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A note on currency option pricing
A Note on Currency Option Pricing
SANJAY K. NAWALKHA and DONALD R. CHAMBERS
This paper clarifies and extends recent currency option pricing research that attempts to incorporate stochastic domestic and foreign interest rates. We give an alternative perspective on the currency option pricing model of Hilliard, Madura and Tucker (1991) by identifying it with the constant forward rate volatility term structure model of Heath, Jarrow and Morton (1992). The arithmetic random walk process for the instantaneous short rate, considered by Hilliard, Madura and Tucker (199 1) for the empirical testing of their currency option pricing model, is shown to imply an unreasonable shape for the term structure of interest rates. We show that the pricing of currency options can be consistent with an initially observed term StrUCNre of interest rates (both domestic and foreign) and independent of investors’ preferences.
I. INTRODUCTION AND SUMMARY Currency options may be viewed as options to exchange a domestic bond and a foreign bond. Early work on currency option pricing models includes the work of Biger and Hull (1983), Garman and Kohlhagen (1983) and Grabbe (1983). These studies produced the widely recognized and seminal currency option pricing model found in most texts on options. The traditional currency option pricing model assumes, unfortunately, that the interest rate on the domestic currency and the interest rate on the foreign currency are both non-stochastic. Recently, research has been increasingly focused on currency option pricing models that incorporate stochastic domestic and foreign term structures in option pricing. The purpose of this paper is to extend recent research on currency options. We attempt to clarify and extend the currency option pricing model of Hilliard, Madura and Tucker (1991) by demonstrating that it can be made consistent with initially observed term structures of interest rates (both domestic and foreign).
II.
THE HILLIARD, MADURA AND TUCKER MODEL
Recently, Hill&d, Madura and Tucker (HMT) (1991) derived a currency option pricing model assuming stochastic domestic and foreign interest rates. By constructing a risk-free hedge following Grabbe (1983), invoking the risk neutrality argument of Cox and Ross (1976), and by identifying Vasicek’s (1977) term structure model as the appropriate bond pricing model,
Sanjay K. Nawalkha
l
Assistant Professor, Department of Economics
N. Charles St., Baltimore, MD 21201; Donald R. Chambers of Finance, Lafayette
l
and Finance, University of Baltimore,
Walter E. Hanson/KPMG
1420
Peat Marwick Professor
College, 208 Simon Center, Easton, PA, 18042-1776.
International Review of Financial Analysis, Vol. 4, No. 1,1995, pp. 81-84. Copyright 0 1994 by JAI Press Inc., All rights of reproduction
ISSN: 1057-5219
in any form reserved.
81
82
INTERNATIONAL
REVIEW OF FINANCIAL ANALYSIS / Vol. 4(l)
HMT derive a closed-form currency option pricing model under stochastic interest rates. For the empirical testing of their model, HMT consider a special case of Vasicek (1977) by assuming that both the domestic and foreign instantaneous short rates follow a random arithmetic walk of the following form: h(r) = &Z(t),
(1)
where dZ(t) is a standard Wiener process and B > 0 is a constant. The above process is a special case of the Ornstein-Uhlenbeck process given by Vasicek (1977), assuming that the speed of mean reversion equals zero. HMT (Equation 22,199l) claim that, under this special case, the time t price of a bond maturing at a future date T can be given in closed-form as follows:
W,7l = exp(-2 4th where T = T - t, and r(t) is the instantaneous
(2)
short rate at time t.
HMT justify using the short-rate process given by equation (1) for the empirical testing of their model, since it avoids the non-linear estimation of the coefficient of the speed of mean reversion contained in Vasicek’s original model. Unfortunately, it can be demonstrated that the bond price specification given in equation (2) is not the correct closed-form solution consistent with the short-rate process given by equation (1). A direct simplification of equation (27) of Vasicek (1977) shows that, under zero mean reversion, the bond price is given in closed-form as follows: B(t,T) = exp[-‘t r(t) - 012*/2 + 02t3/6],
(3)
where Eis the market price as in Vasicek (1977). Equation (3) is the correct closed-form solution of the bond price when the instantaneous short rate follows a random arithmetic walk (equation (1)). A similar solution for the bond price has been given by Met-ton (1970), assuming that the instantaneous short rate follows an arithmetic brownian motion with constant drift and a constant diffusion coefficient. Note that there is a significant difference between the bond price specifications given by equation (2) and equation (3). According to equation (3), as ‘t tends to infinity, the bond price tends to infinity and not zero. A similar unreasonable result is noted by Merton (1970) and Ingersoll, Skelton and Weil(1978), because of the occurrence of negative interest rates. Below, we identify the currency option pricing model of HMT with the Ho and Lee (1986) term structure model in its continuous time limit given by Heath, Jarrow and Morton (HJM) (1992). Although the Ho and Lee (1986) model allows the occurrence of negative interest rates, it is consistent with an initially observed term structure of interest rates. As shown by HJM, in the continuous-time limit, the Ho and Lee model reduces to a constant forward rate volatility term structure model. The constant forward rate volatility term structure model leads to a currency option pricing formula identical to the one originally proposed by HMT with respect to the arithmetic random process for the short rate, except that the domestic and foreign bond pricing function are consistent with the corresponding initially observed term structures of interest rates.
83
A Note on Currency Option Pricing
Assuming that certain regularity conditions (Conditions 2 and 3, HJM) are satisfied, the domestic and the foreign instantaneous short-rate processes consistent with the constant forward rate volatility model of HJM can be given as follows: dq(t) = @I dt + 61 dZl(t), drz(t) = p2 dt + 62 dZz(t),
(4)
where the subscripts 1 and 2 denote the domestic and foreign variables, l_tX= dfx(O,t)/dt - oxgx(t) + o$, for x = 1, 2,f(O,t) is the instantaneous forward rate for term t, and g(t) is the market price of risk as defined in HJM. The essential difference between the arithmetic random walk model and the constant forward rate volatility term structure model of HJM can be seen by comparing equation (1) with equation (4). When the short rate follows an arithmetic random walk given by equation (l), the drift term is zero such that it implies a specific bond price function given by equation (3). On the other hand, for the constant forward rate volatility term structure model, the drift term for the short-rate process is assumed to be time-dependent (see equation (4)) such that the model is consistent with an initially observed term structure of forward rates, which is independent of the market price of risk. Of course, the contemporaneous covariance structure of the domestic bond return, the foreign bond return, and the spot rate process is identical under the arithmetic random walk model and the constant forward rate volatility model; therefore, equation (4) leads to the same currency option pricing formula as obtained by HMT using equation (1).
REFERENCES Biger, N., & Hull, J. (1983). The valuation of currency options. Financial Management, 12 (Spring), 24-28. Cox, J., & Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3 (Jan/March), 145-166. Garman, M., & Kohlhagen, S. (1983). Foreign currency option values. Journal of International Money and Finance, 2,231-237. Grabbe, 0. (1983). The pricing of call and put options on foreign exchange. Journal of International Money and Finance,2 (December), 239-253. Heath, D., Jarrow, R., & Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 60 (January), 77-105. Hilliard, J. E., Madura, J., & Tucker, A. (1991). Currency option pricing with stochastic domestic and foreign interest rates. Journal of Financial and Quantitative Analysis, 26 (June), 139-151. Ho, T. S., & Lee, S. (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41 (December), 1011-1030. Ingersoll, J. E., Skelton, J., & Weil, R. L. (1978). Duration forty years later. Journal ofFinancial and Quantitative Analysis, 13 (November), 627-650.
84
INTERNATIONAL
REVIEW OF FINANCIAL ANALYSIS/Vol.
4(l)
Merton, R. C. (1970). A dynamic general equilibrium model of the asset market and its application to thepricing of the capital structure of thejrm. Working Paper, Massachusetts Institute of Technology. Vasicek, 0. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177-188.
SANJAY K. NAWALKHA and DONALD R. CHAMBERS
This paper clarifies and extends recent currency option pricing research that attempts to incorporate stochastic domestic and foreign interest rates. We give an alternative perspective on the currency option pricing model of Hilliard, Madura and Tucker (1991) by identifying it with the constant forward rate volatility term structure model of Heath, Jarrow and Morton (1992). The arithmetic random walk process for the instantaneous short rate, considered by Hilliard, Madura and Tucker (199 1) for the empirical testing of their currency option pricing model, is shown to imply an unreasonable shape for the term structure of interest rates. We show that the pricing of currency options can be consistent with an initially observed term StrUCNre of interest rates (both domestic and foreign) and independent of investors’ preferences.
I. INTRODUCTION AND SUMMARY Currency options may be viewed as options to exchange a domestic bond and a foreign bond. Early work on currency option pricing models includes the work of Biger and Hull (1983), Garman and Kohlhagen (1983) and Grabbe (1983). These studies produced the widely recognized and seminal currency option pricing model found in most texts on options. The traditional currency option pricing model assumes, unfortunately, that the interest rate on the domestic currency and the interest rate on the foreign currency are both non-stochastic. Recently, research has been increasingly focused on currency option pricing models that incorporate stochastic domestic and foreign term structures in option pricing. The purpose of this paper is to extend recent research on currency options. We attempt to clarify and extend the currency option pricing model of Hilliard, Madura and Tucker (1991) by demonstrating that it can be made consistent with initially observed term structures of interest rates (both domestic and foreign).
II.
THE HILLIARD, MADURA AND TUCKER MODEL
Recently, Hill&d, Madura and Tucker (HMT) (1991) derived a currency option pricing model assuming stochastic domestic and foreign interest rates. By constructing a risk-free hedge following Grabbe (1983), invoking the risk neutrality argument of Cox and Ross (1976), and by identifying Vasicek’s (1977) term structure model as the appropriate bond pricing model,
Sanjay K. Nawalkha
l
Assistant Professor, Department of Economics
N. Charles St., Baltimore, MD 21201; Donald R. Chambers of Finance, Lafayette
l
and Finance, University of Baltimore,
Walter E. Hanson/KPMG
1420
Peat Marwick Professor
College, 208 Simon Center, Easton, PA, 18042-1776.
International Review of Financial Analysis, Vol. 4, No. 1,1995, pp. 81-84. Copyright 0 1994 by JAI Press Inc., All rights of reproduction
ISSN: 1057-5219
in any form reserved.
81
82
INTERNATIONAL
REVIEW OF FINANCIAL ANALYSIS / Vol. 4(l)
HMT derive a closed-form currency option pricing model under stochastic interest rates. For the empirical testing of their model, HMT consider a special case of Vasicek (1977) by assuming that both the domestic and foreign instantaneous short rates follow a random arithmetic walk of the following form: h(r) = &Z(t),
(1)
where dZ(t) is a standard Wiener process and B > 0 is a constant. The above process is a special case of the Ornstein-Uhlenbeck process given by Vasicek (1977), assuming that the speed of mean reversion equals zero. HMT (Equation 22,199l) claim that, under this special case, the time t price of a bond maturing at a future date T can be given in closed-form as follows:
W,7l = exp(-2 4th where T = T - t, and r(t) is the instantaneous
(2)
short rate at time t.
HMT justify using the short-rate process given by equation (1) for the empirical testing of their model, since it avoids the non-linear estimation of the coefficient of the speed of mean reversion contained in Vasicek’s original model. Unfortunately, it can be demonstrated that the bond price specification given in equation (2) is not the correct closed-form solution consistent with the short-rate process given by equation (1). A direct simplification of equation (27) of Vasicek (1977) shows that, under zero mean reversion, the bond price is given in closed-form as follows: B(t,T) = exp[-‘t r(t) - 012*/2 + 02t3/6],
(3)
where Eis the market price as in Vasicek (1977). Equation (3) is the correct closed-form solution of the bond price when the instantaneous short rate follows a random arithmetic walk (equation (1)). A similar solution for the bond price has been given by Met-ton (1970), assuming that the instantaneous short rate follows an arithmetic brownian motion with constant drift and a constant diffusion coefficient. Note that there is a significant difference between the bond price specifications given by equation (2) and equation (3). According to equation (3), as ‘t tends to infinity, the bond price tends to infinity and not zero. A similar unreasonable result is noted by Merton (1970) and Ingersoll, Skelton and Weil(1978), because of the occurrence of negative interest rates. Below, we identify the currency option pricing model of HMT with the Ho and Lee (1986) term structure model in its continuous time limit given by Heath, Jarrow and Morton (HJM) (1992). Although the Ho and Lee (1986) model allows the occurrence of negative interest rates, it is consistent with an initially observed term structure of interest rates. As shown by HJM, in the continuous-time limit, the Ho and Lee model reduces to a constant forward rate volatility term structure model. The constant forward rate volatility term structure model leads to a currency option pricing formula identical to the one originally proposed by HMT with respect to the arithmetic random process for the short rate, except that the domestic and foreign bond pricing function are consistent with the corresponding initially observed term structures of interest rates.
83
A Note on Currency Option Pricing
Assuming that certain regularity conditions (Conditions 2 and 3, HJM) are satisfied, the domestic and the foreign instantaneous short-rate processes consistent with the constant forward rate volatility model of HJM can be given as follows: dq(t) = @I dt + 61 dZl(t), drz(t) = p2 dt + 62 dZz(t),
(4)
where the subscripts 1 and 2 denote the domestic and foreign variables, l_tX= dfx(O,t)/dt - oxgx(t) + o$, for x = 1, 2,f(O,t) is the instantaneous forward rate for term t, and g(t) is the market price of risk as defined in HJM. The essential difference between the arithmetic random walk model and the constant forward rate volatility term structure model of HJM can be seen by comparing equation (1) with equation (4). When the short rate follows an arithmetic random walk given by equation (l), the drift term is zero such that it implies a specific bond price function given by equation (3). On the other hand, for the constant forward rate volatility term structure model, the drift term for the short-rate process is assumed to be time-dependent (see equation (4)) such that the model is consistent with an initially observed term structure of forward rates, which is independent of the market price of risk. Of course, the contemporaneous covariance structure of the domestic bond return, the foreign bond return, and the spot rate process is identical under the arithmetic random walk model and the constant forward rate volatility model; therefore, equation (4) leads to the same currency option pricing formula as obtained by HMT using equation (1).
REFERENCES Biger, N., & Hull, J. (1983). The valuation of currency options. Financial Management, 12 (Spring), 24-28. Cox, J., & Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3 (Jan/March), 145-166. Garman, M., & Kohlhagen, S. (1983). Foreign currency option values. Journal of International Money and Finance, 2,231-237. Grabbe, 0. (1983). The pricing of call and put options on foreign exchange. Journal of International Money and Finance,2 (December), 239-253. Heath, D., Jarrow, R., & Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 60 (January), 77-105. Hilliard, J. E., Madura, J., & Tucker, A. (1991). Currency option pricing with stochastic domestic and foreign interest rates. Journal of Financial and Quantitative Analysis, 26 (June), 139-151. Ho, T. S., & Lee, S. (1986). Term structure movements and pricing interest rate contingent claims. Journal of Finance, 41 (December), 1011-1030. Ingersoll, J. E., Skelton, J., & Weil, R. L. (1978). Duration forty years later. Journal ofFinancial and Quantitative Analysis, 13 (November), 627-650.
84
INTERNATIONAL
REVIEW OF FINANCIAL ANALYSIS/Vol.
4(l)
Merton, R. C. (1970). A dynamic general equilibrium model of the asset market and its application to thepricing of the capital structure of thejrm. Working Paper, Massachusetts Institute of Technology. Vasicek, 0. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177-188.