# Option Pricing for Path-Dependent Options with Assets Exposed to Multiple Defaults Risk

Authors: Taoshun He
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In the present paper, we derive analytical formulas for barrier and lookback options with underlying assets exposed to multiple defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price drop to zero and the exogenous counterparty default risk induces a drop in the asset price, but the asset can still be traded after this default time. An original technique is developed to valuate the barrier and lookback options by first conditioning on the predefault and the afterdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with the default-free option model and exogenous counterparty default risk option model.


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## Body

## 1. Introduction

Barrier and lookback options are among the most popular path-dependent derivatives traded in exchanges and over-the-counter markets worldwide. The barrier option is a financial derivative contract that is activated or deactivated when the price of the underlying asset crosses a certain level (called barrier) from above or below. And the standard floating lookback call (put) option confers the holder the right to buy (sell) an asset at its lowest (highest) price during the life of the contract. For a complete description of these and other related contracts, refer to Hull [1]. There has been extensive research in option pricing for path-dependent option. For example, Merton [2] and Goldman et al. [3] derive closed-form solutions for barrier and lookback options in the standard Black-Scholes model. Davydov and Linetsky [4] derive analytical formulas for the prices of path-dependent options, such as barrier and lookback options, with the asset price process under Constant Elasticity of Variance (CEV) diffusion model. Kou et al. [5] present analytical solutions for two-dimensional Laplace transforms of barrier option prices under a double exponential jump diffusion model.Multiple defaults risks include exogenous counterparty default risk and endogenous default risk, where the exogenous counterparty default risk may not be unique. However, for simplicity, we assume that the exogenous counterparty default risk is sole in the following parts. In the financial market, an exogenous counterparty default usually has important influences in various contexts. In terms of credit spreads, one observes, in general, a positive jump of the default intensity which is called the contagious jump (see [6]). According to asset values for a firm, the default of a counterparty will in general induce a drop of its value process (refer to [7]). Jiao et al. analyze the impact of the single exogenous counterparty risk and the multiple exogenous counterparty risk on the optimal investment problem, we can refer to [8, 9] for more detail. In this paper, we study the impact of the multiple defaults risk on option pricing problem. In particular, we focus on the pricing of path-dependent option with the underlying asset subject to multiple defaults risk such that the instantaneous loss of the asset at the exogenous counterparty default time and the asset price instantaneously become zero at endogenous default time.Ma et al., in [10], obtain that the explicit valuation of European options with the asset exposed to exogenous counterparty default risk. Yan derives analytical formulas for lookback and barrier options on underlying assets that are subject to an exogenous counterparty risk in [11]. The explicit pricing formulas for European option with asset exposed to multiple defaults risk is given by He [12]. However, to the best of our knowledge, the derivation of the analytic formula for pricing barrier and lookback options under the multiple defaults risk model has not been performed in the previous literature. The main difficultly lies in that the distribution of the first passage time is difficult to derive owing to the multiple defaults and the continuous trading of the underlying asset after the exogenous counterparty default time. We use the conditional density approach of default, which is particularly suitable to study what goes on after the default and was adopted by Jiao and Pham [8] for the optimal investment problem, to derive the explicit distribution of the first passage time and then obtain the analytic formulas for valuation of the barrier and lookback options. We also compare the pricing results of the multiple defaults risk model with Merton’s [2] default-free option model and Yan’s [11] exogenous counterparty default risk option model.The organization of the rest of this paper is as follows. First, a financial model is introduced in Section2. Next, an analytical formula for barrier option on underlying asset that can be exposed to multiple defaults risk is derived in Section 3. Then, in Section 4 we derive the formula for pricing lookback options under this model. Finally, we conclude the paper in final Section.


## 2. Financial Model

In this section, we consider a financial market model with a risk asset (stock) subject to multiple defaults risks. We denote the stock byStt∈0,T, the dynamic of the stock is affected by not only the possibility of the exogenous counterparty default but also the possibility of the endogenous default. However, this stock still exists and can be traded after the exogenous counterparty default.AssumeΩ,G,ℙ is a complete probability space satisfying the usual conditions. Let Wtt∈0,T be a Brownian motion with horizon T<∞ on the probability space Ω,G,ℙ and denote by F=ℱtt∈0,T the natural filtration of W. Let τ1 and τ2 be both, almost surely, nonnegative random variables on Ω,G,ℙ, representing the stock of the exogenous counterparty default time and the endogenous default time, respectively. Then, ℋt1t∈0,T is defined by ℋt1≔σHu1:u≤t, where Ht1≔1τ1≤t, which equals 0 if τ1>t and 1 otherwise. Similar, ℋt2t∈0,T is defined by ℋt2≔σHu2:u≤t, where Ht2≔1τ2≤t. Denote ℍ1=ℋt1t∈0,T and ℍ2=ℋt2t∈0,T, then the progressively enlarged filtration G=Gtt∈0,T=F∨ℍ1∨ℍ2, representing the structure of information available for the investors over 0,T. The market model is given by the following stochastic differential equation (SDE):(1)dStSt−=μtdt+σtdWt−γtdHt1−dHt2,0≤t≤T,where μt, σt, and γt are G-predictable processes. μt and σt are the drift rate and volatility rate of the stock S, respectively, and γt is the (percentage) loss on the stock price induced by the defaults of the counterparty. At default time τ1, the stock price S is reduced by a percentage of γt. However, the stock price S falls to zero at default time τ2.Let us define the following (mutually exclusive and exhaustive) events ordering the default times:(2)A=T<τ1≤τ2,B=T<τ2≤τ1,C=τ2≤τ1≤T,D=τ2≤T<τ1,E=τ1≤T<τ2,F=τ1≤τ2≤T.Then, according to Pham [13], the dynamic of stock price process (1) can be decomposed to the following four situations under physical measure:Situation 1: if the stock is in absence of any default in the life of the option, i.e., the default times satisfyA∪B, then we have(3)dStF=StFμtFdt+σtFdWt,0≤t≤T.Situation 2: if the default times satisfyC∪D, then we have(4)dStF=StFμtFdt+σtFdWt,0≤t<τ2,Std2τ2=0,τ2≤t≤T.Situation 3: if the stock has only exogenous counterparty default in the life of the option, i.e., the default times satisfyE, then we obtain(5)dStF=StFμtFdt+σtFdWt,0≤t<τ1,dStd1τ1=Std1τ1μtd1τ1dt+σtd1τ1dWt,τ1<t≤T,Sτ1d1τ1=Sτ1−F1−γtF.Situation 4: if the stock has both endogenous default and exogenous counterparty default in the life of the option and the exogenous default time is early than the endogenous default time, i.e., the default times satisfyF, then we obtain(6)dStF=StFμtFdt+σtFdWt,0≤t<τ1,dStd1τ1=Std1τ1μtd1τ1dt+σtd1τ1dWt,τ1<t<τ2,Std1,2τ1,τ2=0,τ2≤t≤T,Sτ1d1τ1=Sτ1−F1−γtF,where μtF,σtF,γtF,andStF are F-adapted process and μtd1τ1,σtd1τ1,andStd1τ1 are ℱt⊗ℬ0,t-measurable functions for all t∈0,T. When the counterparty default, the drift, and volatility coefficient μ,σ of the stock price switch from μtF,σtF to μtd1τ1,σtd1τ1, the after default coefficients may depend on the default time τ1. However, when the stock itself default, the drift and diffusion coefficient μ,σ of the stock price switch from μtF,σtF to 0,0 due to the stock price identically vanishing. Here, for simplicity we assume that (7)μtF=μ1,σtF=σ1,μtdτ1=μ2,σtd1τ1=σ2,γtF=γ,with μ1,σ1,μ2,andσ2 are nonnegative constants and the distribution of γγ<1 fixed. Moreover γ,τ1,τ2, and Wt are independent and τ1andτ2 are all the exponential variables with parameter λ1andλ2, respectively. For more details refer to Jiao and Pham [8].Assume thatr is a risk-free interest rate and denote n=Eγ. Let us define the G-adapted process:(8)θt=μt−r−λ1n1t≤τ1−λ21t≤τ2σt,0≤t≤T.By assumingE∫0T1/2θt2dt<∞, we define a probability measure ℚ which is equivalent to ℙ on Ω,G with Radon–Nikodym density:(9)dℚdℙ=ZTG=exp−∫0TθtdWt−12∫0Tθt2dt,under which, by Girsanov’s theorem, Wt˜=Wt+∫0tθudu is a ℚ,G-Brownian motion. And thus we can rewrite (1) as follows:(10)dSt=St−r+λ1n1t≤τ1+λ21t≤τ2dt+σtdW˜t−γtdHt1−dHt2,0≤t≤T,that is, by changing measure, the four situations to decompose of the stock price St under the physical measure ℙ can be transformed into the corresponding following four forms under the equivalent martingale measure ℚ:Situation I: if the stock is in absence of any default in the life of the option, i.e., the default times satisfyA∪B, then we have(11)dStF=StFr+λ1n+λ2dt+σ1dW˜t,0≤t≤T.Situation II: if the default times satisfyC∪D, then we obtain(12)dStF=StFr+λ1n+λ2dt+σ1dW˜t,0≤t<τ2,Std2τ2=0,τ2≤t≤T.Situation III: if the stock has only exogenous counterparty default in the life of the option, i.e., the default times satisfyE, then we have(13)dStF=StFr+λ1n+λ2dt+σ1dW˜t,0≤t<τ1,dStd1τ1=Std1τ1r+λ2dt+σ2dW˜t,τ1<t≤T,Sτ1d1τ1=Sτ1−F1−γ.Situation IV: if the stock has both endogenous default and exogenous counterparty default in the life of the option and the exogenous default time is early than the endogenous default time, i.e., the default times satisfyF, then we obtain(14)dStF=StFr+λ1n+λ2dt+σ1dW˜t,0≤t<τ1,dStd1τ1=Std1τ1r+λ2dt+σ2dW˜t,τ1<t<τ2,Std1,2τ1,τ2=0,τ2≤t≤T,Sτ1d1τ1=Sτ1−F1−γ.In practice, we may assumeγ is a discrete random variable to simplify the computation; in what follows, we further assume that γ takes value γi with probability pi for i=1,2,3, where 0<γ1<1 (loss), γ2=0 (no change), and γ3<0 (gain).


## 3. Analytic Formula for Pricing Barrier Options

In this section we derive an analytic formula for pricing barrier options under the model (1). The barrier options include up-and-out, up-and-in, down-and-out, and down-and-in puts and calls. Since the approaches for deriving the formulas for pricing these kinds of barrier options are similar, we only study the up-and-out barrier call in this section.Consider an up-and-out barrier call option expiring at timeT, with strike price K and barrier level B. We assume that K<B and denote the maximum of the stock price up time to T by(15)YT=max0≤t≤TSt.Then, the option knocks out (i.e., payoff equals to 0) if and only ifYT>B, on the other hand, the option pay off is ST−K+ when YT≤B. In other words, the payoff of the option is(16)VTuo=ST−K+1YT≤B=ST−K1ST≥K,YT≤B.Thus, the risk-neutral price of the up-and-out barrier call option at initial time is(17)V0uo=E˜e−rTST−K1ST≥K,YT≤B.According to compute (17), we obtain the risk-neutral price of the up-and-out barrier call option at time 0 under multiple defaults risks model as follows.Theorem 1.
If0<S0≤B, then the risk-neutral price of an up-and-out barrier call option at time 0 under model (1) is given by(18)V0uo=e−λ1+λ2TV˜0uo+e−λ2T∑i=13pi∫0Tλ1e−λ1+rt∫−∞1/σ12lnB/S0∫w+1/σ12lnB/S0φγi,w⋅eλ2TΦδ^2+φγi,wK−Φδ^2+φγi,wB−e−rT−tKΦδ^2−φγi,wK−Φδ^2−φγi,wB−eλ2TBφγi,wB−2r+λ2/σ22Φδ^2+B2Kφγi,w−Φδ^2+Bφγi,w+e−rT−tKφγi,wB1−2r+λ2/σ22Φδ^1−B2Kφγi,w−Φδ^1−Bφγi,w⋅22m−wt2πteα1w−1/2α12t−1/2t2m−w2dmdwdt,where Φ is the standard normal distribution function and(19)V˜0uo=S0eλ1n+λ2TΦδ^1+S0K−Φδ^1+S0B−e−rTKΦδ^1−S0K−Φδ^1−S0B−eλ1n+λ2TBS0B−2r+λ1n+λ2/σ12Φδ^1+B2KS0−Φδ^1+BS0+e−rTKS0B1−2r+λ1n+λ2/σ12Φδ^1−B2KS0−Φδ^1−BS0,with(20)δ^1±s=1σ1Tlns+r+λ1n+λ2±σ122T,δ^2±s=1σ2T−tlns+r+λ2±σ222T−t,α1=1σ1r+λ1n+λ2−σ122,φγi,w=S01−γieσ1w.Proof.
See Appendix A.Remark 1.
(1)
Ifλ2=0, i.e., there is no exist endogenous default risk in model (1), then the risk-neutral price of the up-and-out barrier call option at time 0 under this model becomes
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Ifλ1=0 and λ2=0, i.e., there is no any default in model (1), then the risk-neutral price of up-and-out barrier call option at time 0 under this model becomes(24)V0uo=S0Φδ^1+S0K−Φδ^1+S0B−e−rTKΦδ^1−S0K−Φδ^1−S0B−BS0B−2r/σ12Φδ^1+B2KS0−Φδ^1+BS0+e−rTKS0B1−2r/σ12Φδ^1−B2KS0−Φδ^1−BS0,withδ^1±s=1/σ1Tlns+r±σ12/2T. It is obvious that (24) is standard Black–Scholes formula for up-and-out call option.


## 4. Analytic Formula for Pricing Lookback Options

In this section, we price a floating strike lookback option, whose payoff is the difference between the maximum asset price over the life time and the asset price at expiration.By formula (15), the risk-neutral price of the lookback option at initial time can be written as follows:(25)V0=E˜e−rTYT−ST=E˜e−rTYT−E˜e−rTST.According to the calculation of (25), we obtain the following theorem.Theorem 2.
The risk-neutral price of the lookback option at initial time under model (1) is given by(26)V0=S0e−r+λ2TV˜0+S0eλ1n−λ1T1+σ122r+λ1n+λ2Φα1+σ1T+S0e−r+λ1+λ2T1−σ122r+λ1n+λ21−Φα1T+S0e−rTV^0+λ2S0e−rT∫0Ter+λ1n−λ1u1+σ122r+λ1n+λ2Φα1+σ1u+e−λ1+λ2u1−σ122r+λ1n+λ21−Φα1udu−S0,where(27)V˜0=∑i=13pi∫0tTλ1e−λ1t∫0+∞∫w+∞eσ1mΦδ˜1++1−γi1+σ222r+λ2⋅eσ1w+rT−tΦδ˜2−σ222r+λ21−γi1−2r+λ2/σ22eσ1w+2σ1r+λ2m−w/σ22⋅Φδ˜1−22m−wt2πteα1w−1/2α12t−1/2t2m−w2dmdwdt,V^0=∫0Tλ2e−λ2u∫0uλ1e−λ1t∫0+∞∫w+∞eσ1mΦδ˜1++1−γi1+σ222r+λ2eσ1w+rT−tΦδ˜2−σ222r+λ21−γi1−2r+λ2/σ22eσ1w+2σ1r+λ2m−w/σ22Φδ˜1−⋅22m−wt2πteα1w−1/2α12t−1/2t2m−w2dmdwdt,with(28)α1=1σ1r+λ1n+λ2−σ122,β2=r+λ2−σ222,δ˜1±=±σ1m−w−ln1−γi−β2T−tσ2T−t,δ˜2=−σ1m−w+ln1−γi+β2+σ22T−tσ2T−t.Proof.
See Appendix B.Remark 2.
(1)
Ifλ2=0, i.e., there is no exist endogenous default risk in model (1), then the risk-neutral price of the lookback option at time 0 under this model becomes
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It is obvious that the value of lookback option at time 0 under model (1) is the same as the one at time 0 with the stock exposed to counterparty risk (see Yan [11]).
(2)
Ifλ1=0 and λ2=0, i.e., there is no any default in model (1), then the risk-neutral price of lookback option at time 0 under this model becomes
(32)
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, with α1=1/σ1r−σ12/2. It is obvious that (32) is standard Black–Scholes formula for lookback option.


## 5. Conclusions

The explicit analytical formulas for European call and put options with asset exposed to multiple defaults risks have been derived. However, it is still very challenging to obtain the explicit analytical formulas for path-dependent options under this model. This is because the multiple default risks cause the difficultly in deriving the density of the first passage time for the maximum asset price. In this paper, the conditional density approach, which is developed by Jiao and Pham [8] for optimal investment, is utilized to overcome the difficulty and derive the formulas for lookback and barrier options when the underlying asset is subject to multiple defaults risks. Future research lies in deriving analytic formulas for the path-dependent options with two underlying assets exposed to loop contagion risks.



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