A Linear Regression Approach for Determining Option Pricing for Currency-Rate Diffusion Model with Dependent Stochastic Volatility, Stochastic Interest Rate, and Return Processes ()
1. 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),
= exchange rate process over the time interval:
, where u = number of domestic currency units, e.g., $, per unit of foreign currency = $-price of foreign currency.
As interest rate
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 interest-rate parity condition we have
, 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
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 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.
2. Currency Spot Option
Given the spot rate
, consider the present value of option
(1)
where K is the known strike price and
is a mean-reverting stochastic process given in (2) below.
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.
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
(3) and domestic interest rate
process (4), and foreign interest rate
is a known constant.
(2)
(3)
(4)
(5)
Equation (5) is obtained from Equation (2) by the application of Ito calculus [7] .
Assumption:
, where
and
and
are independent Brownian processes.
, where ![]()
and
and
are independent Brownian processes. (6)
From the assumption above, the return processes
are correlated with
and that
are standard Brownian motion processes.
Then it follows, see [2] [3] , that the distributions of
and
are Gaussian processes.
Alternatively,
and
may be expressed as:
(7)
![]()
where
is the long-term mean and where
.
(8)
Remark 1:
From (8), choosing
and that is small in value, we can make
negligible.
If, alternatively, we assume that
has a square root process [8] , then the random variable H(u) distribution is non-central
. For simplicity we chose the mean-reverting process model (3).
![]()
![]()
where
and ![]()
(9)
Assuming
, and
(10)
The Brownian motion processes
and
are as follows:
, where
(11)
In addition, the Brownian motion processes
and
under Q are independent.
Remark 2:
: the volatility process.
It follows from [2] that the distribution of
is:
(12)
Alternatively,
may be expressed as
(13)
where
and
.
See [9] for a similar assumption. See also [2] and [3] .
Note that
has a normal distribution with mean 0 and variance s, so
can be written as
, where
is a standard normal variable. Then
can be written as a quadratic function of
plus a residual term
. {See Proposition 1 below}.
For
, we define a volatility process
.
Define
, as the average standard
deviation in the case of uncorrelated Brownian motion process
[See [10] , p. 182].
Proposition 1:
![]()
(14)
where
(15)
Proof: See Appendix B.
We consider a mean-reverting Gaussian process model (2), the volatility stochastic processes
and the processes,
and
in (3) to be correlated; where
is a standard Brownian motion return process. In addition, in (3), we define the volatility
as a mean reverting Gaussian process with
as its long-term mean.
Assumption 1:
(16)
In (4), we define the domestic interest rate process
as a mean reverting Gaussian process with
as its long-term mean. The process
is such that the return process
is a correlated standard Brownian motion process to
. The foreign interest rate
is a constant ![]()
Assumption 2:
It follows from [2] that the distribution of
:
(17)
Now we use the results obtained in Proposition 1 to derive an explicit expression for
![]()
Proposition 2:
(18)
Remark 3:
From the expression for
. the stochastic terms
modifies
and the constant term
modifies
with the addition of
and the constant terms
modifies
with the addition of
.
Then, using the results in [2] , Proposition 1 and those in Appendix A and Appendix B we have:
(19)
Therefore,
![]()
![]()
![]()
Remark 4:
Note that
in this paper is an updated version from the
in [2] ,
due to our treatment of a stochastic interest rate: ![]()
![]()
![]()
In the case of
.
, where
(20)
(21)
where
(22)
![]()
![]()
is provided in (B1)
![]()
Case 1:
;
![]()
Let ![]()
![]()
Let
.
Assumption 3:
and
are independent random variables.
Assumption 4:
.
Assumption 5:
and
.
If Assumptions (4) and (5) hold, then the conditional risk-neutral distribution of
is:
Proposition 3:
![]()
(23)
where
(24)
If
, then the roots of the equation defined in (24) are equal so that
, then there exists a value
such that
.
In other words,
is the lowest value for the conditional random variable
.
Remark 5:
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
,
. Suppose
, Conditional risk-neutral distribution of
is as follows:
![]()
![]()
where
![]()
Example 1
![]()
:
:
Then
![]()
Remark 6:
From the expression for
![]()
the stochastic terms
modify the term
and the constant terms
modifies
with the addition of
.
Proof:
Apply a proof similar to the one in Appendix A of [2] using the result for
in Appendix B of the current paper. See also Proposition 4.
Remark 7:
Assume
, which implies that
.
If Assumption (3) holds then the conditional risk-neutral distribution of
is:
(25)
where
(26)
If
, then the roots of the equation defined in (26) are equal so that
, then there exists a value
such that
.
In other words,
is the lowest value of the conditional random variable
.
Call option price:
![]()
Proposition 4:
![]()
where from Proposition 1
![]()
See Appendix B.
![]()
Remark 8:
Given the formula for
, the stochastic expression
modifies the function
and the constant terms
, modifies
with the addition of
.
(27)
Let
![]()
(28)
Then
![]()
Hedge Ratio:
![]()
D-Neutral Portfolio
Delta-Neutral Portfolio
Consider the following portfolio that includes a short position of one European call and a long position of delta units of the domestic currency.
The portfolio of delta-neutral positions is defined as:
![]()
We obtain below Conditional Risk-neutral Distribution function of
(29)
by considering the cases of: h = 1, 0 and −1
We use a discrete approximation (see [2] , (28)).
Suppose
, which implies
.
Again, we consider the Equations (1)-(4) to define Example 1 below.
(30)
(31))
(32)
(33)
Let
![]()
Then,
:
And
:
If Assumption (2) holds then the unconditional risk-neutral distribution of
and
are independent random variables.
Then Figure 1 depicts the unconditional risk-neutral distribution of
.
Remark 9:
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 Figure 2, Figure 3 ( [5] , p. 204). Sudden changes in their values may occur because 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.
![]()
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 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] . The results are applicable to bond option models ( [5] , p. 783).
Appendix A
![]()
![]()
is the regression coefficient.
![]()
(2A1)
![]()
Then the regression equation is
(2A2)
Assumption 6:
(Approximately) (2A3)
Note that
and
.
![]()
Assumption 7:
(Approximately)
(2A4)
Proof of Proposition 1:
![]()
![]()
Appendix A from [2]
![]()
where
![]()
![]()
where
![]()
Appendix B
![]()
See [13] .
![]()
because
![]()
Let ![]()
where ![]()
Let
![]()
Let
![]()
where applying Wilk’s linear regression [14] , we get
(B1)