A Linear Regression Approach for Determining Option Pricing for Currency Rate Diffusion Model with Dependent Stochastic Volatility, Stochastic Interest Rate, and Return Processes

(1)

http://www.scirp.org/journal/jmf

ISSN Online: 2162-2442 ISSN Print: 2162-2434

DOI: 10.4236/jmf.2018.81013 Feb. 28, 2018 161 Journal of Mathematical Finance

A Linear Regression Approach for Determining

Option Pricing for Currency-Rate Diffusion

Model with Dependent Stochastic Volatility,

Stochastic Interest Rate, and Return Processes

Raj Jagannathan

Department of Management Sciences, Tippie College of Business, The University of Iowa, Iowa City, USA

Abstract

A three-factor exchange-rate diffusion model that includes three stochastical-ly-dependent Brownian motion processes, namely, the domestic interest rate process, volatility process and return process is considered. A linear regres-sion approach that derives explicit expresregres-sions for the distribution function of log return of foreign exchange rate is derived. Subsequently, a closed form workable formula for the call option price that has an algebraic ex-pression similar to a Black-Scholes model, which facilitates easier study, is discussed.

Keywords

Option Pricing, Interest-Rate Parity Condition, Black-Scholes Model, Linear Regression Approach, Spot Option, Ito Calculus

1. Introduction

A foreign exchange rate depends on the supply and demand dynamics of a cur-rency. The exchange rate is a function of trade balance, the interest rate differen-tial and differendifferen-tial inflation expectations between the two countries [1][2].

Let S(u), u≥0 = exchange rate process over the time interval: u≥0, where

u = number of domestic currency units, e.g., $, per unit of foreign currency = $-price of foreign currency.

As interest rate rD

( )

u increases, $ appreciates because investors prefer

$-denominated bonds. Assuming a frictionless, arbitrage-free continuous-time economy in [1], we define a diffusion process model for S(u). In addition, using

How to cite this paper: Jagannathan, R. (2018) A Linear Regression Approach for Determining Option Pricing for Curren-cy-Rate Diffusion Model with Dependent Stochastic Volatility, Stochastic Interest Rate, and Return Processes. Journal of Mathematical Finance, 8, 161-177.

https://doi.org/10.4236/jmf.2018.81013

Received: August 14, 2017 Accepted: February 25, 2018 Published: February 28, 2018

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jmf.2018.81013 162 Journal of Mathematical Finance

interest-rate parity condition we have

( )

0( ( ) ( ))

d 0e

h

D F

r u r u u

S h S

− ∫

= _{, see}_[1]_.

In the following section, the formula for valuations of currency spot options is considered, where we obtain a closed form formula for the call option price that has a simple algebraic expression, which is similar to the call option price ex-pression of a Black-Scholes model, making it much easier to compute its value and study. As in [2], we can define an implied volatility function and derive its skewness property.

Subsequently, the proposed three-factor exchange-rate diffusion model is discussed, such that the stochastic volatility process and the stochastic domestic interest rate process each have a stochastically dependent Brownian motion re-turn process.

In the next section, a linear regression approach that derives explicit expres-sions for the distribution function of lnS s

( )

is treated.

Foreign exchange rate option modeling is the subject of several well-known papers and in chapters within [3][4][5][6]. Leveraging Heston’s model [4] for this application would introduce complexity due to the need to numerically in-tegrate conditional characteristic functions obtained as solutions of nonlinear pdf to derive the call option prices. An equivalent two-factor Black-Derman-Toy model [2] can be formulated with introduction of H(u).

The method suggested in this paper results in Black-Scholes type formula for call option pricing, which is easily computable.

Finally, we provide concluding remarks and suggestions for future direction.

2. Currency Spot Option

Given the spot rate S

( )

0 =s0, consider the present value of option

(

)

( )

(

( )

)

*

0 0

0

, ,* exp d *

s D

C s K =E  − r u u S s −K+ 

   



∫

 (1)

where K is the known strike price and

(

r_D

( )

u u, ≥0

)

is a mean-reverting

sto-chastic process given in (2) below. S s

( )

is the value of the exchange rate at the option’s maturity price. The option to purchase foreign currency over the coun-ter can be exercised when S(s) > the strike price exchange rate K.

3. A Diffusion Process Model

A continuous-time risk-adjusted and risk-neutral exchange rate model, under a Martingale Measure Q, is defined below as a diffusion process (2), mean-reverting stochastic processes: Volatility

{

H u u

( )

, ≥0

}

(3) and domestic interest rate

( )

D

r t process (4), and foreign interest rate r_F is a known constant.

( )

(

( )

)

( )

(

( )

)

( )

2

1

d

d d

2

D F

S u H u

r u r u H u B u

S u υ

 

=_ − + _ + +

(3)

( )

(

( )

)

1

( )

dH u =α H u −θ du+ηdV u , ,θ η≥0,α >0 (3)

( )

(

( )

)

2

( )

dr_D u =β r_D u −λ du+ζdV u , ,λ ζ ≥0,β >0 (4)

( )

2

(

( )

)

1

( )

d ln d d

2

D F

S u r u r υ  u H u υ B u

∴ =_ − − _ + +

  (5)

Equation (5) is obtained from Equation (2) by the application of Ito calculus [7]. Assumption:

( )

1 1

V t =ρB t +δC t , where δ _ ₁₋ρ2 and

( )

1

B t and C t

( )

are

in-dependent Brownian processes.

( )

2 1 1 1 1

V t =ρB t +δC t , where 2

1 1 1

δ = −ρ

and B t1

( )

and C t1

( )

are independent Brownian processes. (6)

From the assumption above, the return processes V_j

( )

u ,u≥0,j=1, 2 are

correlated with B u1

( )

and that Vj

( )

t ,j=1, 2 are standard Brownian motion processes.

Then it follows, see [2] [3], that the distributions of H u

( )

and rD

( )

u are

Gaussian processes.

Alternatively, H u

( )

and rD

( )

u may be expressed as:

( )

( ) ( )

( )

1 0

d .

d d .

u u

H u q u t V t

H u q u t B t C t

ψ

ψ ρ δ = +

∴ = + _ + _

∫

(7)

( )

(

)

( )

(

)

(

)

e 1 e

e , 0,

u u

u

E H u q u α α

α

κ θ θ κ θ α

− −

−

= = + −

= + − >

where θ_{is the long-term mean and where} H

_{( )}

0 =κ .

( )

(

)

( )

(

)

2

2 2 2 2 2

0 0

1 e

var d e d

2

So 0

u u u

t H

H H

H u t t t

P H u

α α

σ ψ η η

α µ

σ

−

−  − 

= = = _ _

   

< = Φ −_ _  

∫



(8)

Remark 1:

From (8), choosing θ >0 and that is small in value, we can make

( )

(

0

)

P H u < negligible.

If, alternatively, we assume that

{

H u

( )

,u≥0

}

has a square root process [8],

then the random variable H(u) distribution is non-central χ2

( )

_. . For simplicity

we chose the mean-reverting process model (3).

( )

(

)

1

( )

1e

(

1 e

)

(

1

)

e , 0

u u u

D

E r u =q u =κ −β +λ − −β = +λ κ −λ −β β >

( )

(

)

2

( )

1 0

var d

u D

r u =  ψ t u_



∫



where 1

( )

e , 0

u

u β u

ψ ₌ζ − _> _and

_{( )}

1

0

D

r =κ

( )

1

( )

1

( )

1 1

( )

1 1

( )

0

d d

u D

r u q u ψ t ρ B t δ C t

(4)

Assuming rD

( )

0 =κ1, and

( )

(

)

(

)

( )

(

)

1

1 1 1

0

e , , 0, 0

e 1 e e

1 e d

t

u u u

s s

t t u

q u

Q s q u u s

β

β β β

β

ψ θ θ β

κ λ λ κ λ κ λ λ

β

−

− − −

−

= ≥ ≤ ≤

= + − = + − − =

_∫

= − +

(10)

The Brownian motion processes V u1

( )

and B u1

( )

are as follows:

( )

1 1 1

V u =ρB u +δC u , where δ ₌ ₁₋ρ2 (11)

In addition, the Brownian motion processes B u1

( )

and C u1

( )

under Q are

independent. Remark 2:

( )

, 0

H u u≥ : the volatility process.

It follows from [2] that the distribution of H u

( )

is:

( )

(

₂

)

1 2

~ , 1 e , 0 .

2

u

H u N q u η α u s

α

−

 

− ≤ ≤

 

  (12)

Alternatively, H u

( )

may be expressed as

( )

( ) ( )

( )

1 0

d

d d

u u

H u q u t V t

q u t B t C t

ψ

ψ ρ δ = +

= + _ + _

∫

(13)

where 2

1

δ _ −ρ and ψ

( )

u =ηe−αu, ,η α≥0.

See [9] for a similar assumption. See also [2] and [3].

Note that B s1

( )

has a normal distribution with mean 0 and variance s, so

( )

1

B s can be written as B s₁

( )

=Z s

( )

s, where Z s

( )

is a standard normal

variable. Then lnX s

( )

can be written as a quadratic function of

( )

1

( )

, 0

B s

Z s s

s

= > _{plus a residual term} ε

( )

s . {See Proposition 1 below}.

For d

ξ

=dC t

( )

, 0≤ ≤t u, 0≤ ≤u s, we define a volatility process

( )

( ) ( )

0

d , 0

u u

V ξ =_V = ψ t C t ≤ ≤u s_



∫

.

Define

(

( )

)

(

)

1 2

2 2

0

1 e 1

d 2

u s

V s u V

s

α

η σ

α

−

  −  

   

 _ _ 



∫



  , as the average standard

deviation in the case of uncorrelated Brownian motion process

( )

{

C u , 0≤ ≤u s

}

ξ [See [10], p. 182].

Proposition 1:

( )

1

( )

1

( )

1 1

( )

1 1

( )

0

d d ;

u D

r u =q u +

∫

ψ t _ρ B t +δ C t _

( )

(

( )

)

( )

1 1 1 1 1 1

0 0 0 0

1 1 1 1 1 1 1 3

d d d d d

.

s s s u

D

r u u q u u u t B t C t

Q s B s G s s V e s

ψ ρ δ

ρ χ δ

∴ = + _ + _

= + − + +

(5)

where

( )

(

)

(

)

( )

(

)

( ) ( )

( )

(

) ( )

1 1 1

0

1 1

0

1

1 1

1 e d

1 e

d 1

!

1 1 !

s s

t j

s

j j

Q s q u u s

s

G s t

j j

s

j j

s

β

κ λ λ β

ζ ζ β

β β

ζ β β χ

−

− _∞

+

=

+

∞ ₊

=

−

= − +

 

−

 

= −

 

 

 

−

 

 ₊ 

 

=

∫

∑

∫

∑



 (15)

Proof: See Appendix B.

We consider a mean-reverting Gaussian process model (2), the volatility sto-chastic processes

{

H u

( )

, 0≤ ≤u s

}

and the processes, V u1

( )

and B u1

( )

in

(3) to be correlated; where V u1

( )

is a standard Brownian motion return

process. In addition, in (3), we define the volatility H u

( )

εR as a mean

revert-ing Gaussian process with θ_{as its long-term mean.}

Assumption 1:

( )

2

1 1 1 1 , where 1 1 1

V t =ρB t +δC t δ = −ρ (16)

In (4), we define the domestic interest rate process rD

( )

u as a mean

revert-ing Gaussian process with λ_{as its long-term mean. The process}

( )

{

r_D u , 0≤ ≤u s

}

is such that the return process V₂

( )

u is a correlated

stan-dard Brownian motion process to B u1

( )

. The foreign interest rate rF

( )

u is a constant rf

Assumption 2:

It follows from [2] that the distribution of rD

( )

u :

( )

(

₂

)

1 2

1

~ , 1 e , 0 .

2

u D

r u N q u ζ β u s

β

−

 

− ≤ ≤

 

  (17)

Now we use the results obtained in Proposition 1 to derive an explicit expres-sion for

( )

0

d ln ln ln

s

S u = S s − S

∫

Proposition 2:

( )

(

)

( )

(

( )

)

( )

0

2

0 1

0 0

1 1

0 0

d ln

d d

2

d d d ;

s

s s

D F

s u

S u

r u r u s q u B u

t B t C t B u

υ _{υ θ}

ψ ρ δ

= − − + _ + + _

 

+ _ + _

 

∫

∫ ∫

(18)

Remark 3:

From the expression for

( )

1

( )

1 1

( ) ( )

(

1 1

( )

)

1 1 0

d

s D

r u u=Q s +ρB s G s −χ s +δV

(6)

( )

(

( )

)

1B s1 G s1 1 s

ρ −χ _modifies n s

( )

and the constant term Q s₁

( )

+δ_{1 1}V

modifies *

( )

,

p s h with the addition of ρ₁B s₁

( )

(

G s₁

( )

−χ₁

( )

s

)

and the

con-stant terms Q s1

( )

+δ1 1V modifies

( )

*

,

p s h with the addition of Q s₁

( )

+δ_{1 1}V .

Then, using the results in [2], Proposition 1 and those in Appendix A and Appendix B we have:

( )

(

) ( )

( ) ( )

( )

( ) ( )

(

( )

)

( )

( ) ( )

2

0 1 1 1 1

2 1

2 2 1

1 1 1 1 1 1 1 3

2

ln ln

2

2 2

, 2

F

S s s s r s B s s B s e s

B s s

s e s Q s

B s G s s V Y e s

m s

Z s Z s n s p s s U s s

υ _{υ θ} _γ

γ ρ

ρ χ δ

ε ξ ε = − − + + +_ + _

   

+ _ − _+ +

   

 

+ − + +

+ + + +

 

(19)

Therefore,

( )

2

( ) ( )

( )

, ,

2 m s

U sξ =Z s +Z s n s +p s

( )

2

( )

;

m s =ργ s s

( )

(

)

1

( )

3

( )

(

1

(

1

( )

1

( )

)

1 1

)

;

n s =_

υ θ δ

+ + V +

γ

s +

γ

s +

ρ

G s −

χ

s +

δ

V _ s

Remark 4:

Note that n s

( )

in this paper is an updated version from the n s

( )

in [2], due to our treatment of a stochastic interest rate:

( )

0

d

s D

r u u

∫

( )

2

( )

0 2 1

1

ln ;

2 2 F

s

p s = s − υ s−ργ s +ε s +Q s −r s

( )

(

)

(

)

(

)

(

)

(

)

2 2

2

0

2

2 2

0

2 2

1 e d 2

1 e 1

d 2

2 1 e

1

, 4

u s

s

V s u

u s

s

V s

α

η σ

α η

α η α

α

−

 − 

 

=

 

 

 − 

 

=

 

 

− − =

∫



In the case of C u u

( )

, ≥0.

( )

(

)

2

(

2

)

2 2

1 2 1

2 1 e

1

4

s

V s V

s

β

ζ

β

σ

β

−

− −

= _ _{, where} C u1

( )

,u≥0 (20)

( )

(

)

( )

(

( )

)

( )

2

2 3 2 1 1 1

2

0 2 1 1 1

;

1 ln

2 2 F

m s s s

n s V s s G s s s

s

p s S s s s Q s r s V

ργ

υ θ δ γ λ γ ω ρ χ

υ ργ ε δ

=

 

=_ + + + + + + + − _

= − − + + − +

(21)

( )

s e s1

( )

e2

( )

s e3

( )

s

(7)

( )

(

)

(

)

( )

(

)

(

)

( )

(

)

(

)

(

)

1 1 1 1 1 0 1 0 0

2 2 2

0 0 0

2 2 1 e d ; 1 e 1 d ; 1 e 2 2

d d d

1 e 2 . s s s s u

s u s

s

Q s q u u s

s q u u

s s

s t t u u

s s s s β α α α κ λ λ β κ θ γ α η η γ ψ α α η α − − − − − − = = + − − = = − = = − − =

∫

∫ ∫

∫

(22)

( )

(

)

(

)

2 2 2

1

1 e 1 e

; 2

s

Var e s

s α α κ θ α α − − _ ₋ _{ } ₋ _    = − _ _{ }− _       

( )

(

)

2 2

(

)

(

)

2

2 2 1 e 2 1 e

2 4

s s

Var e s η η α η η α

α α α α

− −     = − −  − − −     

( )

(

3

)

Var e s is provided in (B1)

( ) ( )

(

i , j

)

0; , 1, 2, 3,

Cov e s e s = i j= i≠ j

Case 1:

ρ

>0;

( )

(

)

(

( )

)

( )

(

( )

)

( )

(

( )

)

0 2

1 1 0 1

0 0 0

1 1 1 1 1 1 1

0 0 0

d ln

d d d

2

d d d d ;

s

s s s

D F

s u s

S u

r u r u s q u u q u B u

t B t C t B u t B u C u

υ _λ _{υ θ}

ψ ρ δ ψ ρ δ

= − − + + + _ + + _   _ _ + _ + _ + _ + _  

∫

∫ ∫

∫

Let S

( )

0 =s0

( )

(

)

( )

( ) ( )

( )

( ) ( )

(

( )

)

( )

2 0 1 2 1

1 1 1 2 2

1 1 1 1 1 1 1 3

ln ln

2

2 2

.

F

S s S s r s V B s

B s s

s B s e s s e s

Q s B s G s s V e s

υ _{υ θ δ}

γ γ ρ

ρ χ δ

= − − + + +     +_ +  _+ _ − _+        + + − + +

Let

( )

2

( ) ( )

ln ,

2 m s

S s Z s +Z s n s +p s +ε s U sξ +ε s .

Assumption 3: U s

( )

,ξ and ε

( )

s are independent random variables.

Assumption 4: 2

( )

( ) ( )

(

)

, 2 , 0

n sξ − m s p s h −ω ≥ .

Assumption 5: m s

( )

≠0 and m s

( )

<1.

If Assumptions (4) and (5) hold, then the conditional risk-neutral distribution of lnS s

( )

is:

Proposition 3: ( )

(

)

( )

(

)

(

( )

)

(

( )

)

(

)

(

)

ln 1 2 1 ,

1 , , ,

ln |

1, , ,

S s

F h

z h z h h

P S s s h

h

ω

ω ω ρ ξ ω ε

ω ω ρ ξ

∗

−

− Φ + Φ ≥

= ≥ = =

(8)

( )

(

)

(

( )

)

( )

(

)

(

( )

)

(

)

(

)

1 2

ln

, , ,

, ln |

0, , .

S s

z h z h h

F h P S s s h

h

ω ω ρ ξ

ω ω ε

ω ω ρ ξ

∗

Φ − Φ ≥

∴ = ≤ = =

≤

(23) where

( ) ( )

( )

_{( )}

( ) ( )

(

)

(

)

( )

_{( )}

2

1 2

2 *

, 2

, , ,

,

2

n s n s m s p s h

z z h z h

m s

n s

h h p s

m s

ξ ω

ξ ξ

ω ρ

− ± − + −

= =

= + −

(24)

If 2

( )

( ) ( )

(

*

)

, 2 0

n sξ − m s p s − −h ω = , then the roots of the equation defined

in (24) are equal so that 1 2

( )

_{( )}

, n s

z z z

m s

ξ ξ

∗

= = = − , then there exists a value

(

, ,h

)

ω ρ ξ∗ _{such that}

( )

(

) ( )

( )

(

_ln _, _| _, 2

)

₁

P S s ≥ω ρ∗ h ε s =h V s =ξ = .

In other words, *

(

)

, ,h

ω ρ ξ _{is the lowest value for the conditional random}

variable ln

(

S s

( )

)

.

Remark 5:

Since we know the CDF of lnS(s) we can estimate the parameters of the un-derlying model (2)-(5).

Case 2: Conditional Risk-neutral Distribution function of

( ) ( )

( )

(

lnS s |ε s =h V s, =ξ

)

, m s

( )

<0. Suppose

ρ

<0, Conditional risk-neutral

distribution of _ln_{S s}

( )

_|

(

ε

( )

_s ₌_{h V}_, 2

( )

_s ₌ξ

)

is as follows:

( )

(

)

( )

(

)

( )

(

)

(

( )

)

(

)

(

)

ln |

2 1

1 ,

ln | ( ) ,

, , ,

0, , ,

S s

F h

P S s s h V s

z h z h h

h

ξ ω

ω ε ξ

ω ω ρ ξ ω ω ρ ξ

∗

−

= ≥ = =

Φ − Φ ≤

 =  _≥



( )

(

)

( )

(

)

(

( )

)

(

)

(

)

2 1

ln |

1 , , ,

,

1, ,

S s

z h z h h

F h

h ξ

ω ω ρ ξ ω

ω ω ρ

∗

− Φ + Φ ≤

∴ =

≥

where

( ) ( )

( )

2

( )

_{( )}

( ) ( )

(

)

1 2

, , 2

, , , n s n s m s p s h

z z h z h

m s

ξ ξ ω

ξ ξ − ± − + −

= =

Example 1

( )

(

( )

)

( )

(

( )

)

( ) ( )

( )

(

( )

)

( ) ( )

( )

(

( )

)

( ) ( )

2 1

1

d d d , , 0, 0, 0 ;

d 0.3 0.02 d 0.04d , 0 0.03;

d d , 0 ;

d 0.1 d 0.3 , 0 0.6

D D D

r u r u u V u r

r u r u u V u r

H u H u u V u H

β λ ζ λ ζ β κ

α θ η κ

= − + ≥ > =

= − + =

(9)

( )

D

r u :

β λ ζ rD

( )

0 =κ1 ρ1

0.3 0.2 0.4 0.03 0.6

( )

H u :

α θ η ρ κ rf

1 0.1 0.3 −0.8 0.6 0.8

Then

( )

*

1 1 1

, ln 0 ln

p s h  p s + −h Q s − s + K−ρY

Remark 6:

From the expression for

( )

1

( )

1 1

( ) ( )

(

1 1

( )

)

1 1 0

d .

s D

∫

the stochastic terms ρ1B s1

( )

(

G s1

( )

−χ1

( )

s

)

modify the term n s

( )

and the

constant terms Q s1

( )

+δ1 1V modifies

( )

* _,

p s h with the addition of

( )

1 1 1

Q s +δV.

Proof:

Apply a proof similar to the one in Appendix A of [2] using the result for

( )

0

d

s D

r u u

∫

in Appendix B of the current paper. See also Proposition 4. Remark 7:

Assume

ρ

>0, which implies that m s

( )

>0.

If Assumption (3) holds then the conditional risk-neutral distribution of

( ) ( )

( )

{

lnS s |ε s =h V s, =ξ

}

is:

( )

(

)

( )

(

)

(

( )

)

(

( )

)

(

)

(

)

( )

(

)

(

( )

)

( )

(

)

(

( )

)

(

)

(

)

ln

1 2

ln

1 ,

1 , , ,

ln |

1, , ,

, , ,

, ln |

0, , ,

S s

F h

z h z h h

P S s s h

h

z h z h h

F h P S s s h

h ξ

ω

ω ω ρ ξ ω ε

ω ω ρ ξ

ω ω ε

ω ω ρ ξ

∗

−

− Φ + Φ ≥

= ≥ = =

≤

Φ − Φ ≥

∴ = ≤ = =

≤

(25) where

( ) ( )

( )

_{( )}

( ) ( )

(

)

(

)

( )

_{( )}

2

1 2

2 *

, 2

, , ,

,

2

n s n s m s p s h

z z h z h

m s

n s

h h p s

m s

ξ ω

ξ ξ

ω ρ

− ± − + −

= =

= + −

(26)

If _n2

( )

_s_,ξ ₋₂_{m s}

( ) ( )

(

_{p s} _{− −}_h ω*

)

₌₀, then the roots of the equation defined

in (26) are equal so that 1 2

( )

_{( )}

, n s

z z z

m s

ξ ξ

∗

(10)

(

, ,h

)

ω ρ ξ∗ _{such that} _P

(

_ln_{S s}

_{( )}

_≥ω ρ∗

₍

_,_h

_{) ( )}

_|ε _s ₌_{h V}_,2

_{( )}

_s ₌ξ

)

₌₁.

In other words, *

(

)

, ,h

ω ρ ξ _{is the lowest value of the conditional random}

variable

{

lnS s

( ) ( )

|

ε

s =h

σ

ε( )s

}

.

Call option price:

( )

(

)

( )

*

0

, , , 0 exp d ln ln

s Q

D

c s S s K S =E _− r u u _{ }+ S s − K+_

  

∫

 Proposition 4:

(

)

( )

(

)

( )

(

)

( )

(

( )

)

( )

(

)

( )

(

( )

)

( )

(

)

( )

(

)

( )

0 0 0 d 0 2 1

0 1 2 1 2

2 _d

*

, , , , , 0

e |

0, for ln , ;

1 1

1

1 1 1

exp , e | ln ln

2 1 s

D

s d

r u u Q

r u u

C s K r h

E S s K s h

K h

z h m s n s z h m s n s

s

m s m s m s

n s

p s h KE S s K

m s η ρ ε ω ρ ∗ − ₊ ∗ − ∫ ∫ <     = − =     >  _  ₋ ₋   ₋ ₋ _      

=_ _Φ − Φ __

    −  _ _ − _ _ − __        ∗_ _ + __− ≥ −                      _ _ 

where from Proposition 1

( )

1

( )

1 1

( ) ( )

(

1 1

( )

)

1 1 0

d .

s D

∫

See Appendix B.

( )

{

( ) ( )

}

( )

*

0 1 1 1

, | ln ln

p s h = p s ε s =h + s − K+_Q s +δV_

Remark 8:

Given the formula for

( )

1

( )

1 1

( ) ( )

(

1 1

( )

)

1 1 0

d

s D

∫

, the stochastic expression

( )

(

( )

)

1B s1 G s1 1 s

ρ −χ modifies the function n s

( )

and the constant terms

( )

1 1 1

Q s +δV, modifies p*

( )

s h, with the addition of Q s₁

( )

+δ_{1 1}V.

( )

(

)

(

( )

)

{

}

( ) ( ) 0 2 2 1 2 2 ₁ 1 d 2

1 1 1 1 1

2

e | ln ln

exp e d

e d ;

s D h h h h

r u u

z _z

z

z _{A s} _{A s z z} z

K S s K

K Q s V G s s z s z

K z

δ χ ρ

− − + − ∫    _≥      = + + − =

∫

(27) Let

( )

(

( )

)

( )

(

( )

)

( )

1 1 1 1 1 1 1 1

1 1 1

;

A s z G s s B s G s s z s

A s Q s V

ρ χ ρ χ

δ

− = −

= +



( )

2

( )

2

( )

₁2

( )

1 1 2 2 ;

2 8

A s

z

(11)

Then

( ) ( )

( )

2

2 ₁

1

2

2 1

2 1 1 1

e d

2 2 exp ;

8 h

z A s A s z z z

h h

K z

A s

z A s z A s A s

+ −

 

 

= Φ_ _ − _− Φ_ − __ _ − _

 

∫

Hedge Ratio:

( )

(

( )

)

( )

(

)

( )

(

( )

)

( )

(

)

( )

₍

( )

_{( )}

₎

(

)

(

)

* 0

2 1

1 2 1 2

2

* *

*

.

1 1

1

1 1 1

exp , , ln , ;

2 1

0, ln , ;

C s

z h m s n s z h m s n s

m s m s m s

n s

p s h K h

m s

K h

ω ρ

ω ρ ∂

∆ = ∂

 _  ₋ ₋   ₋ ₋ _  _Φ − Φ _  ₋ _  ₋   ₋ _

   

 



=   

∗ _ + _ < −

 _ _



≥ 

∆_{-Neutral Portfolio}

Delta-Neutral Portfolio

Consider the following portfolio that includes a short position of one Euro-pean call and a long position of delta units of the domestic currency.

The portfolio of delta-neutral positions is defined as:

( )

*

0 Hedge ratio 0

P=s∆− ⇒c P =

We obtain below Conditional Risk-neutral Distribution function of

( ) ( )

( )

(

lnS s |ε s =hσε,V s =ξ

)

,m s

( )

<0 (29)

by considering the cases of: h = 1, 0 and −1 We use a discrete approximation (see [2], (28)). Suppose

ρ

<0, which implies m s

( )

≤0.

Again, we consider the Equations (1)-(4) to define Example 1 below.

( )

(

( )

)

( )

2

1

d

d d ;

2

D F

S u H u

r u r u H u B u

S u υ

 

=_ − + _ + +

  (30)

( )

(

( )

)

1

( )

dH u =α H u −θ du+ηdV u , ,θ η≥0,α >0; (31)

( )

(

( )

)

2

( )

dr_D u =β r_D u −λ du+ζdV u , ,λ ζ ≥0,β >0; (32)

( )

2

(

( )

)

1

( )

d ln d d ;

2

D F

S u r u r υ  u H u υ B u

∴ =_ − − _ + +

  (33)

Let

( )

(

( )

)

( )

(

( )

)

( ) ( )

( )

(

( )

)

( ) ( )

( )

(

( )

)

( ) ( )

2 1

1

d d d , , 0, 0, 0 ;

d 0.3 0.02 d 0.04d , 0 0.03;

d d , 0 ;

d 0.1 d 0.3 , 0 0.6

D D D

r u r u u V u r

r u r u u V u r

H u H u u V u H

β λ ζ λ ς β κ

α θ η κ

= − + ≥ > =

= − + =

(12)

Then, H u

( )

:

υ α η θ ρ κ δ s H

( )

0

0.08 1 0.2 0.1 −0.8 0.1 0.6 0.5 0.06

And rD

( )

u :

β θ1 λ δ1 ζ ρ1 rf

0.3 0.2 0.1 0.5 0.04 3 4 0.06

If Assumption (2) holds then the unconditional risk-neutral distribution of

( ) ( )

(

2

( )

)

lnS s | ε s =h V, s =ξ and U s

( )

,

ξ

are independent random variables.

Then Figure 1 depicts the unconditional risk-neutral distribution of

( )

{

ln |

}

ln(S s( )

(

|

)

P S s ≤u h =F u h .

Remark 9:

[image:12.595.207.542.81.185.2] [image:12.595.243.511.356.489.2]

Future movement of values of risk-free interest rate and volatility are uncer-tain and as they increase, they affect call option values as depicted in the above

[image:12.595.244.508.537.691.2]

Figure 2, Figure 3 ([5], p. 204). Sudden changes in their values may occur be-cause of economic shock. See the models suggested in [11][12].

Figure 1. Unconditional risk-neutral CDF of lnS(s), strike price (cents) k

from 1.1 to 16.2.

Figure 2. Unconditional call option price with strike price k (cents) from 1.1 to 26.

0 0.1 0.2 0.3 0.4 0.5

1.

2

2.

7

4.

2

5.

7

7.

2

8.

7

10.

2

11.

7

13.

2

14.

7

16.

2

Unconditional Risk Neutral Cumulative

Distribution Function of ln(S(s))

Unconditional Risk Neutral CDF

0 5 10 15 20 25 30 35 40

1.

1

2.

5 4

5.

5 7

8.

5 ₁₀

11.

5 ₁₃

14.

5 ₁₆

17.

5 ₁₉

20.

5 ₂₂

23.

5 ₂₅

Unconditional Call Option Price

(13)

[image:13.595.243.511.74.207.2]

Figure 3. Unconditional hedge ratio with strike price k (cents) from 1.1 to 26.

4. Conclusion

We define a three-factor exchange-rate diffusion model with 1) stochastic vola-tility process, 2) stochastic domestic interest rate process, and 3) return process which are Brownian motion return processes that are stochastically dependent. Further generalization is possible with the assumption of domestic and foreign stochastic interest rate processes which are subject to economic shocks [11][12]. The results are applicable to bond option models ([5], p. 783).

References

[1] Krugman, P.R. and Obstfeld, M. (2000) International Economics. 3rd Edition, Ad-dison Wesley, Boston, 360-362.

[2] Jagannathan, R. (2016) A Linear Regression Approach to Two-Factor Option Pric-ing Model. Journal of Math Finance, 6, 317-337.

[3] Vasicek, O. (1977) An Equilibrium Characterization of the Term Structure. Journal Financial Economics, 5, 177-188.https://doi.org/10.1016/0304-405X(77)90016-2 [4] Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility

with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-344.https://doi.org/10.1093/rfs/6.2.327

[5] Hull, J.C. (2009) Options, Futures and Derivatives. Pearson, Prentice Hall, Upper Saddle River.

[6] Garman M.B. and Kohlhagen, S.W. (1983) Foreign Exchange Option Values. Jour-nal of InternatioJour-nal Money and Finance, 2, 231-237.

https://doi.org/10.1016/S0261-5606(83)80001-1

[7] Oksendal, B. (2003) Stochastic Differential Equations—An Introduction with Ap-plications. Springer, New York.

[8] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A Theory of the Term Structure of Interest Rates. Econometrics, 53, 385-407.https://doi.org/10.2307/1911242 [9] Trolle, A.B. and Schwarz, E.S. (2009) A General Stochastic Volatility Model for the

Pricing of the Interest Rate Derivatives. The Review of Financial Studies, 22, 2007-2057. https://doi.org/10.1093/rfs/hhn040

[10] Duffie, D. (1996) Dynamic Asset Pricing Theory. 2nd Edition, Princeton University Press, Princeton.

[11] Merton, R.C. (1976) Option Pricing When the Underlying Stock Returns Are

Dis-0 50 100 150 200 250 300 350

1.

1 3 5 7 9 ₁₁ ₁₃ ₁₅ ₁₇ ₁₉ ₂₁ ₂₃ ₂₅

Unconditional Hedge Ratio

(14)

DOI: 10.4236/jmf.2018.81013 174 Journal of Mathematical Finance continuous. Journal of Financial Economics, 3, 125-144.

[12] Jagannathan, R. (2008) A Class of Asset Pricing Models Governed by Subordinated Processes That Signal Economic Shocks. Journal of Economic Dynamics and Con-trol, 32, 3820-3846.https://doi.org/10.1016/j.jedc.2008.04.004

[13] Bakshi, G., Cao, C. and Chen, Z. (1997) Empirical Performance of Alternative Op-tion Pricing Models. The Review of Financial Studies, 52, 2003-2049.

https://doi.org/10.1111/j.1540-6261.1997.tb02749.x

(15)

Appendix A

( )

(

)

(

)

(

)

2 2 2 2

0 0 0

2 2

1 e

2 2 1 e 2

d d d

1 e 2

.

s

s u s u

s

s u t t u

s s s

s

α α

α

γ ψ η η

α α

α η

α

− − −

−

= = =

− − =

∫ ∫

∫

( )

2 2

0 0

2

d d

s u

s u t t

s

γ =

_{∫ ∫}

ψ

is the regression coefficient.

( )

(

( )

)

( ) ( )

2 2 2 1 1

0 0

2

d d .

s u

e s t s B t B u

s ψ γ

=

_{∫ ∫}

−

( )

(

2

)

0;

(

1

( )

)

0,

(

1

( ) ( )

, 2

)

0

E e s = E e s = Cov e s e s = (2A1)

( )

(

)

(

( )

)

2 ₂

( )

₂

( )

2 2 2 2

0 0 0 0

2

d d d d

s u s u

Var e s u t s t t t u s

s

ψ γ ψ γ

=

_{∫ ∫}

− =

_{∫ ∫}

−

Then the regression equation is

( )

** 2

2 1 2 2 ;

2

s

s s B s e s

ψ

=

γ

_ − _+

  (2A2)

Assumption 6:

( )

(

1 2

(

( )

)

2 ~ 0, 2

e s N Var e s (Approximately) (2A3)

Note that Cov e s

(

1

( ) ( )

,e2 s

)

=0 and

( )

(

1 2

(

( )

)

1 ~ 0, 2

e s  N Var e s .

( )

(

)

(

( )

)

(

1/2

)

(

( )

)

(

( )

)

1 2 1 2

Var e s Var e s Var e s Var e s

∴ + = +

Assumption 7:

( )

(

( )

)

(

( )

)

1 2

)

1 2 ~ 0, 1 2

s e s e s N Var e s Var e s

ε = + + _{(Approximately)}

(2A4) Proof of Proposition 1:

( )

{

( )

}

( )

(

( )

)

(

)

( )

2

0 0

0 1 1 1

0

d ln d

2

d d d ;

f

s s

d

u

S u u r u r

q u t B t C t B u

υ

υ θ ψ ρ δ

 

= − −

  _ _

 

 

+ + + + +

∫

( )

{

}

(

) ( )

( ) ( )

( )

(

)

( )

( ) ( )

( )

2

0 1

2 1

1 1 1 2 2 1

2 1 2

0 0

2 1

1 1 1 1 2 2

1

ln ln d

2

2 2

1

ln d

2 2 2

.

2 2

f

s d

s

d

S s r u r u s B s

B s s

s B s e s s e s VB s

B s s

s r u u s

B s s

s B s e s VB s s e s

υ υ θ

γ γ ρ δ

υ υ θ ρ

γ δ ρ γ

= + − − +_ +

   



+_ +  _+ _ − _+ + _

   

 

 

= + − + + _ − _

 

   

+_ + _+ +  _ − _+ 

   

 

∫

(16)

Appendix A from [2]

( ) ( ) ( )

( )

( ) ( )

( )

1 1 2 1 1 2

0 0 0

2

2 1 2

d d d

2

s u s

B u t B t s B u B u e s

s

s B s e s

ψ γ

γ

= +

  = _ − _+

 

∫

where

( )

(

)

(

)

(

)

2 2 2

0 0 0

2 2 2

2 2 1 e

d d d

1 e 1 e

2 2

s u s u

s s

s u t t u

s s

α

α α

γ ψ η

α α α

η η

α α

−

= =

− − − −

= =

∫ ∫

∫

( )

(

*

( )

)

2 2

0

d ,

s

e s =

∫

ψ u −γ s u

where

( )

(

)

(

( )

)

( )

(

)

(

)

* 0

2

* 2

2 2

0

2 2

2 0

d

d d

1 e

1 e 2

d 2

u s

s

s u

u t t

Var e s u t u s

s u

s

α α

ψ ψ

ψ γ

α

η η

α α

−

=

= −

 _{− −} 

 −  _ _

= _ _ −

 

  _ _

∫

Appendix B

( )

( ) ( )

(

( )

)

( )

1 1 2

0

1 1 1 1 1 1

0

1 1 1 1 1 1

0 0 0 0

1 1 1 1 1 1 1 3

1 1

0

d

d d ;

d d d d d

d ;

u

D

u

s s s u

D

s

r u q u t V t

q u t B t C t

r u u q u u u t B t C t

Q s B s G s s V e s

Q s q u u

ψ

ψ ρ δ

ρ χ δ

= +

= + _ + _

∴ = + _ + _

= + − + +

=

∫

∫ ∫

∫

See [13].

( )

2 3

2! 3!

s s

G s s

ϑ β

 

= _ − + _

 

because

( )

2 3

0

d

2! 3!

s _u _u

G s u u

s

ϑ β

 

= _ − + + _

 

∫



Let 1

( ) ( )

1

( ) ( )

1 4

( )

0

d

u

t B t u B u e u

ψ

γ

+

∫



where 4

( )

2 2 2

1 0

d .

u

e u u u

(17)

Let

( )

(

)

( ) ( )

(

( )

)

2 3

0 0

1 1 1

0 0 0

1

1 e

d d

2! 3!

d d d

u

v v

s s s

v v

G u u u v

u

u B u u B u G u B s G s G u B u

B s G s s

β

ϑ _ϑ

υ γ

β β

ρ γ ρ ρ

ρ χ

−

−  

= = _ − + + _

 

 

= = _ − _

 

= −

∫

 

Let

( ) ( )

( )

3

( )

0 0

d d ;

s s

G u B u

χ

s B u +e s

∫



∫

where applying Wilk’s linear regression [14], we get

( )

(

)

3

2 2 2

0

2 3

0 0 0

0

d

1 e

d d d

d

2! 3!

s

e

t

s s u

s

s G u u s

G u u u t

t u u

s u u

s s s

β

σ χ

ϑ

β ϑ

χ

β

−

= −

−

 

= = =  − + + 

 

∫

∫ ∫

∫

