Equilibrium pricing of currency options under a discontinuous model in a two-country economy

Appendix A. Proofs of Theorems

Proof of Theorem 1. As pointed out in formula (A3) in (Bakshi and Chen 1997), the risk premium on any contingent claim is determined by the Euler equation by solving the agent’s first-order condition. If the contingent claim is a bond, similarly we have

Et[e−ρΔtm(t)m(t+Δt)(ΔB(t)B(t)−R(t)Δt)]=0.

Where

m(t)m(t+Δt)=1−(Δm(t)m(t))c+(Δm(t)m(t))c2+(11+km−1)ΔQm(t)+o(Δt).

Notice that B(t, τ; Y, Z) is a function related with variable t and Y, Z, there is no jump in the process of B(t, τ; Y, Z). We write

dB=∂B∂tdt+∂B∂YdY+∂B∂ZdZ+12∂2B∂Y2(dY)2+12∂2B∂Z2(dZ)2+∂2B∂Y∂ZdYdZ,

then we can get the discredited form ΔB. Combine the above equations, let Δt→0 and we obtain the PDE of the price of B(t, τ) as

−∂B∂τ+∂B∂Y(θy−kyY−σ1σyY)+∂B∂Z(θZ−kZZ−σ2σZZ)+12∂2B∂Y2σy2Y+12∂2B∂Z2σZ2Z=RB,

Solve the above PDE with the terminal condition B(t, 0; Y, Z)=1, we can get a closed form solution of B(t, τ; Y, Z) expressed in Formula (10).■

Proof of Theorem 2. Applying Ito’s lemma with respect to C(t), we get:

dC(t)=∂C∂tdt+∂C∂ee(dee)c+∂C∂YY(dYY)c+∂C∂ZZ(dZZ)c+12∂2C∂e2e2(dee)c2+12∂2C∂Y2Y2(dYY)c2+12∂2C∂Z2Z2(dZZ)c2+∂2C∂Y∂eYe(dYY)c(dee)c+∂2C∂Z∂eZe(dZZ)c(dee)c+[C(t, τ; Y, Z, e(1+km))−C(t, τ; Y, Z, e)]dQm+[C(t, τ; Y, Z, e(11+km∗))−C(t, τ; Y, Z, e)]dQm∗.

We write

dC(t)C(t)=(dC(t)C(t))c+kCdQm(t)+kC∗dQm∗(t),

kC=C(t, τ; Y, Z, e(1+km))−C(t, τ; Y, Z, e)C(t, τ; Y, Z, e).

Substitute the discredited form ΔC into the following equation

Et[e−ρΔtm(t)m(t+Δt)(ΔC(t)C(t)−R(t)Δt)]=0,

then let Δt→0 and we will obtain the PIDE of (13).■

Proof of Theorem 3. First, we let L(t)=ln[e(t)], then the variable e in (13) is replaced by the variable L as follows:

Similar to (Heston 1993) and (Bakshi and Madan 2000), by analogy with the Black-Scholes formula, we choose a solution as the following form

Denote Π1^=B∗Π1 and Π2^=BΠ2, then insert (A.2) into (A.1), we obtain the following two PIDEs:

According to the definition of an option, Π₁ and Π₂ are subject to the terminal condition:

Although it is difficult to compute Π_j directly, we can compute the characteristic function f_j of Π_j. Let f1^=B∗f1 and f2^=Bf2, with the F-K theorem, we get the PIDE of fj^ as follows:

with the terminal condition f1^(t, 0, Y, Z, L, ϕ)=E(eiϕL),

and,

with the terminal condition f2^(t, 0, Y, Z, L, ϕ)=E(eiϕL).

Since the coefficients are linear with respect to V and V*, we follow the idea of (Heston 1993; Bakshi and Chen 1997), combine the terminal condition, and choose the form of f1^ is:

Inserting (A.8) into (A.6), we can get three ODEs:

−∂X(τ)∂τ+(iϕσyσe−ky−σyσ1∗)X(τ)+12σy2X2(τ)+(1+iϕ)(η1−η1∗+σ1∗2−σ12+12iϕσe2)−η1+σ12=0,−∂X∗(τ)∂τ+(iϕσzσe∗−kz−σzσ2∗)X∗(τ)+12σz2X∗2(τ)+(1+iϕ)(η2−η2∗+σ2∗2−σ22+12iϕσe∗2)−η2+σ22=0,−∂u(τ)∂τ+θyX(τ)+θzX∗(τ)+[(μm−μm∗)−(λmkm¯−λm∗km∗¯)]iϕ−ρ−μm∗+λm∗km∗¯+λmE[eiϕln(1+km)−1]+λm∗E[e−(1+iϕ)ln(1+km∗)−1]=0, with the initial conditions X(0)=X*(0)=u(0)=0.

We solve the above ODEs to obtain (16). Using the same approach, we choose:

Inserting (A.9) into (A.7), we can get three ODEs:

−∂W(τ)∂τ+iϕσyσe−ky−σ1σy)W(τ)+12σy2W2(τ)+iϕ(η1−η1∗+σ1∗2−σ12−12σe2)−η1+σ12−12ϕ2σe2=0,−∂W∗(τ)∂τ+(iϕσzσe∗−kz−σ2σz)W∗(τ)+12σz2W∗2(τ)+iϕ(η2−η2∗+σ2∗2−σ22−12σe∗2)−η2+σ22−12ϕ2σe∗2=0,−∂V(τ)∂τ+θyW(τ)+θzW∗(τ)+[(μm−μm∗)−(λmkm¯−λm∗km∗¯)]iϕ−ρ−μm+λmkm+λmE[e−(1−iϕ)ln(1+km)−1]+λm∗E[e−iϕln(1+km∗)−1]=0,

with the initial conditions W(0)=W*(0)=V(0)=0.

We solve the above ODEs to arrive at (17).     ■