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

A three-factor exchange-rate diffusion model that includes three stochastically-dependent Brownian motion processes, namely, the domestic interest rate process, volatility process and return process is considered. A linear regression approach that derives explicit expressions 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 expression similar to a Black-Scholes model, which facilitates easier study, is discussed.


Introduction
A foreign exchange rate depends on the supply and demand dynamics of a currency.The exchange rate is a function of trade balance, the interest rate differential and differential inflation expectations between the two countries [1] [2].
Let S(u), 0 u ≥ = exchange rate process over the time interval: 0 u ≥ , where u = number of domestic currency units, e.g., $, per unit of foreign currency = $-price of foreign currency.
As interest rate ( ) D r 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 , 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 expression 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 return process.
In the next section, a linear regression approach that derives explicit expressions for the distribution function of ( ) 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 integrate 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.

Currency Spot Option
Given the spot rate ( ) where K is the known strike price and S s is the value of the exchange rate at the option's maturity price.The option to purchase foreign currency over the counter can be exercised when S(s) > the strike price exchange rate K.

A Diffusion Process Model
Equation ( 5) is obtained from Equation ( 2) by the application of Ito calculus [7].Assumption: ( ) ( ) ( ) , where ( ) ( ) ( ) , where From the assumption above, the return processes  where θ is the long-term mean and where ( ) Remark 1: From (8), choosing 0 θ > and that is small in value, we can make ( ) ( ) If, alternatively, we assume that has a square root process [8], then the random variable H(u) distribution is non-central ( )

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

( )
H u may be expressed as where See [9] for a similar assumption.See also [2] and [3].

Note that
( ) B s has a normal distribution with mean 0 and variance s, so ( ) B s can be written as , where ( ) Z s is a standard normal variable.Then ( ) ln X s can be written as a quadratic function of ( ) ( ) . {See Proposition 1 below}.
For ( ) ( ) ( ) ( ) ( ) ( ) We consider a mean-reverting Gaussian process model ( 2), the volatility stochastic processes and the processes, ( ) V u is a standard Brownian motion return process.In addition, in (3), we define the volatility as a mean reverting Gaussian process with θ as its long-term mean.
Assumption 1: ( ) ( ) ( ) , where In ( 4), we define the domestic interest rate process ( ) D r u as a mean revert- ing Gaussian process with λ as its long-term mean.The process is such that the return process ( ) V u is a correlated standard Brownian motion process to ( ) It follows from [2] that the distribution of ( ) ( ) Remark 3: From the expression for n s and the constant term ( ) and the constant terms ( ) , p s h with the addition of ( ) Then, using the results in [2], Proposition 1 and those in Appendix A and Appendix B we have: ; Note that ( ) n s in this paper is an updated version from the ( ) due to our treatment of a stochastic interest rate: ( ) Var e s is provided in (B1)

Cov e s e s i j i j = = ≠
Case 1:

s n s m s p s h z z h z h m s n s h h p s m s
, 2 0 n s m s p s h ξ ω − − − =, then the roots of the equation defined in (24) are equal so that , , then there exists a value ( ) In other words, is as follows:

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
where Given the formula for n s and the constant terms ( ) , p s h with the addition of ( )

Conclusion
We define a three-factor exchange-rate diffusion model with 1) stochastic volatility 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].

Unconditional Hedge Ratio
Unconditional Hedge Ratio

1 C
t are independent Brownian processes.
as the average standard deviation in the case of uncorrelated Brownian motion process 17)Now we use the results obtained in Proposition 1 to derive an explicit expression for Since we know the CDF of lnS(s) we can estimate the parameters of the underlying model (2)-(5).Case 2: Conditional Risk-neutral Distribution function of then the roots of the equation defined in (26) are equal so that Future movement of values of risk-free interest rate and volatility are uncertain and as they increase, they affect call option values as depicted in the above Figure2, Figure3 ([5], p. 204).Sudden changes in their values may occur because of economic shock.See the models suggested in[11] [12].

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

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