پرش سیستماتیک خطرات در یک اقتصاد کوچک باز : ارزیابی تعادل همزمان گزینه ها در پرتفوی بازار و نرخ ارز

The valuation of stock options and currency options has witnessed an explosion of new development in the past 20 years. These models, set up either in a partial equilibrium or a general equilibrium framework, have certainly enriched our understanding of option valuation in one way or the other. However, the main drawback of these models is that stock options and currency options are analyzed in separate contexts. The co-movement of the stock market and the currency market is absent from the option valuation analysis. Such co-movement is extremely important and is best illustrated by the Southeast Asian financial crisis. To overcome this drawback, this paper uses an equilibrium model to investigate the joint dynamics of the exchange rate and the market portfolio in a small open monetary economy with jump-diffusion money supplies and aggregate dividends. It is shown that the exchange rate and the market portfolio are strongly correlated since both are driven by the same economic fundamentals. Furthermore, options on the exchange rate and the market portfolio are evaluated in the same equilibrium context. The analysis shows that parameters describing the same economic fundamentals have very different effects on currency and stock options

Derivatives valuation has witnessed an explosion of new development in the past 20 years. Examples for stock option valuations include Black and Scholes (1973), Merton (1976), Cox and Ross (1976), Hull and White (1987), Bailey and Stulz (1989) and Naik and Lee (1990). Examples for currency option models include Biger and Hull (1983), Garman and Kohlhagen (1983), Grabbe (1983), Chesney and Scott (1989), Amin and Jarrow (1991), Heston (1993), Bates (1996) and Bakshi and Chen (1997). The references listed here are by no means exhaustive. These models, set up either in a partial equilibrium or a general equilibrium framework, have certainly enriched our understanding of option valuation in one way or the other. However, the main drawback of these models is that stock options and currency options are analyzed in separate contexts. The co-movement of the stock market and the currency market is absent from the option valuation analysis. Such co-movement is extremely important and is best illustrated by the recent Southeast Asian financial crisis, which has swamped small economies like Thailand, Indonesia, Malaysia and Korea. During the crisis, the dramatic currency devaluations were always accompanied by sharp decreases in their corresponding stock markets. As shown in Table 1, the 1997 average return on Southeast Asia's currency and the stock market is about -45%. The 1998 drastic devaluation of the Russia ruble and Russia's stock market only adds more evidence to the co-movement. Such evidence suggests that the stock market and the currency market are affected by the same fundamental economic factors. Failure to incorporate such simultaneous reactions to changes in the same fundamental economic factors would misguide the derivative valuations. Table 1. Summary of currency and stock index performance Country 1997 returns on currency 1997 returns on the stock index (%) (%) Thailand −45 −54 Indonesia −56 −37 Malaysia −35 −52 Korea −47 −38 Average −46 −45 Table options The second drawback of the existing models is best summarized by Jorion (1988, pp. 427-428): Many financial models rely heavily on the assumption of a particular stochastic process, while relatively little attention is paid to the empirical fit of the postulated distribution. As a result, models like option pricing models are applied indiscriminately to various markets such as the stock market and the foreign exchange market when the underlying processes may be fundamentally different. Obviously, the information arrival process in the foreign exchange market differs from that in the stock market, since exchange rates are directly influenced by monetary polices that do not have apparent counterparts in the stock market. It is important to directly investigate the effect of monetary policy changes on exchange rates and hence on currency options. Such analysis can only be carried out in a general equilibrium framework where the relation between exchange rates and monetary policies can be endogenized. In fact, indiscriminately applying the Black and Scholes (1973) formula to both stock options and currency options yields the opposite pricing bias pattern. The Black-Scholes formula generally overprices out-of-the-money stock call options and underprices in-the-money stock call options (MacBeth and Merville, 1979), but it usually underprices out-of-the-money currency calls (Bodurtha and Courtadon, 1987). Another problem of applying stock option models to currency options is that the assumptions for stock option models may not be valid for currency options. For example, a number of scholars, such as Bodurtha and Courtadon (1987), Jorion (1988) and Dumas et al. (1995), suggest that currency options should be priced with Merton's (1976) mixed jump–diffusion stock option model since jumps have been found in exchange rates.1 The key problem with this application is that the jump risk in Merton's model is assumed to be uncorrelated with the market. Such an assumption of uncorrelated jump risk may be reasonable if the concern were firm specific stocks, but is problematic for currency markets. Since the exchange rate reflects one nation's purchasing power relative to another nation, the exchange rate is inherently correlated with aggregate fundamental forces that affect the market. The main objective of this paper is to overcome these drawbacks, by simultaneously analyzing option valuations for the exchange rate and the market portfolio in a small open economy with systematic and non-systematic jump risks. I employ a continuous-time extension of the Lucas (1978) asset pricing model to a small open monetary economy, where money has a non-trivial role in the agents' utility function. Based on utility maximization, the equilibrium analysis endogenizes the precise relationship between the exchange rate and the market portfolio which are functions of the same fundamental forces. The explicit modelling of the relationship between the exchange rate and monetary policies also helps to uncover the distinct nature of the exchange rate process that differs from the stock price process. Under the logarithmic utility function, the equilibrium exchange rate, expressed as the relative price of foreign currency in terms of home currency, is affected by the domestic money supply, aggregate dividends and the level of investments in foreign assets. In contrast to the exchange rate, the real price of the domestic stock is affected by aggregate dividends and the level of investments in foreign assets. This equilibrium formulation also enables me to price options on the exchange rate and stock accordingly. Comparative analysis shows that currency options and stock options are affected differently by the parameters underlying economic fundamentals. In addition, this paper also addresses the analog of the so-called Siegel's paradox in currency option valuation with systematic jump risks, which is illustrated by Dumas et al. (1995). 2 The current model is obviously different from the existing partial equilibrium option models in which the exchange rate or the stock price is exogenous. As pointed out by Bailey and Stulz (1989), the arbitrary choice of the exogenous process for any security price in the partial equilibrium models is unlikely to be consistent with the equilibrium conditions or to provide important insights into how derivative prices may respond to changes in any fundamental economic variables. Though the current model shares the equilibrium approach with the existing equilibrium option models, the key difference is that the current model simultaneously analyzes currency option and stock option valuation in a consistent manner. Moreover, the focus here is on a small open economy, which is different from a closed pure-exchange economy as in Naik and Lee (1990) for stock option valuation, or a two-country setting as in Bakshi and Chen (1997) for currency option valuation. The remainder of this paper is organized as follows. Section 2 describes the economy and presents the equilibrium results. Section 3 examines the endogenized exchange rate and the price of the market portfolio, and derives equilibrium prices for European currency and stock options from the view of the domestic risk-averse agent. Section 4 identifies the adjustments on the risk-neutral process of the exchange rate that help to solve the analog of Siegel's paradox in currency options. Section 5 extends the model to allow for a correlation between the money supply and aggregate dividends. Section 6 concludes the paper and the appendices provide the necessary proofs.

Derivatives valuation has witnessed an explosion of new development in the past 20 years. Examples for stock option valuations include Black and Scholes (1973), Merton (1976), Cox and Ross (1976), Hull and White (1987), Bailey and Stulz (1989) and Naik and Lee (1990). Examples for currency option models include Biger and Hull (1983), Garman and Kohlhagen (1983), Grabbe (1983), Chesney and Scott (1989), Amin and Jarrow (1991), Heston (1993), Bates (1996) and Bakshi and Chen (1997). The references listed here are by no means exhaustive. These models, set up either in a partial equilibrium or a general equilibrium framework, have certainly enriched our understanding of option valuation in one way or the other. However, the main drawback of these models is that stock options and currency options are analyzed in separate contexts. The co-movement of the stock market and the currency market is absent from the option valuation analysis. Such co-movement is extremely important and is best illustrated by the recent Southeast Asian financial crisis, which has swamped small economies like Thailand, Indonesia, Malaysia and Korea. During the crisis, the dramatic currency devaluations were always accompanied by sharp decreases in their corresponding stock markets. As shown in Table 1, the 1997 average return on Southeast Asia's currency and the stock market is about -45%. The 1998 drastic devaluation of the Russia ruble and Russia's stock market only adds more evidence to the co-movement. Such evidence suggests that the stock market and the currency market are affected by the same fundamental economic factors. Failure to incorporate such simultaneous reactions to changes in the same fundamental economic factors would misguide the derivative valuations. Table 1. Summary of currency and stock index performance Country 1997 returns on currency 1997 returns on the stock index (%) (%) Thailand −45 −54 Indonesia −56 −37 Malaysia −35 −52 Korea −47 −38 Average −46 −45 Table options The second drawback of the existing models is best summarized by Jorion (1988, pp. 427-428): Many financial models rely heavily on the assumption of a particular stochastic process, while relatively little attention is paid to the empirical fit of the postulated distribution. As a result, models like option pricing models are applied indiscriminately to various markets such as the stock market and the foreign exchange market when the underlying processes may be fundamentally different. Obviously, the information arrival process in the foreign exchange market differs from that in the stock market, since exchange rates are directly influenced by monetary polices that do not have apparent counterparts in the stock market. It is important to directly investigate the effect of monetary policy changes on exchange rates and hence on currency options. Such analysis can only be carried out in a general equilibrium framework where the relation between exchange rates and monetary policies can be endogenized. In fact, indiscriminately applying the Black and Scholes (1973) formula to both stock options and currency options yields the opposite pricing bias pattern. The Black-Scholes formula generally overprices out-of-the-money stock call options and underprices in-the-money stock call options (MacBeth and Merville, 1979), but it usually underprices out-of-the-money currency calls (Bodurtha and Courtadon, 1987). Another problem of applying stock option models to currency options is that the assumptions for stock option models may not be valid for currency options. For example, a number of scholars, such as Bodurtha and Courtadon (1987), Jorion (1988) and Dumas et al. (1995), suggest that currency options should be priced with Merton's (1976) mixed jump–diffusion stock option model since jumps have been found in exchange rates.1 The key problem with this application is that the jump risk in Merton's model is assumed to be uncorrelated with the market. Such an assumption of uncorrelated jump risk may be reasonable if the concern were firm specific stocks, but is problematic for currency markets. Since the exchange rate reflects one nation's purchasing power relative to another nation, the exchange rate is inherently correlated with aggregate fundamental forces that affect the market. The main objective of this paper is to overcome these drawbacks, by simultaneously analyzing option valuations for the exchange rate and the market portfolio in a small open economy with systematic and non-systematic jump risks. I employ a continuous-time extension of the Lucas (1978) asset pricing model to a small open monetary economy, where money has a non-trivial role in the agents' utility function. Based on utility maximization, the equilibrium analysis endogenizes the precise relationship between the exchange rate and the market portfolio which are functions of the same fundamental forces. The explicit modelling of the relationship between the exchange rate and monetary policies also helps to uncover the distinct nature of the exchange rate process that differs from the stock price process. Under the logarithmic utility function, the equilibrium exchange rate, expressed as the relative price of foreign currency in terms of home currency, is affected by the domestic money supply, aggregate dividends and the level of investments in foreign assets. In contrast to the exchange rate, the real price of the domestic stock is affected by aggregate dividends and the level of investments in foreign assets. This equilibrium formulation also enables me to price options on the exchange rate and stock accordingly. Comparative analysis shows that currency options and stock options are affected differently by the parameters underlying economic fundamentals. In addition, this paper also addresses the analog of the so-called Siegel's paradox in currency option valuation with systematic jump risks, which is illustrated by Dumas et al. (1995). 2 The current model is obviously different from the existing partial equilibrium option models in which the exchange rate or the stock price is exogenous. As pointed out by Bailey and Stulz (1989), the arbitrary choice of the exogenous process for any security price in the partial equilibrium models is unlikely to be consistent with the equilibrium conditions or to provide important insights into how derivative prices may respond to changes in any fundamental economic variables. Though the current model shares the equilibrium approach with the existing equilibrium option models, the key difference is that the current model simultaneously analyzes currency option and stock option valuation in a consistent manner. Moreover, the focus here is on a small open economy, which is different from a closed pure-exchange economy as in Naik and Lee (1990) for stock option valuation, or a two-country setting as in Bakshi and Chen (1997) for currency option valuation. The remainder of this paper is organized as follows. Section 2 describes the economy and presents the equilibrium results. Section 3 examines the endogenized exchange rate and the price of the market portfolio, and derives equilibrium prices for European currency and stock options from the view of the domestic risk-averse agent. Section 4 identifies the adjustments on the risk-neutral process of the exchange rate that help to solve the analog of Siegel's paradox in currency options. Section 5 extends the model to allow for a correlation between the money supply and aggregate dividends. Section 6 concludes the paper and the appendices provide the necessary proofs.