# Pricing Formula for Exotic Options with Assets Exposed to Counterparty Risk

Authors: Li Yan
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper gives analytical formulas for lookback and barrier options on underlying assets that are exposed to a counterparty risk. The counterparty risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to valuate the lookback and barrier options by first conditioning on the predefault and the postdefault time and then obtain the unconditional analytic formulas for their prices.


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## Body

## 1. Introduction

Lookback and barrier options are among the most popular path-dependent options traded in exchanges and over-the-counter markets. A standard floating lookback call (put) option gives the holder the right to buy (sell) an asset (e.g., stock, index, exchange rate, and interest rate) at its lowest (highest) price during the life of the contract. In other words, the payoffs of a floating lookback call/put are, respectively,ST-min0≤t≤TSt and max0≤t≤TSt-ST. A barrier option is a financial derivatives contract that is activated (“in”) or deactivated (“out”) when the price of the underlying asset crosses a certain level (i.e., a barrier) from above (“down”) or below (“up”). Lookback and barrier options have been useful in the “real options" literature. For example, Longstaff [1] approximates the marketability values of a security as a continuous-time lookback option. Barrier options are inherent in the reduced-form approach to credit risk, where the default time of a company is modeled as the first hitting time, or barrier breaching time, of the asset process of the company; see Merton [2], Black and Cox [3], Leland and Toft [4], and Chen and Kou [5]. For a review of the literature of both continuous and discrete lookback options and barrier options, please refer to Kou [6] and the references therein.In the financial market, a 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, e.g., Jarrow and Yu [7]). In terms of asset (or stock) values for a firm, the default of a counterparty will in general induce a drop of its value process (see, e.g., El Karoui et al. [8]). Jiao and Pham [9] analyzed the impact of this counterparty risk on the optimal investment problem. In this paper, we study the impact of the counterparty risk on option pricing problems. In particular, we focus on the pricing of lookback and barrier options when the underlying asset is subject to the default of a counterparty and the instantaneous loss of the asset at the default time.The explicit valuation of vanilla European options with this counterparty default risk was partly given by Ma et al. [10]. However, the derivation of the analytic formula for pricing lookback and barrier options with this default risk model has not been done in the previous literature. The main difficulty lies in that the distribution of the first passage time is difficult to derive due to the default and the continuous trading of the underlying asset after the default time. We consider 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 [9] for the optimal investment problem. We derive the explicit distribution of the first passage time and then obtain the analytic formulas for valuation of the lookback and barrier options.The outline of the paper is organized as follows. In Section2, we introduce the financial models. In Section 3, we derive the distribution of the first passage time and then the formula for pricing lookback options. In Section 4, we derive the formula for pricing barrier options. Conclusions are given in the final section.


## 2. Financial Models

In this section, we consider a financial market model with a risky asset subject to counterparty risk: the dynamics of the risky asset is affected by the possibility of the counterparty defaulting. However, this stock still exists and can be traded after the default.LetW=(Wt)t∈[0,T] be a Brownian motion with horizon T<∞ on the probability space (Ω,G,P) and denote by F=(Ft)t∈[0,T] the natural filtration of W. Let τ, an almost surely finite nonnegative random variable on (Ω,G,P), represent the default time. Before τ, the filtration F represents the information accessible to the investors. After τ, the investors add this new information τ to the reference filtration F. Introduce Dt=1{τ≤t},0≤t≤T, and let D=(Dt)t∈[0,T] be the filtration generated by this jump process, and G=(Gt)t∈[0,T] the progressively enlarged filtration F∨D, representing the structure of information available for the investors over [0,T].The stock price process is governed by the following dynamic:(1)dSt=St-μtdt+σtdWt-γtdDt,0≤t≤T,where μ, σ, and γ are G-predictable processes. From [11], any G-predictable process φ can be written in the form:(2)φt=φtF1t<τ+φtdτ1t≥τ,0≤t≤T,with φtF  F-adapted, and φtd(θ) is measurable with respect to Ft⊗B(R+), for all t∈[0,T]. Therefore, the dynamic (1) can be rewritten as(3)dStF=StFμtFdt+σtFdWt,0≤t<τ,(4)dStdτ=Stdτμtdτdt+σtdτdWt,τ<t≤T,(5)Sτdτ=Sτ-F1-γτF,where μtF, σtF, γF, StF are F-adapted processes, and μtd(τ), σtd(τ), Std(τ) are Ft⊗B([0,t))-measurable functions for all t∈[0,T]. The process γ represents the (proportional) loss on the stock price induced by the default of the counterparty; by misuse of notation, we shall thus identify γ in (1) with the F-adapted process γF in (5). When the counterparty defaults, the drift, and diffusion coefficients (μ,σ) of the stock price switch from (μF,σF) to (μd(τ),σd(τ)), the postdefault coefficients may depend on the default time τ. Here for simplicity we assume that(6)μtF=μ1,σtF=σ1,μtdτ=μ2,σtdτ=σ2,γτF=γ,where μ1,σ1 are nonnegative constants, μtd(θ), σtd(θ) are only deterministic functions of θ∈R+, in [9], for example μtd(θ)=μF(θ/T),σtd(θ)=σF(2-θ/T), θ∈[0,T], which have the following economic interpretation. The ratio between the pre- and postdefault rate of return is smaller than one and increases linearly with the default time: the postdefault rate of return drops to zero, when the default time occurs near the initial date and converges to the predefault rate of return, when the default time occurs near the finite investment horizon. We have a similar interpretation for the volatility whose ratio is larger than one, which decreases linearly with the default time, converging to the double (resp., initial) value of the predefault volatility, when the default time goes to the initial (resp., terminal horizon) time. The distribution of γ(γ≤1) is fixed. Moreover γ, τ, and Wt are independent and τ is an exponential variable with parameter λ. For more general set-ups, see Jiao and Pham [9].Assume thatr is a risk-free interest rate. Let us define the G-adapted process(7)βt=μt-r-nλIt≤τσt,0≤t≤T.By assuming Eexp⁡∫0T(1/2)βt2dt<∞, we define a probability measure P~ which is equivalent to P on (Ω,G) with Radon-Nikodym density:(8)dP~dP=ZTG=exp⁡-∫0TβtdWt-12∫0Tβt2dt,under which, by Girsanov’s theorem, W~t=Wt+∫0tβudu is a (P~,G)-Brownian motion. Therefore we can rewrite (1) as follows:(9)dSt=St-r+nλIt≤τdt+σtdW~t-γtdDt,0≤t≤T,That is, by changing measure, the dynamics (3)–(5) for asset price St under the physical measure P can be transformed into the following form under the equivalent martingale measure P~:(10)dStF=StFr+λndt+σ1dW~t,0≤t<τ,dStdτ=Stdτrdt+σ2dW~t,τ<t≤T,Sτdτ=Sτ-F1-γ,where n=E(γ). In practice, we may assume that γ is a discrete random variable to simplify the computation; for example, we may assume that γ takes value γi with probability pi for i=1,2, where 0<γ1≤1 (loss) and γ2=0 (no change).


## 3. Analytic Formula for Pricing Lookback Options

In this section, we derive an analytical formula for pricing 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.Using Ito’s lemma, the solution of the SDEs (10) can be written as(11)StF=S0eσ1W^tF,0≤t<τ,Stdτ=Sτdτeσ2W^td-W^τd,τ<t≤T,Sτdτ=Sτ-F1-γ,where(12)W^tF≡α1t+W~t,W^td≡α2t+W~t,α1≡1σ1r+λn-σ122,α2≡1σ2r-σ222.Denote the maximum of the asset price up to time T by(13)YT=max0≤t≤T⁡St.Then at time 0, the risk-neutral price of the lookback option is given by(14)V0=Ee-rTYT-ST=Ee-rTYT-Ee-rTST.Theorem 1.
The risk-neutral price of the lookback option at time 0 under models (3)–(5) is given by(15)V0=S0eλn-λT1+σ122r+λnΦα1+σ1T+S0e-r+λT1-σ122r+λn1-Φα1T+S0e-rT∑i=12pi∫0Tλe-λt∫0∞∫w∞eσ1mΦd~1++1-γi1+σ222reσ1w+rT-tΦd~2-σ222r1-γiσ22-2r/σ22eσ1w+2rσ1/σ22m-wΦd~1-22m-wt2πteα1w-1/2α12t-1/2t2m-w2dmdwdt-S0,where Φ is the distribution function of the standard normal random variable, and(16)β1=r+λn-σ122,β2=r-σ222,d~1±=±σ1m-w-ln⁡1-γi-β2T-tσ2T-t,d~2=-σ1m-w+ln⁡1-γi+β2+σ22T-tσ2T-t.Proof.
The risk-neutral price of the lookback option (14) can be rewritten as (17)V0=Ee-rTYT1τ≥T+Ee-rTYT1τ<T-Ee-rTST.The distribution function of the random variable ST is given by [10](18)FS=P~ST≤S=e-λTΦln⁡S/S0-aTbT+∑i=12pi∫0Tλe-λtΦ1btln⁡SS01-γi-atdtwith (19)at=r+λn-σ122t+r-12σ22T-t,bt=σ12t+σ22T-t.Using the distribution function F(S) and the following identity (20)∫0∞SdΦ1Bln⁡SC-A=CeA+B2/2we can compute(21)Ee-rTST=S0e-rTer+λn-λT+∑i=12pi∫0Tλ1-γieλn-λt+rTdt=S0.Next we aim to calculate(22)Ee-rTYT=Ee-rTYT1τ≥T+Ee-rTYT1τ<T.Define (23)M^TF≡max0≤t≤T⁡W^tF,YTF≡max0≤t≤T⁡StF.Then we have (24)YTF=max0≤t≤T⁡S0eσ1W^tF=S0eσ1M^TF. Hence the first term of the right-hand side of (22) can be calculated as(25)Ee-rTYT1τ≥T=e-rT∫0∞λe-λtEYT1τ≥T∣τ=tdt=e-rT∫T∞λe-λtEYTF∣τ=tdt=e-r+λTEYTF.According to [12], the density under P~ for the random variable M^TF is given by(26)f~M^TFm=22πTe-1/2Tm-α1T2-2α1e2α1mΦ-m-α1TT,m≥0.So we have (27)EYTF=∫0∞S0eσ1m22πTe-1/2Tm-α1T2-2α1e2α1mΦ-m-α1TTdm=2S0er+λnT∫0∞12πTe-m-α1+σ1T2/2Tdm-S01-σ122r+λn∫0∞Φ-m-α1TTdeσ1+2α1m=2S0er+λnTΦα1+σ1T-S01-σ122r+λn-Φ-α1T+er+λnTΦα1+σ1T=S0er+λnT1+σ122r+λnΦα1+σ1T+S01-σ122r+λn1-Φα1T. Hence the value of the expectation in (25) can be calculated as(28)Ee-rTYT1τ≥T=S0eλn-λT1+σ122r+λnΦα1+σ1T+S0e-r+λT1-σ122r+λn1-Φα1T.Denote (29)M^T-td≡max0≤t≤T-t⁡W^td,Ytd≡maxt≤s≤T⁡Ssdt.Then(30)Ytd=maxt≤s≤T⁡Stdteσ2W^sd-W^td,(31)YtdStdt=emaxt≤s≤T⁡σ2W^sd-W^td,which is independent of Ft. Then we calculate the second term of the right-hand side of (22) as(32)Ee-rTYT1τ<T=e-rTEmax⁡YτF,Yτd1τ<T(33)=e-rT∫0Tλe-λtEmax⁡YτF,Yτd∣τ=tdt=e-rT∫0Tλe-λtE∑i=12piStdtEmax⁡YtFStdt,YtdStdt∣Ft,γdt.
We shall use the independence lemma (see [12]) to calculate the above expectations. To this end, we define (34)Gt,x=Emax⁡x,YtdStdt∣γ=γi,RT-t,m=P~M^T-td≤m. According to [12], we have (35)P~M^T-td≤m=Φm-α2T-tT-t-e2α2mΦ-m-α2T-tT-t,m≥0. Then(36)Gt,x=Emax⁡x,YtdStdt∣γ=γi=Eexp⁡max⁡ln⁡x,maxt≤s≤T⁡σ2W^sd-W^td∣γ=γi=Eexp⁡maxln⁡x,σ2max0≤s≤T-t⁡W^sd∣γ=γi=Eexp⁡max⁡ln⁡x,σ2M^T-td∣γ=γi=Ex1ln⁡x>σ2M^T-td∣γ=γi+Eeσ2M^T-td1ln⁡x≤σ2M^T-td∣γ=γi=xRT-t,1σ2ln⁡x+∫ln⁡x/σ2∞eσ2mRmT-t,mdm=xΦd1+-x2β2/σ22Φd1-+1+σ222rerT-tΦd2+β2rx2r/σ22Φd1-=xΦd1++1+σ222rerT-tΦd2-σ222rx2r/σ22Φd1-,where (37)β2≡r-σ222,d1±≡±ln⁡x-β2T-tσ2T-t,d2≡-ln⁡x+β2+σ22T-tσ2T-t.By the independence lemma (see [12]), we have(38)Emax⁡YtFStdt,YtdStdt∣Ft,γ=Gt,YtFStdt.According to [12], the joint density function under P~ of (W^tF,M^tF) involved in YtF and Std(t) (see (10)) is(39)f~W^tF,M^tFm,w=22m-wt2πteα1w-1/2α12t-1/2t2m-w2,w≤m,m≥0.Substituting (38) and (36) into (32) and using (39), we derive that(40)Ee-rTYT1τ<T=e-rT∫0Tλe-λt·∑i=12piEYtFStdt2r/σ22Φd1-YtFΦd1++Stdt1+σ222rerT-tΦd2-Stdtσ222rYtFStdt2r/σ22Φd1-dt=S0e-rT∑i=12pi∫0Tλe-λt∫0∞∫w∞eσ1mΦd~1+1+σ222r+1-γi1+σ222reσ1w+rT-tΦd~21+σ222r-σ222r1-γiσ22-2r/σ22eσ1w+2rσ1/σ22m-wΦd~1-·22m-wt2πteα1w-1/2α12t-1/2t2m-w2dmdwdt,where d~1± and d~2 are defined in Theorem 1. Combining (28), (40), and (22) gives the value of E[e-rTYT], and then using (21) completes the proof of this theorem.


## 4. Analytic Formula for Pricing Barrier Options

In this section we derive an analytic formula for pricing barrier options under the counterparty risk models (10). 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 with expiry dateT, strike price K, and barrier level B and we assume that K<B. The solution to the stochastic differential equation for the asset price is given by (11), and the maximum of the asset price YT is defined by (13). The option knocks out (i.e., pays off 0) if and only if YT>B. If YT≤B, then the option pays off (ST-K)+. In order words, the payoff of the option is(41)VTuo=ST-K+1YT≤B=ST-K1ST≥K,YT≤B.At time 0, the risk-neutral price of the up-and-out barrier call option is(42)V0uo=Ee-rTST-K1ST≥K,YT≤B.Theorem 2.
The risk-neutral price of an up-and-out barrier call option at time 0 under models (3)–(5) is given by (43)V0uo=e-λTV~0uo+∑i=12pi·∫0Tλe-λ+rt∫01/σ1ln⁡B/S0∫w1/σ1ln⁡B/S0Φd^2-φγi,wK-Φd^2-φγi,wBφγi,w·Φd^2+φγi,wK-Φd^2+φγi,wB-e-rT-tKΦd^2-φγi,wK-Φd^2-φγi,wB-Bφγi,wB-2r/σ22Φd^2+B2Kφγi,w-Φd^2+Bφγi,w+e-rT-tKφγi,wB1-2r/σ22·Φd^2-φγi,wK-Φd^2-φγi,wB·22m-wt2πteα1w-1/2α12t-1/2t2m-w2dmdwdt,where (44)V~0uo=S0eλnTΦd^1+S0K-Φd^1+S0B-e-rTKΦd^1-S0K-Φd^1-S0B-BS0B-2r+λn/σ12Φd^1+B2KS0-Φd^1+BS0+e-rTKS0B-2r+λn/σ12+1Φd^1-S0K-Φd^1-S0B,with (45)d^1±s=1σ1Tln⁡s+r+λn±12σ12T,d^2±s=1σ2T-tln⁡s+r±12σ22T-t,φγi,w=S01-γieσ1w.Proof.
We rewrite (42) as follows:(46)V0uo=Ee-rTST-K1ST≥K,YT≤B1τ>T+Ee-rTST-K1ST≥K,YT≤B1τ≤T.The first term on the right-hand side of (46) is calculated as(47)Ee-rTST-K1ST≥K,YT≤B1τ>T=e-λTEe-rTSTF-K1STF≥K,YTF≤B≕e-λTV~0uo.Notice that V~0uo corresponds to the case when there is no default of the counterparty. Then we can calculate it using a similar technique as in [12] and obtain that (48)V~0uo=S0eλnTΦd^1+S0K-Φd^1+S0B-e-rTKΦd^1-S0K-Φd^1-S0B-BS0B-2r+λn/σ12Φd^1+B2KS0-Φd^1+BS0+e-rTKS0B-2r+λn/σ12+1Φd^1-S0K-Φd^1-S0B,where d^1±(s) are defined in Theorem 2.
Then we compute the second term of (46):(49)Ee-rTST-K1ST≥K,YT≤B1τ≤T=∫0Tλe-λtEe-rTST-K1ST≥K,YτF≤B,Yτd≤B∣τ=tdt=∫0Tλe-λtE1YtF≤BEe-rTSTdt-K1STdt≥K,Ytd≤B∣Ftdt.Using the definitions of (11) and (31), we have (50)Stdt=S01-γeσ1W^tF,STdtStdt=eσ2W^Td-W^td,YtdStdt=eσ2maxt≤s≤T⁡W^sd-W^td.It is obvious that STd(t)/Std(t) and Ytd/Std(t) are independent on Ft. Letting x=Std(t), we calculate(51)Ee-rTSTdt-K1STdt≥K,Ytd≤B∣Ft=∑i=12piEe-rTxeσ2W^Td-W^td-K1xeσ2W^Td-W^td≥K,xeσ2maxt≤s≤T⁡W^sd-W^td≤B∣γ=γi=∑i=12piEe-rTxeσ2W^T-td-K1xeσ2W^T-td≥K,xeσ2max0≤s≤T-t⁡W^sd≤B∣γ=γi=∑i=12pie-rtEe-rT-txeσ2W^T-td-K1W^T-td≥k,M^T-td≤b∣γ=γi,where (52)k=1σ2ln⁡Kx,b=1σ2ln⁡Bx.Notice that the expectation(53)Ee-rT-txeσ2W^T-td-K1W^T-td≥k,M^T-td≤b∣γ=γi corresponds to the case when there is no default of the counterparty. Therefore using the techniques in [12], we have(54)Ee-rT-txeσ2W^T-td-K1W^T-td≥k,M^T-td≤b∣γ=γi=xΦd^2+xK-Φd^2+xB-e-rT-tKΦd^2-xK-Φd^2-xB-BxB-2r/σ22Φd^2+B2Kx-Φd^2+Bx+e-rT-tKxB-2r/σ22+1Φd^2-xK-Φd^2-xB,where d^2±(s) are defined in Theorem 2.
Substituting (54) and x=Std(t) into (51) and using (39), we can continue to calculate (49) as(55)Ee-rTST-K1ST≥K,YT≤B1τ≤T=∑i=12pi∫0Tλe-λ+rt∫01/σ1ln⁡B/S0∫w1/σ1ln⁡B/S0Φd^2-φγi,wK-Φd^2-φγi,wBφγi,w·Φd^2+φγi,wK-Φd^2+φγi,wB-e-rT-tKΦd^2-φγi,wK-Φd^2-φγi,wB-Bφγi,wB-2r/σ22Φd^2+B2Kφγi,w-Φd^2+Bφγi,w+e-rT-tKφγi,wB1-2r/σ22·Φd^2-φγi,wK-Φd^2-φγi,wB·22m-wt2πteα1w-1/2α12t-1/2t2m-w2dmdwdt,where φ(γi,w) is defined in Theorem 2. Combining (47) and (55), we obtain the formula for V0uo.


## 5. Conclusions

This paper derives analytic formulas for lookback and barrier options when the underlying asset is subject to counterparty risk. The counterparty default risk induces a drop in the price of the risky asset (stock), and the stock can still be traded after this default time. This kind of counterparty risk causes difficulty in deriving the density of the first passage time for the maximum asset price. The conditional density approach, which is developed by Jiao and Pham [9] for optimal investment, is utilized to overcome the difficulty and derive the formulas for the lookback and barrier options. Future research lies in deriving analytic formulas for the exotic options with two underlying assets exposed to bilateral counterparty risk.



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