, we have ρ2G=σ1ρ12+σXρ2X/σG. In the same way, according to σGρGV=σ1ρ1V+σXρVX, we have ρGV=σ1ρ1V+σXρVX/σG. On the contrary, Nt is still a Poisson process with intensity λ, and ZGt=ZXt+Z1t is a normal distribution with mean μX+μ1 and variance γX2+γ12.
By using the orthogonal transformation, we have(9)dGt=rd−k1λ−kXλGtdt+σGGtdWGt+eZGt−−1Gt−dNt,dVt=rd−kVλVtdt+σVρGVVtdBGt+σV1−ρGV2VtdBVt+eZVt−−1Vt−dNt,dS2t=rd−k2λS2tdt+σ2ρ2VS2tρGVdBGt+1−ρGV2dBVt+σ21−ρ2V2S2tdB2t+eZ2t−−1S2t−dNt,where dBGt, dB2t, and dBVt are independent Brownian motions under the risk-neutral probability measure P.
Define a new probability measureQ, which is equivalent to a probability measure P.(10)Δt=dQdP=EtpS2α2TES2α2T.
From the driving equation of risk assetsS2t (9), the following equation can be obtained by transformation:(11)dlnS2t=dS2tS2t−121S22tdS2t⋅dS2t+lnS2t−lnS2t−dNt=rd−σ222−k2λdt+σ2ρ2v⋅ρGV⋅dBGt+σ2ρ2v1−ρGV2⋅dBVt+σ21−ρ2V2dB2t+lnS2t−+eZ2t−−1S2t−−lnS2t−dNt=rd−σ222−k2λdt+σ2ρ2vρGVdBGt+σ2ρ2v1−ρGV2⋅dBVt+σ21−ρ2V2dB2t+Z2t−dNt.
To integrate directly, we have(12)S2α2t=S2α20⋅expα2rd−σ222−k2λt+α2σ2ρ2vρGVBGt+α2σ2ρ2v1−ρGV2⋅BVt+α2σ21−ρ2V2B2t+α2∑kNtZ2τk.
Then, to calculate its expectation, we have(13)EPS2α2T=S2α20⋅expα2rd−σ222−k2λ+α22σ222+λeα2μ2+1/2α22γ22−1T.
Therefore, the new probability measure can be obtained:(14)Δt=EtpS2α2TES2α2T=EPS2α2TES2α2TFt=Eexp−α2σ222T+α2σ2ρ2v⋅ρGV⋅BGT+α2σ2ρ2v1−ρGV2⋅BVT+α2σ21−ρ2V2B2T+α2∑kNTZ2τk−λeα2μ2+1/2α22γ22−1TFt.
According to Girsanov theorem,BGQt, B2Qt, and BVQt are Brownian motions under the probability measure Q, and(15)BGQt=BGt−α2σ2ρ2VρGVt,B2Qt=B2t−α2σ21−ρ2V2t,BVQt=BVt−α2σ2ρ2V1−ρGV2t.
On the contrary, after measure transformation, the discontinuity term also changes correspondingly, whereNQt is also the Poisson process, but its strength is λQ=λeα2μ2+1/2α22γ22, Z2,kQ is normal distribution which has mean μ2+α2γ22 and variance γ22, and ZG,kQ and ZV,kQ are normal distribution whose parameters are unchanged.
The value equation of the dual-currency power exchange options is(16)C∗=e−rTEλ1Gα1T−λ2S2α2T+IVT≥L∗+1−αVTLIVT<L∗=e−rTEλ2S2α2TEλ2S2α2TEλ2S2α2Tλ1Gα1Tλ2S2α2T−1+IVT≥L∗+1−αVTLIVT<L∗.
That is to say,(17)C∗=e−rTEλ2S2α2T⋅EQλ1Gα1Tλ2S2α2T−1+IVT≥L∗+1−αVTLIVT<L∗=e−rTEλ2S2α2T⋅∑n=0∞EQλ1Gα1Tλ2S2α2T−1+IVT≥L∗+1−αVTLIVT<L∗NQt=n⋅PNQt=n=e−rTEλ2S2α2T⋅∑n=0∞λ^Tne−λ^Tn!EQλ1Gα1Tλ2S2α2T−1+⋅IVT≥L∗+1−αVTLIVT<L∗Zn=e−rTEλ2S2α2T⋅∑n=0∞λ^Tne−λ^Tn!⋅Cn,where Cn satisfy(18)Cn=EQλ1Gα1T,nλ2S2α2Τ,n−1Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n≥L∗+1−αVT,nLIλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n<L∗=EQλ1Gα1T,nλ2S2α2Τ,n⋅Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n≥L∗−EQIλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n≥L∗+1−αLEQλ1Gα1T,nλ2S2α2Τ,n⋅VT,n⋅Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n<L∗−1−αLEQVT,n⋅Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n<L∗.
Next, the solutions of the three stochastic differential equations in (9) are calculated. According to the Girsanov theorem, the correlation between the BGQt, B2Qt, and BVQt with the new probability measure Q and BGt, B2t, and BVt with the risk-neutral measure P is equation (15). By using the logarithmic transformation, we have(19)lnGt,n=lnG0+rd−σG22−k1λ−kXλ+α2σ2σGρ2VρGVt+σGBGQt+∑k=1nZGQτk,lnVt,n=lnV0+rd−σV22−kVλ+α2σVσ2ρ2Vt+σVρGVBGQt+σV1−ρGV2BVQt+∑k=1nZVQτk,lnS2t,n=lnS20+rd−σ222−k2λ+α2σ22t+σ2ρ2VρGVBGQt+σ2ρ2V1−ρGV2BVQt+σ21−ρ2V2B2Qt+∑k=1nZ2Qτk,lnS2t,n=lnS20+rd−σ222−k2λ+α2σ22t+σ2ρ2VρGVBGQt+σ2ρ2V1−ρGV2BVQt+σ21−ρ2V2B2Qt+∑k=1nZ2Qτk,lnGα1T,nS2α2T,n=α1lnG0−α2lnS20+α1rd−σG22−k1λ−kXλ+α2σ2σGρ2VρGVt−α2rd−σ222−k2λ+α2σ22t+α1σG−α2σ2ρ2VρGVBGQt−α2σ2ρ2V1−ρGV2BVQt−α2σ21−ρ2V2B2Qt+α1∑k=1nZGQτk−α2∑k=1nZ2Qτk,where the mathematical expectations of lnGα1T,n/S2α2T,n and lnVT,n are(20)M1=ElnGα1T,nS2α2T,n=α1lnG0−α2lnS20+α1rd−σG22−k1λ−kXλ+α2σ2σGρ2VρGV⋅T−α2rd−σ222−k2λ+α2σ22⋅T+nα1μG−nα2μ2+α2γ22,M2=ElnVT,n=lnV0+rd−σV22−kVλ+α2σVσ2ρ2V⋅T+nμV.
Further, the variance and covariance oflnGα1T,n/S2α2T,n and lnVT,n are(21)H1=VarlnGα1T,nS2α2T,n=α12σG2T+α22σ22T−2α1α2σGσ2ρ2VρGVT+nα12γG2+nα22γ22,H2=VarlnVT,n=σV2⋅T+nγV2,R12=CovlnGα1T,nS2α2T,n,lnVT,n=α1σGσVρGV−α2σ2σVρ2V⋅T.
SincelnGα1T,n/S2α2T,n and lnVT,n are two normal random variables with the above properties, for simplicity, it can be transformed into the following form:(22)lnGα1T,nS2α2T,n=M1+H1ξ1,lnVT,n=M2+H2ξ2,where ξ1,ξ2 is a two-dimensional standard normally distributed random variable with correlation coefficient ρ=R12/H1⋅H2. As a result,(23)K1=EQλ1Gα1T,nλ2S2α2T,n⋅Iλ1Gα1T,n/λ2S2α2T,n≥1,VT,n≥L∗=λ1λ2eM1+1/2H1N2M1−lnλ2/λ1+H1H1,M2−lnL∗+ρH1H2H2,ρ,K2=EQIλ1Gα1T,n/λ2S2α2T,n≥1,VT,n≥L∗=N2M1−lnλ2/λ1H1,M2−lnL∗H2,ρ,K3=1−αLEQλ1Gα1T,nλ2S2α2T,n⋅VT,n⋅Iλ1Gα1T,n/λ2S2α2T,n≥1,VT,n<L∗=λ1λ2⋅1−αL⋅eM1+M2+1/2H1+1/2H2+ρH1H2⋅N2M1−lnλ2/λ1+H1+ρH1H2H1,−lnL∗−M2−ρH1H2−H2H2,−ρ.
Similarly,(24)K4=1−αLEQVT,n⋅Iλ1Gα1T,n/λ2S2α2T,n≥1,VT,n<L∗=1−αLeM2+1/2H2N2M1−lnλ2/λ1+ρH1H2H1,−M2−lnL∗+H2H2,−ρ,and finally, we get the pricing formula of the first kind of dual-currency power exchange options with jump risk.


### 3.2. Solution of the Pricing Model ofType 2

For the power-exchange option model of the second kind of dual currency, the following conclusions can be obtained by applying the similar method.Theorem 2.
Power exchange options with double risks of exchange rate and default under the second kind of jump risk have the following pricing formula:(25)C∗∗=e−rdTλ2EXTS2α2T⋅∑n=0∞λQ^Tne−λQ^Tn!⋅K1′−K2′+K3′−K4′=λ2e−rdT⋅X0S2α20⋅expα2rd−σX22−σ222−ρ2Xσ2σXT−12α22σX2+2σXσ2ρ2VρVX+σ22T+α2−kVλ−k2λT+λeμX+1/2γX2−1T+λeα2μ2+1/2α22γ22−1T⋅∑n=0∞λQ^Tne−λQ^Tn!⋅K1′−K2′+K3′−K4′,where(26)K1′=λ1λ2eM1′+1/2H1′N2M1′−lnλ2/λ1+H1′H1′,M2′−lnL∗+ρ′H1′H2′H2′,ρ′,K2′=N2M1′−lnλ2/λ1H1′,M2′−lnL∗H2′,ρ′,K3′=λ1λ2⋅1−αL⋅eM1′+M2′+1/2H1′+1/2H2′+ρ^H1′H2′⋅N2M1′−lnλ2/λ1+H1′+ρ′H1′H2′H1′,−lnL∗−M2′−ρ′H1′H2′−H2′H2′,−ρ′,K4′=1−αL⋅eM2′+1/2H2′⋅N2M1′−lnλ2/λ1+ρ′H1′H2′H1′,−M2′−lnL∗+H2′H2′,−ρ′.


## 3.1. Solution of the Pricing Model ofType 1

According to the principle of no arbitrage and using ITO’s lemma, we get the following conclusions for the first kind of the dual-currency power exchange option model.Theorem 1.
Pricing formula of dual-currency power exchange options under the first kind of jump risk is(6)C∗=e−rdTEλ2S2α2Τ⋅∑n=0∞λQTne−λQTn!K1−K2+K3−K4=λ2e−rdT⋅S2α20⋅expα2rd−σ222−k2λ+α22σ222T⋅expλeα2μ2+1/2α22γ22−1T⋅∑n=0∞λQTne−λQTn!K1−K2+K3−K4,where(7)K1=λ1λ2eM1+1/2H1N2M1−lnλ2/λ1+H1H1,M2−lnL∗+ρH1H2H2,ρ,K2=N2M1−lnλ2/λ1H1,M2−lnL∗H2,ρ,K3=λ1λ2⋅1−αL⋅eM1+M2+1/2H1+1/2H2+ρH1H2⋅N2M1−lnλ2/λ1+H1+ρH1H2H1,−lnL∗−M2−ρH1H2−H2H2,−ρ,K4=1−αLeM2+1/2H2N2M1−lnλ2/λ1+ρH1H2H1,−M2−lnL∗+H2H2,−ρ.Proof.
LetGt=Xt⋅S1t, so Gt satisfy the following random process:(8)dGt=dXtS1t=XtdS1t+S1tdXt+dS1tdXt+XtS1t−Xt−S1t−dNt=XtS1trf−ρ1Xσ1σX−k1λdt+σ1dW1t+S1tXtrd−rf−kXλdt+σXdWXt+σ1S1tσXXtdW1tdWXt+eZXt−−1Xt−+Xt−⋅eZ1t−−1S1t−+S1t−−Xt−S1t−dNt=Gtrd−k1λ−kXλdt+Gtσ1dW1t+σXdWXt+eZXt−+Z1t−−1Gt−dNt=rd−k1λ−kXλGtdt+σGGtdWGt+eZGt−−1Gt−dNt,where WGt is the Brownian motion under risk-neutral measure and σG2=σ12+σX2−2σ1σXρ1X.
According toσGρ2G=σ1ρ12+σXρ2X, we have ρ2G=σ1ρ12+σXρ2X/σG. In the same way, according to σGρGV=σ1ρ1V+σXρVX, we have ρGV=σ1ρ1V+σXρVX/σG. On the contrary, Nt is still a Poisson process with intensity λ, and ZGt=ZXt+Z1t is a normal distribution with mean μX+μ1 and variance γX2+γ12.
By using the orthogonal transformation, we have(9)dGt=rd−k1λ−kXλGtdt+σGGtdWGt+eZGt−−1Gt−dNt,dVt=rd−kVλVtdt+σVρGVVtdBGt+σV1−ρGV2VtdBVt+eZVt−−1Vt−dNt,dS2t=rd−k2λS2tdt+σ2ρ2VS2tρGVdBGt+1−ρGV2dBVt+σ21−ρ2V2S2tdB2t+eZ2t−−1S2t−dNt,where dBGt, dB2t, and dBVt are independent Brownian motions under the risk-neutral probability measure P.
Define a new probability measureQ, which is equivalent to a probability measure P.(10)Δt=dQdP=EtpS2α2TES2α2T.
From the driving equation of risk assetsS2t (9), the following equation can be obtained by transformation:(11)dlnS2t=dS2tS2t−121S22tdS2t⋅dS2t+lnS2t−lnS2t−dNt=rd−σ222−k2λdt+σ2ρ2v⋅ρGV⋅dBGt+σ2ρ2v1−ρGV2⋅dBVt+σ21−ρ2V2dB2t+lnS2t−+eZ2t−−1S2t−−lnS2t−dNt=rd−σ222−k2λdt+σ2ρ2vρGVdBGt+σ2ρ2v1−ρGV2⋅dBVt+σ21−ρ2V2dB2t+Z2t−dNt.
To integrate directly, we have(12)S2α2t=S2α20⋅expα2rd−σ222−k2λt+α2σ2ρ2vρGVBGt+α2σ2ρ2v1−ρGV2⋅BVt+α2σ21−ρ2V2B2t+α2∑kNtZ2τk.
Then, to calculate its expectation, we have(13)EPS2α2T=S2α20⋅expα2rd−σ222−k2λ+α22σ222+λeα2μ2+1/2α22γ22−1T.
Therefore, the new probability measure can be obtained:(14)Δt=EtpS2α2TES2α2T=EPS2α2TES2α2TFt=Eexp−α2σ222T+α2σ2ρ2v⋅ρGV⋅BGT+α2σ2ρ2v1−ρGV2⋅BVT+α2σ21−ρ2V2B2T+α2∑kNTZ2τk−λeα2μ2+1/2α22γ22−1TFt.
According to Girsanov theorem,BGQt, B2Qt, and BVQt are Brownian motions under the probability measure Q, and(15)BGQt=BGt−α2σ2ρ2VρGVt,B2Qt=B2t−α2σ21−ρ2V2t,BVQt=BVt−α2σ2ρ2V1−ρGV2t.
On the contrary, after measure transformation, the discontinuity term also changes correspondingly, whereNQt is also the Poisson process, but its strength is λQ=λeα2μ2+1/2α22γ22, Z2,kQ is normal distribution which has mean μ2+α2γ22 and variance γ22, and ZG,kQ and ZV,kQ are normal distribution whose parameters are unchanged.
The value equation of the dual-currency power exchange options is(16)C∗=e−rTEλ1Gα1T−λ2S2α2T+IVT≥L∗+1−αVTLIVT<L∗=e−rTEλ2S2α2TEλ2S2α2TEλ2S2α2Tλ1Gα1Tλ2S2α2T−1+IVT≥L∗+1−αVTLIVT<L∗.
That is to say,(17)C∗=e−rTEλ2S2α2T⋅EQλ1Gα1Tλ2S2α2T−1+IVT≥L∗+1−αVTLIVT<L∗=e−rTEλ2S2α2T⋅∑n=0∞EQλ1Gα1Tλ2S2α2T−1+IVT≥L∗+1−αVTLIVT<L∗NQt=n⋅PNQt=n=e−rTEλ2S2α2T⋅∑n=0∞λ^Tne−λ^Tn!EQλ1Gα1Tλ2S2α2T−1+⋅IVT≥L∗+1−αVTLIVT<L∗Zn=e−rTEλ2S2α2T⋅∑n=0∞λ^Tne−λ^Tn!⋅Cn,where Cn satisfy(18)Cn=EQλ1Gα1T,nλ2S2α2Τ,n−1Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n≥L∗+1−αVT,nLIλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n<L∗=EQλ1Gα1T,nλ2S2α2Τ,n⋅Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n≥L∗−EQIλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n≥L∗+1−αLEQλ1Gα1T,nλ2S2α2Τ,n⋅VT,n⋅Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n<L∗−1−αLEQVT,n⋅Iλ1Gα1T,n/λ2S2α2Τ,n≥1,VT,n<L∗.
Next, the solutions of the three stochastic differential equations in (9) are calculated. According to the Girsanov theorem, the correlation between the BGQt, B2Qt, and BVQt with the new probability measure Q and BGt, B2t, and BVt with the risk-neutral measure P is equation (15). By using the logarithmic transformation, we have(19)lnGt,n=lnG0+rd−σG22−k1λ−kXλ+α2σ2σGρ2VρGVt+σGBGQt+∑k=1nZGQτk,lnVt,n=lnV0+rd−σV22−kVλ+α2σVσ2ρ2Vt+σVρGVBGQt+σV1−ρGV2BVQt+∑k=1nZVQτk,lnS2t,n=lnS20+rd−σ222−k2λ+α2σ22t+σ2ρ2VρGVBGQt+σ2ρ2V1−ρGV2BVQt+σ21−ρ2V2B2Qt+∑k=1nZ2Qτk,lnS2t,n=lnS20+rd−σ222−k2λ+α2σ22t+σ2ρ2VρGVBGQt+σ2ρ2V1−ρGV2BVQt+σ21−ρ2V2B2Qt+∑k=1nZ2Qτk,lnGα1T,nS2α2T,n=α1lnG0−α2lnS20+α1rd−σG22−k1λ−kXλ+α2σ2σGρ2VρGVt−α2rd−σ222−k2λ+α2σ22t+α1σG−α2σ2ρ2VρGVBGQt−α2σ2ρ2V1−ρGV2BVQt−α2σ21−ρ2V2B2Qt+α1∑k=1nZGQτk−α2∑k=1nZ2Qτk,where the mathematical expectations of lnGα1T,n/S2α2T,n and lnVT,n are(20)M1=ElnGα1T,nS2α2T,n=α1lnG0−α2lnS20+α1rd−σG22−k1λ−kXλ+α2σ2σGρ2VρGV⋅T−α2rd−σ222−k2λ+α2σ22⋅T+nα1μG−nα2μ2+α2γ22,M2=ElnVT,n=lnV0+rd−σV22−kVλ+α2σVσ2ρ2V⋅T+nμV.
Further, the variance and covariance oflnGα1T,n/S2α2T,n and lnVT,n are(21)H1=VarlnGα1T,nS2α2T,n=α12σG2T+α22σ22T−2α1α2σGσ2ρ2VρGVT+nα12γG2+nα22γ22,H2=VarlnVT,n=σV2⋅T+nγV2,R12=CovlnGα1T,nS2α2T,n,lnVT,n=α1σGσVρGV−α2σ2σVρ2V⋅T.
SincelnGα1T,n/S2α2T,n and lnVT,n are two normal random variables with the above properties, for simplicity, it can be transformed into the following form:(22)lnGα1T,nS2α2T,n=M1+H1ξ1,lnVT,n=M2+H2ξ2,where ξ1,ξ2 is a two-dimensional standard normally distributed random variable with correlation coefficient ρ=R12/H1⋅H2. As a result,(23)K1=EQλ1Gα1T,nλ2S2α2T,n⋅Iλ1Gα1T,n/λ2S2α2T,n≥1,VT,n≥L∗=λ1λ2eM1+1/2H1N2M1−lnλ2/λ1+H1H1,M2−lnL∗+ρH1H2H2,ρ,K2=EQIλ1Gα1T,n/λ2S2α2T,n≥1,VT,n≥L∗=N2M1−lnλ2/λ1H1,M2−lnL∗H2,ρ,K3=1−αLEQλ1Gα1T,nλ2S2α2T,n⋅VT,n⋅Iλ1Gα1T,n/λ2S2α2T,n≥1,VT,n<L∗=λ1λ2⋅1−αL⋅eM1+M2+1/2H1+1/2H2+ρH1H2⋅N2M1−lnλ2/λ1+H1+ρH1H2H1,−lnL∗−M2−ρH1H2−H2H2,−ρ.
Similarly,(24)K4=1−αLEQVT,n⋅Iλ1Gα1T,n/λ2S2α2T,n≥1,VT,n<L∗=1−αLeM2+1/2H2N2M1−lnλ2/λ1+ρH1H2H1,−M2−lnL∗+H2H2,−ρ,and finally, we get the pricing formula of the first kind of dual-currency power exchange options with jump risk.


## 3.2. Solution of the Pricing Model ofType 2

For the power-exchange option model of the second kind of dual currency, the following conclusions can be obtained by applying the similar method.Theorem 2.
Power exchange options with double risks of exchange rate and default under the second kind of jump risk have the following pricing formula:(25)C∗∗=e−rdTλ2EXTS2α2T⋅∑n=0∞λQ^Tne−λQ^Tn!⋅K1′−K2′+K3′−K4′=λ2e−rdT⋅X0S2α20⋅expα2rd−σX22−σ222−ρ2Xσ2σXT−12α22σX2+2σXσ2ρ2VρVX+σ22T+α2−kVλ−k2λT+λeμX+1/2γX2−1T+λeα2μ2+1/2α22γ22−1T⋅∑n=0∞λQ^Tne−λQ^Tn!⋅K1′−K2′+K3′−K4′,where(26)K1′=λ1λ2eM1′+1/2H1′N2M1′−lnλ2/λ1+H1′H1′,M2′−lnL∗+ρ′H1′H2′H2′,ρ′,K2′=N2M1′−lnλ2/λ1H1′,M2′−lnL∗H2′,ρ′,K3′=λ1λ2⋅1−αL⋅eM1′+M2′+1/2H1′+1/2H2′+ρ^H1′H2′⋅N2M1′−lnλ2/λ1+H1′+ρ′H1′H2′H1′,−lnL∗−M2′−ρ′H1′H2′−H2′H2′,−ρ′,K4′=1−αL⋅eM2′+1/2H2′⋅N2M1′−lnλ2/λ1+ρ′H1′H2′H1′,−M2′−lnL∗+H2′H2′,−ρ′.


## 4. Numerical Examples

According to the calculation results of two kinds of dual-currency power exchange options, we assign the related parameters in the pricing formula and simulate a numerical case to illustrate how some specific factors affect the price of power exchange options.Firstly, we consider the first characteristic of power exchange options, the exponential factor of magnification, and analyze how the dual-currency power exchange options with exchange rate and default risk will react with the change of option time under the influence of exponential power. Secondly, in order to explore what kind of influence is taken by the jump risk, exchange rate risk, and default risk, respectively, we increase the special circumstances as the comparison sample: no jump risk fragile dual-currency power exchange option, without the fragility of the exchange rate risk jumping power exchange option, and without the risk of default dual-currency bouncing power exchange options, according to the two differences between them were analyzed and determine each factor’s influence on power exchange option price direction and the amplitude of the influence. Finally, for the basic model of power exchange options with the dual risk of exchange rate and default with the jump process studied in this paper, the influence of the change of important parameters related to the jump process on the price of options is considered.As a reference standard for numerical simulation test, Table1 summarizes the assignment of selected basic parameters. The values of the following parameters are mainly derived from those of Wang et al. [12], which are commonly used in the literature by Bakshi et al. [21] and Christoffersen et al. [22]. Without loss of generality, we set the initial price of the underlying asset to be S10=S20=5.0 and the domestic risk-free rate to be r=0.02. Assume that the instantaneous volatility of the underlying asset to be σ1=σ2=0.15 and the jump intensity to be λ=1. Now, we should choose the parameters of the option seller’s assets, and we assume instantaneous volatility σV=0.15, the limit for a company to actually default is 3/4 of the initial value of the seller’s assets V0; that is to say, in the following example, the default barrier L∗=7.5, and Chen [23] found in the literature that the bond recovery rate of 9 different states was about 0.60, so we set the self-weight cost below to be α=0.40. Finally, the option expiration date is assumed to be T=1.0. In Figures 1 and 2 and Table 1, we will change the parameter values accordingly to research the impact of exchange rate risk, counterparty risk, and jump risk on the option price. Other variables maintain the values listed in Table 1.Table 1
Parameter assignment table.
Parameter
Assignment
Initial priceS10
5.0
Constant coefficientλ1
1
Index coefficientα1
2
Volatilityσ1
0.15
Domestic risk-free raterd rd
0.02
Initial rateX
1
Initial assetV
10
Deadweight lossα
0.4
Material default limitL∗
7.5
Creditor’s rightsL
7.5
Correlation coefficientρ1V
0.5
Maturity dateT
1
Intensity of jumpλ
1
Step of jumpμX
0
Standard deviationγ1
0.1
Standard deviationγ2
0.1
Initial priceS20
5.0
Constant coefficientλ2
1
Index coefficientα2
2
Volatilityσ2
0.15
Foreign risk-free raterf
0.03
VolatilityσX
0.15
VolatilityσV
0.15
Correlation coefficientρ1X
0.5
Correlation coefficientρVX
0.2
Correlation coefficientρ2X
0.5
Correlation coefficientρ2V
0.5
Step of jumpμ1
0
Step of jumpμ2
0
Step of jumpμV
0
Standard deviationγX
0.1
Standard deviationγV
0.1Figure 1
Exponential change of the first kind of dual-currency power exchange option price and expiration time.Figure 2
Exponential change of the second kind of dual-currency power exchange option price and expiration time.At the same time, to confirm the value ofn, we should analyze the convergence of the series ∑n=0∞λQ^Tn/n!. By using MATLAB, the following calculation is chosen for accuracy 0.001.Next, we analyze the exponential sensitivity of power exchange options. The basic operations are set the expiration time as a variable on the premise that the data in the above parameter table remain unchanged. Change index term of power exchange options under the triple risk of exchange rate, volatility risk, counterparty default risk, and jump risk. Specifically, the exponential coefficients are set to beα1=α2=1;1.5;2;2.5. Then, observe the relative position of the four curves.Figures1 and 2, respectively, represent the price changes of power exchange options with different index terms related to expiration time T in two types of dual-currency power exchange options. From the trend, the option price increases with the expiration time T. However, due to the difference in the magnification factor of the index term, when the option α becomes larger, it is more sensitive to the expiration time T, which is what we understand the function of leverage in economic terms. In a benign financial environment, index power changes the structure of returns and effectively hedges risks. However, in a vicious financial environment, the higher the leverage, the more losses will be multiplied. If not timely controlled, the single high-leverage behavior can even cause irreconcilable harm to the entire financial market.By comparing Figures1 and 2, no matter which index value is selected, we find that, at time 0, the price of the power exchange option of the first type of dual currency is higher than the price of the power exchange option of the second type of dual currency. However, the second kind of the dual-currency power exchange option is more sensitive than the first kind of the dual-currency power exchange option.Lines in Figures3–6, black, green, red, and blue lines, represent the exchange rate, the risk of default, jumping, three kinds of standard power exchange option, no jump risk, power exchange option, without the risk of default power exchange option, and exchange rate risk of power exchange option of option value with the changing trend of related parameters.Figure 3
The relationship between the expiration time and the price of the power exchange option of the first kind in dual currency.Figure 4
The relationship between the expiration time and the price of the power exchange option of the second kind in dual currency.Figure 5
The relationship between jump strength and power exchange option price of the first kind in dual currency.Figure 6
The relationship between jump strength and power exchange option price of the second kind in dual currency.Figures3 and 4, respectively, represent the changing relationship between the value and expiration time of power swap options in two types of dual currencies. The overall trend is that, regardless of which of the three risks is removed, the option price increases with the expiration time.In Figure3, in the event of a market impact, compared with no jump risk, green line, black line, and red line, blue line all had greater growth, which is similar to our conjecture. As an example, in March 2018, when the trade war between China and the United States was launched, the market sentiment was depressed, and the global stock market fell, and the Shanghai stock index immediately fell below 3200. Many asset prices were suddenly impacted. Adding the same situation to our research when some unforeseen major events occur, it will not only cause the price of the underlying assets to soar or plummet but also take huge fluctuation to the value of the company, even cause frequent fluctuation of exchange rate. By the combined influence of these risk factors, the option price is bound to be higher than the option price without jump risk. A similar conclusion can be drawn from Figure 4, that is, the occurrence of major unexpected events will lead to the increase of option price.Figures5 and 6 show the impact of jump intensity λ on the option price. In terms of the overall trend, except for the nonjump risk model, the jump intensity all influences the other three models to some extent. However, despite the positive or negative impact of jump intensity on the option price, the impact range is almost zero. According to the differences in Figures 5 and 6, for the first type of dual-currency option model, the jump intensity has a negative impact on the option price due to the change of exchange rate; for the second kind of dual-currency option model, the price of the power exchange option increases slightly with the increase of jump intensity.To sum up, through numerical examples, we make some simple verification of the pricing formula and get the following conclusions:(1)
With the change of index term of power exchange option, the price of the power exchange option can more sensitively follow the change of relevant factors.(2)
Exchange rate risk, counterparty default risk, and jump risk all increase the price of power swap options, but their effects are different. The biggest impact is the jump risk, which is the impact of major events, which makes the price curve of power exchange options rise almost in parallel, which is consistent with the impact of major international relation events on cross-border transactions we analyzed.(3)
When other factors remain unchanged, the jump intensity makes the power exchange option price fluctuate up and down, but the fluctuation range is limited.



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*Source: 4268196-2020-03-17.xml*