# Explicit Pricing Formulas for European Option with Asset Exposed to Double Defaults Risk

Authors: Taoshun He
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We derive analytical formulas for European call and put options on underlying assets that are exposed to double defaults risks which include exogenous counterparty default risk and endogenous default risk. The endogenous default risk leads the asset price to 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. A novel technique is developed to evaluate the European call and put options by first conditioning on the predefault and the postdefault time and then obtaining the unconditional analytic formulas for their price. We also compare the pricing results of our model with default-free option model and counterparty default risk option model.


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## Body

## 1. Introduction

Over the past few decades, academic researchers and market practitioners have developed and adopted different models and techniques for pricing European option. The path-breaking work on option valuation was done by Black and Scholes [1] and Merton [2]; their works assumed that the absence of arbitrage opportunities and the asset price dynamics are governed by a Geometric Brownian Motion (GBM). For the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion, a review of the literature can refer to Elliott et al. [3] 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 Jarrow and Yu [4]). In terms of asset values for a firm, the default of a counterparty will generally induces a drop of its value process (see El Karoui et al. [5]). Jiao and Pham analyzed the impact of the single exogenous counterparty risk and the multiple exogenous counterparty risk on the optimal investment problem; for more detail refer to [6, 7]. In this paper, we study the impact of the double defaults risk, which includes exogenous counterparty default risk and endogenous default risk, on option pricing problem. In particular, we focus on the pricing of European option with the underlying asset being subject to double defaults risk such that the instantaneous loss of the asset at the exogenous counterparty default time and the asset price instantaneous become to zero at endogenous default time.The explicit valuation of European options with assets exposed to exogenous counterparty default risk was partly given by Ma et al. [8]. Yan derived analytical formulas for lookback and barrier options on underlying assets that are subject to an exogenous counterparty risk (see Yan [9]). However, the derivation of the analytic formula for pricing European call and put options under the double defaults risk model has not been done in the previous literature. The main difficulty lies in the derivation of the distribution of the stock price at expire time T under the double 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 [6] for the optimal investment problem, to derive the explicit distribution of the stock price at expire time T and then obtain the analytic formulas for valuation of the European call and put options. We also compare the pricing results of double defaults risk model with Black and Scholes [1] default-free option model and Ma et al. [8] counterparty default risk option model.The rest of this paper is organized as follows. In the next section, we introduce the financial model and change from the actual probability measureP to risk-neutral probability measure P~ in model of financial market. In Section 3, we derive the distribution of the stock price at expire time T and then the formula for pricing European call and put options. Conclusions are given in the final section.


## 2. Model Setting and Change of Measure

In this section, we consider a financial market model with a risk asset (stock) subject to double default risks. We denote the stock by(St)t∈[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.Assume(Ω,G,P) is a complete probability space satisfying the usual conditions. Let (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 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 (Ht1)t∈[0,T] is defined by Ht1≔σ(Hu1:u≤t), where Ht1≔1{τ1≤t} which equals 0 if τ1>t and 1 otherwise, and denote H1=(Ht1)t∈[0,T]. Similarly, (Ht2)t∈[0,T] is defined by Ht2≔σ(Hu2:u≤t), where Ht2≔1{τ2≤t}, and denote H2=(Ht2)t∈[0,T]. Denote by G=(Gt)t∈[0,T] the progressively enlarged filtration G=F∨H1∨H2, representing the structure of information available for the investors over [0,T]. The market model is given by the following SDE (stochastic differential equation):(1)dStSt-=μtdt+σtdWt-γtdHt1-dHt2,0≤t≤T,where μt, σt, and γt are G-predictable processes. μt and σt are 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. Denote τ^1≡min⁡(τ1,τ2) and τ^2≡max⁡(τ1,τ2), according to Pham [10]; any G-predictable process Y=(Yt)t≥0 can be represented as(2)Yt=Yt01t≤τ^1+Yt1,1τ1,γ11τ1<t≤τ2+Yt1,2τ2,γ21τ2<t≤τ1+Yt2τ1,τ2,γ1,γ21t>τ^2,where Yt0 is F-predictable and Yt1,1, Yt1,2 are measurable with respect to Ft⊗B(R+)⊗B(L) and Yt2 is measurable with respect to Ft⊗B(R+2)⊗B(L2) for all t∈[0,T], and L=(-∞,1) represents the possibility of the value of γt. Let us define the following (mutually exclusive and exhaustive) events ordering the default times:(3)A=T<τ1≤τ2,B=T<τ2≤τ1,C=τ2≤τ1≤T,D=τ2≤T<τ1,E=τ1≤T<τ2,F=τ1≤τ2≤T.Therefore, the dynamic of stock price process (1) can be decomposed by the following four situations.Situation i.If the stock is in absence of any default in the life of the option, i.e., the default times satisfy A∪B, then we have(4)dStF=StFμtFdt+σtFdWt,0≤t≤T.Situation ii.If the default times satisfy C∪D, then we have(5)dStF=StFμtFdt+σtFdWt,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 satisfy E, then we obtain(6)dStF=StFμtFdt+σtFdWt,0≤t<τ1,dStd1τ1=Std1τ1μtd1τ1dt+σtd1τ1dWt,τ1<t≤T,Sτ1d1τ1=Sτ1-F1-γtF.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 earlier than the endogenous default time, i.e., the default times satisfy F, then we obtain(7)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,StF are F-adapted process and μtd1(τ1),σtd1(τ1),Std1(τ1) are Ft⊗B([0,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 (μF,σF) to (μtd1(τ1),σtd1(τ1)), the postdefault coefficients may depend on the default time τ1. However, when the stock itself defaults, the drift and diffusion coefficients (μ,σ) of the stock price switch from (μF,σF) to (0,0) due to the stock price identically vanishing. Here for simplicity we assume that(8)μtF=μ1,σtF=σ1,γtF=γ,μtd1τ1=μ2,σtd1τ1=σ2,where μ1,σ1 are nonnegative constants, μtd1(τ1),σtd1(τ1) are only deterministic functions of τ1∈R+, in Jiao and Pham [6], for example, μtd1(τ)=μF·(τ/T),σtd1(τ)=σF·(2-τ/T),τ∈[0,T], which have meaningful economic interpretation. And then we assume that the distribution of γ(γ<1) is fixed. Moreover γ,τ1,τ2, and Wt are independent and τ1,τ2 are all the exponential variables with parameters λ1,λ2, respectively. For more details, refer to Jiao and Pham [6].According to Bielecki and Rutkowski [11] or Elliott et al. [12], we can decompose γHt=Mt+At, where Mt is a martingale and At=∫0t∧τλmds is a bounded variation process and m=E(γ). Thus we obtain that Mt1=γHt1-At1=γHt1-∫0t∧τ1λ1mds=γHt1-∫0tλ1m1{τ1≥s}ds and Mt2=Ht2-∫0tλ21{τ2≥s}ds are all martingale.Assume thatr is a risk-free interest rate; let D(t)=e-rt. Since Ht1 and Ht2 are pure jump process and D(t) is continuous, [D,S](t)=0. We now use Ito’s product rule for jump process and model (1) to obtain (9)DtSt=S0+∫0tDu-dSu+∫0tSu-dDu+D,St=S0+∫0tDu-Su-μ1-rdu+∫0tDu-Su-σ1dWu1-∫0tDu-Su-γdHu1-∫0tDu-Su-dHu2=S0+∫0tDuSuμt-r-λ1m1t≤τ1-λ21t≤τ2du+∫0tDuSuσ1dWu1-∫0tDu-Su-dMu1-∫0tDu-Su-dMu2, which in differential form is(10)dDtSt=Dt-dSt+St-dDt+dDtdSt=Dt-St-μ1-rdt+σ1dWt1-γdHt1-dHt2=DtStμt-r-λ1m1t≤τ1-λ21t≤τ2dt+σ1dWt1-Dt-St-dMt1-Dt-St-dMt2.Since both Mt1 and Mt2 are martingales and D(t-)S(t-) is left continuous, the second term and last term of (10) are all martingales. In order to make discounted stock price be martingale, we would like to rewrite (10) as(11)dDtSt=σ1DtStdWt~1-Dt-St-dMt1-Dt-St-dMt2,where dWt~1=μt-r-λ1m1{t≤τ1}-λ21{t≤τ2}/σ1dt+dWt1≔θtdt+dWt1.Let us define theG-adapted process(12)θt=μt-r-λ1m1t≤τ1-λ21t≤τ2σt,0≤t≤T.By assuming E[∫0T1/2|θt|2dt]<∞, we define a probability measure P~ which is equivalent to P on (Ω,G) with Radon-Nikodym density(13)dP~dP=ZTG=exp⁡-∫0TθtdWt-12∫0Tθt2dt,under which, by Girsanov’s theorem, Wt~=Wt+∫0tθudu is a (P~,G)-Brownian motion. And thus we can rewrite (1) as follows:(14)dSt=St-r+λ1m1t≤τ1+λ21t≤τ2dt+σtdW~t-γtdHt1-dHt2,0≤t≤T.That is, by changing measure, the four situations to decompose the stock price St under the physical measure P can be transformed into the corresponding following four forms under the equivalent martingale measure P~.Situation I.If the stock is in absence of any default in the life of the option, i.e., the default times satisfy A∪B, then we have(15)dStF=StFr+λ1m+λ2dt+σ1dW~t,0≤t≤T.Situation II.If the default times satisfy C∪D, then we obtain(16)dStF=StFr+λ1m+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 satisfy E, then we have(17)dStF=StFr+λ1m+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 earlier than the endogenous default time, i.e., the default times satisfy F, then we obtain(18)dStF=StFr+λ1m+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 assume γ is a discrete random variable to simplify the computation; in what follows, we 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 European Options

Consider a European option, expiring at timeT, with strike price K. In this section, we derive an analytical formula for pricing this option, whose payoff is the difference between the stock price at expiration T and the strike price K. Firstly, we need to compute the distribution function of random variable ST and obtain the following lemma.Lemma 1.
If the dynamic of stock price process follows model (1), then the distribution function of the stock price at expire time T is given by(19)FS=e-λ1+λ2TΦln⁡S/S0-aTbT+1-e-λ2T+e-λ2T∑i=13pi∫0Tλ1e-λ1tΦ1btln⁡SS01-γi-atdt,with a(t)=(r+λ1m+λ2-σ12/2)t+(r+λ2-σ22/2)(T-t), b(t)=σ12t+σ22(T-t), and Φ being the standard normal distribution function.Proof.
LetS≥0; according to the definition of distribution function, we have (20)FS=P~ST≤S=E~1ST≤S=E~1ST≤S1A∪B∪C∪D∪E∪F=E~1ST≤S1A∪B+E~1ST≤S1C∪D+E~1ST≤S1E+E~1ST≤S1F.If the default times satisfy Situation I, then the dynamic of stock price process satisfies the stochastic differential equation (15). By Ito’s lemma and (15), we have (21)STF=S0er+mλ1+λ2-σ12/2T+σ1W~T, and thus(22)E~1ST≤S1A∪B=E~E~1ST≤S1τ1>T,τ2>T∣τ1,τ2=∫T+∞λ2e-λ2u∫T+∞λ1e-λ1tE~1STF≤S∣τ1=t,τ2=udtdu=∫T+∞λ2e-λ2u∫T+∞λ1e-λ1tP~STF≤Sdtdu=e-λ1+λ2TΦln⁡S/S0-r+λ1m+λ2-σ12/2Tσ1T.If the stock has endogenous default in the life of the option, i.e., the default times satisfy Situations II and IV, then the price of the stock at expiration T is zero, and thus we obtain(23)E~1ST≤S1C∪D=E~E~1ST≤S1τ1≤T,τ2≤T∣τ1,τ2=∫0Tλ2e-λ2u∫0Tλ1e-λ1tE~10≤S∣τ1=t,τ2=udtdu=∫0Tλ2e-λ2udu∫0Tλ1e-λ1tdt=1-e-λ1T1-e-λ2T,(24)E~1ST≤S1F=E~E~1ST≤S1τ2≤T<τ1∣τ1,τ2=∫0Tλ2e-λ2u∫T+∞λ1e-λ1tE~10≤S∣τ1=t,τ2=udtdu=∫0Tλ2e-λ2udu∫T+∞λ1e-λ1tdt=e-λ1T1-e-λ2T.If the default times satisfy Situation III, then the dynamic of stock price process satisfies the stochastic differential equation (17). By Ito’s lemma and (17), we can obtain (25)STd1τ1=Sτ1d1τ1er+λ2-σ22/2T-τ1+σ2W~T-W~τ1=S01-γer+λ1m+λ2-σ12/2τ1+r+λ2-σ22/2T-τ1+σ1W~τ1+σ2W~T-W~τ1, and thus(26)E~1ST≤S1E=E~E~1ST≤S1τ1≤T<τ2∣τ1,τ2=∫T+∞λ2e-λ2u∫0Tλ1e-λ1tE~1STd1τ1≤S∣τ1=t,τ2=udtdu=∫T+∞λ2e-λ2u∫0Tλ1e-λ1tP~STd1τ1≤Sdtdu=∫T+∞λ2e-λ2u∫0Tλ1e-λ1tE~P~STd1τ1≤S∣γdtdu=∫T+∞λ2e-λ2u∫0Tλ1e-λ1tE~Φ1btln⁡SS01-γ-at∣γdtdu=e-λ2T∑i=13pi∫0Tλ1e-λ1tΦ1btln⁡SS01-γi-atdt,where a(t)=(r+λ1m+λ2-σ12/2)t+(r+λ2-σ22/2)(T-t), b(t)=σ12t+σ22(T-t), and Φ being the standard normal distribution function. Therefore, combining equalities (22)–(26), we can obtain that the distribution function of ST under model (1) is (19).Combining the distribution function of the stock price at expire timeT (19) and the following identity(27)∫0∞S-K+dΦ1Bln⁡SC-A=CeA+B2/2Φx0+B-KΦx0,where A is a constant, B,C,K are all positive constants, and x0=1/B(A-ln⁡(K/C)), we can compute the value of a call option at time 0 under model (1), summarizing to the following theorem.Theorem 2.
The risk-neutral price of the European call option at time 0 under model (1) is given by(28)C0,S0=S0e-1-mλ1TΦδ0+bT-Ke-r+λ1+λ2TΦδ0+e-r+λ2T∑i=13pi∫0Tλ1e-λ1tS01-γieat+bt2/2Φδit+bt-KΦδitdt,with δ0=1/bT(a(T)-ln⁡(K/S0)) and δi(t)=1/bt(a(t)-ln⁡(K/S0(1-γi))) for i=1,2,3.Proof.
According to the distribution function ofST and noticing the identity ∫0+∞(S-K)+d(1-e-λ2T)=0, we have(29)C0,S0=e-rTE~ST-K+=e-rT∫0+∞S-K+dFS=e-rT∫0+∞S-K+de-λ1+λ2TΦln⁡S/S0-aTbT+e-rT∫0+∞S-K+·de-λ2T∑i=13pi∫0Tλ1e-λ1tΦ1btln⁡SS01-γi-atdt≡I1+I2.It follows from formula (27) that the first term in (29) is(30)I1=S0e-1-mλ1TΦδ0+bT-Ke-r+λ1+λ2TΦδ0,and the second term in (29) is(31)I2=e-r+λ2T∑i=13pi∫0Tλ1e-λ1t∫0+∞S-K+dΦ1btln⁡SS01-γi-atdt=e-r+λ2T∑i=13pi∫0Tλ1e-λ1tS01-γieat+b2t/2Φδi+bt-Ke-r+λ1+λ2TΦδidt,where δ0=1/bT(a(T)-ln⁡(K/S0)) and δi(t)=1/bt(a(t)-ln⁡(K/S0(1-γi))) for i=1,2,3. By (30) and (31), we can obtain (28), immediately. Thus the proof of Theorem 2 is completed.Remark 3.
(1) Ifλ2=0, then the risk-neutral price of the European call option at time 0 under model (1) becomes(32)C0,S0=S0e-1-mλ1TΦδ~0+bT-Ke-r+λ1TΦδ~0+e-rT∑i=13pi∫0Tλ1e-λ1tS01-γieat+bt2/2Φδ~it+bt-KΦδ~itdt,where δ~0=1/bT(a(T)-ln⁡(K/S0)) and δ~i(t)=1/bt(a(t)-ln⁡(K/S0(1-γi))), with a(t)=(r+λ1m-σ12/2)t+(r-σ22/2)(T-t) and b(t)=σ12t+σ22(T-t). It can be seen from Figure 1 that the price of call option decreases along with the reduction of default intensity λ2. Figure 1 also shows that the value of call option at time 0 under this model is the same as the one at time 0 with the stock exposed to counterparty risk (refer to Ma et al. [8]); i.e., the double defaults risk model becomes counterparty default risk model when λ2=0. This is because the default intensity λ2=0 means that endogenous default risk has not occurred during the life of the option.
(2) Ifλ1=0 and λ2=0, then the risk-neutral price of the European call option at time 0 under model (1) becomes(33)C0,S0=S0Φδ+-Ke-rTΦδ-,with δ+=(r+σ12/2)T-ln(K/S0)/σ1T and δ-=(r-σ12/2)T-ln⁡(K/S0)/σ1T. In the same way, it follows from Figure 2 that the double defaults risk model becomes standard Black-Scholes model when λ1=0 and λ2=0.Figure 1
Call option prices versus stock prices for double defaults and counterparty default models withγ1=0.5,γ2=0,γ3=-0.2,p1=0.3,p2=0.5,p3=0.2,T=1,r=0.05,σ1=0.2,σ2=0.4, and K=100.Figure 2
Call option prices versus stock prices for double defaults and Black-Scholes models withγ1=0.5,γ2=0,γ3=-0.2,p1=0.3,p2=0.5,p3=0.2,T=1,r=0.05,σ1=0.2,σ2=0.4, and K=100.Theorem 4.
The risk-neutral price of the European put option at time 0 under model (1) can be obtained by put-call parity as follows:(34)P0,S0=C0,S0+Ke-rT-S0e-1-mλ1T-S0λ1e-r+λ2T∑i=13pi1-γi∫0Teat+b2t/2-λ1tdt.Proof.
According to the following identity,(35)K-ST+-ST-K+=K-ST, we have(36)P0,S0=e-rTE~K-ST+=e-rTE~ST-K++e-rTE~K-ST=e-rTE~K-ST+Ke-rT-e-rT∫0+∞S-KdFS=C0,S0+Ke-rT-e-r+λ1+λ2T∫0+∞SdΦln⁡S/S0-aTbT-e-r+λ2T∫0+∞Sd∑i=13pi∫0Tλ1e-λ1tΦ1btln⁡SS01-γi-atdt≡C0,S0+Ke-rT-I3-I4.Let ln⁡(S/S0)-a(T)/b(T)=t, then the third term in (36) can be calculated as(37)I3=e-r+λ1+λ2T∫-∞+∞S0ebTt+atdΦt=e-r+λ1+λ2T∫-∞+∞S0ebTt+at12πe-t2/2dt=e-r+λ1+λ2TS0eaT+bT2/2∫-∞+∞12πe-t-bT2/2dt=e-r+λ1+λ2TS0er+λ1m+λ2-σ12/2T+σ12/2T=S0e-1-mλ1T,and the last term in (36) can be calculated as(38)I4=e-r+λ2T∑i=13pi∫0Tλ1e-λ1t∫0+∞SdΦ1btln⁡SS01-γi-atdt=e-r+λ2T∑i=13pi∫0Tλ1e-λ1tS01-γieat+b2t/2dt=S0λ1e-r+λ2T∑i=13pi1-γi∫0Teat+b2t/2-λ1tdt.Incorporating (37) and (38) into (36), we obtain formula (34). Thus the proof of this theorem is completed.


## 4. Conclusions

In this paper, we have derived explicit analytical formulas for the price of European call and put options when the underlying asset is subject to double defaults risks. The external counterparty default risk induces a drop in the price of the stock, and the stock price drops to zero when the stock itself defaults. Double defaults risks cause difficulty in deriving the distribution function of the stock price at expire timeT. The conditional density approach (see Jiao and Pham [6]) is utilized to overcome the difficulty and derive the formulas for the price of European option. Many questions remain in option pricing with double defaults risks, for example, the prices of path-dependent options whose payoffs depend on the path of the underlying asset under this model and analytic formulas for the options with two underlying assets exposed to loop contagion risk. We leave these and other questions to the future research.



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