_{1}

^{*}

This paper presents a theoretical model to price foreign currency call options. Currency options are employed in international trade to reduce the risk of loss due to the reduction of revenues obtained in depreciating foreign currency for an exporter, or the escalation of expense from appreciating foreign currency for an importer. Other users include banks and hedge funds who engage in currency speculation. Given the fluctuation of option prices over time, the model describes the distribution of foreign currency as a Weiner process for macroeconomically constrained foreign currencies followed by a Laplace distribution for unconstrained currencies. In a departure from existing currency option models, this model expresses foreign currencies as dependent upon the change in macroeconomic variables, such as inflation, interest rates, and government deficits. The distribution of currency calls is described as a Levy process in the context of an option trader’s risk preferences to account for the multiple discontinuities of a jump process. The paper concludes with three models of price functions of the Weiner process for Euro-related currency options, a Weiner process for stable currency options, and a Levy-Khintchine process for volatile currency calls.

A foreign currency call option permits a buyer to purchase foreign currency at the exercise price for a specific time period for an option premium paid to a call writer. To an exporter facing appreciating foreign currency, the exercise of an option limits foreign currency losses given the purchase of the currency at a cheaper rate. The benefit over purchasing a futures or forward contract is that an option is a nonbinding obligation [

This paper proposes a theoretical model to price currency call options. The model contributes to the literature in three ways. It recognizes that an option is a derived instrument. In itself, the option has minimal value. Its true value is based on the value of the underlying asset, the foreign currency. As the value of an option is partly due to the change in price of the option over its life until expiration, we first present the time distribution of the foreign currency. Countries within the European Union are constrained to maintain stability in foreign exchange rates, whereas those in other countries are not held to any restrictions. This paper presents Weiner processes as two constrained foreign currency distributions, and the Laplace distribution as the trajectory of unconstrained exchange rates. This is in contrast to option-only models ( [

The remainder of this paper is organized as follows: Section 2 is a Review of Literature with a description of Macroeconomic Determinants of Exchange Rates and literature on the Distributions of Foreign Currencies and Foreign Currency Options, Section 3 is a Quantitative Model of Foreign Currency Distributions, Section 4 is a Quantitative Model of Foreign Currency Option Distributions, Section 5 provides a Quantitative Model of Pricing Foreign Currency Options, and Section 6 describes Conclusions and Recommendations for Future Research.

Inflation Rates, Interest Rates, Current Account, and Government Debt. The relative Purchasing Power Parity theory maintains that countries with higher inflation rates experience currency depreciation. As domestic prices rise, each unit of currency diminishes in value, necessitating the use of more units of domestic currency to purchase foreign goods. Consequently, the domestic currency will depreciate ( [

A country with high government debt will be challenged to retain foreign capital. Foreign investors will sell their government bonds depreciating the exchange rate. [

Terms of Trade, Political Stability and Speculation. The Harrod-Balassa-Samuelson model posits that the rise in productivity of manufactured exports leads to higher wages. To compete for labor, businesses producing nontradable goods raise wages. The rise in prices of both exports and nontradables results in an increase in currency values, ceteris paribus. [

Additionally, a country with less political conflict attracts foreign investors, increasing its exchange rate. Speculation exists in that [

Stochastic foreign currency distributions were historically considered to be similar to distributions describing the movements of stocks. Consequently, the first class of distributions was the stationary Paretian stable or Student t distribution, as in [

We conclude that a distribution that accounts for leptokurtosis with frequent jumps regularly rather than during a few time periods may be the optimal choice. The [

How do currency option distributions differ from currency distributions? A call option on foreign currency, for a small investment, promises substantial upside potential, beyond the profits from the increases in the foreign currency. The higher profit potential suggests higher risk for the option. The arrival rate of information is lumpy and discontinuous. Option volumes contain forecasts of future events that affect currency values including tariffs, capital flight, or tax cuts which increase government debt. As option trading firms use superior forecasting tools and have access to the expertise of seasoned traders, they obtain accurate forecasts of macroeconomic variables. Positive news suggests that the exchange rate will rise above the forecasted forward rate, while negative news indicates an actual exchange rate below the forward rate. This may explain the prevalence of small jumps as in the 1980s with U.S. dollar depreciation. In anticipation of dollar depreciation, the distribution of currency exchange rates became skewed to the left [

Certain exchange rates had a unique reaction to macroeconomic news, such as inflation announcements in a low inflationary environment, creating a small downward jump in dollar values, while in a high-inflationary environment, such an announcement would cause a large jump in currency values. Such disparate reactions are captured in fat-tailed leptokurtic distributions. In the 1980s, skewness and kurtosis were captured by a drift term in a Brownian motion along with jumps ( [

We surmise that option distributions are dependent upon foreign currency distributions, yet differ from currency distributions in that the jumps are steeper―a fact that has been empirically observed, but not accounted for theoretically in the literature. Further, the literature does not distinguish between the more stable and volatile foreign currencies. We posit that stable and volatile currencies will have different distributions, and create separate models for them.

Σ)σ^{2}

A continuous normal distribution function may be described as,

f ( x | μ , σ ) = 1 / 2 Π V a r exp − [ ( x − μ ) / 2 V a r 2 ] (1)

Applying the European Union’s requirements, x transforms to x_{1}, x_{2}, and x_{3}, where, x_{1} = inflation rate, x_{2} = long-term interest rate, x_{3} = government debt.

f ( μ σ ) = ( 2 Π σ 1 ) exp − 1 exp − ( x 1 − μ 1 ) / 2 σ 1 exp 2 ] exp 2 + 1 / 2 Π σ 2 exp 2 − ( x 2 − μ 2 ) / 2 σ 2 exp 2 ] exp 2 + 1 / 2 Π σ 3 exp 2 ] exp 2 (2)

Taking the first derivative of Equation (1),

f ′ ( μ σ ) = − ( x − μ ) 2 / 2 σ 2 ( 1 / 2 Π σ 2 ) = ( x − μ ) 2 / 2 2 Π σ 3 (3)

where, σ^{3} = skewness,

Expanding to the first derivative of equation (2) yields,

f ′ ( x | μ σ ) = − ( x 1 t − μ 1 t ) 2 / 2 2 Π σ 1 t 3 − ( x 2 t − μ 2 t ) 2 / 2 2 Π σ 2 t 3 − ( x 3 t − μ 3 t ) 2 / 2 2 σ 3 t 3 (4)

where, σ 1 t 3 = skewness due to the Euro being affected by an unexpected increase or decrease in the inflation rate, σ 2 t 3 = skewness due to the Euro being affected by an unexpected increase or decrease in the long-term interest rate, σ 3 t 3 = skewness due to the Euro being affected by an unexpected increase or decrease in government debt.

Given the narrow band within which macroeconomic variables in European Union member states must vary, any unexpected changes in inflation rates, interest rates, or government debt that move these values outside of the band will be forced into the band, so that the three forms of skewness and kurtosis will theoretically tend to 0. If the function at time t = 1, is extended to all time periods, from t = 1, through t = n.

Distribution of the Euro for all time periods,

∫ [ − ( x 1 t − μ 1 t ) 2 / 2 2 Π σ 1 t 3 − ( x 2 t − μ 2 t ) 2 / 2 2 Π σ 2 t 3 − ( x 3 t − μ 3 t ) 2 / 2 2 σ 3 t 3 ] (5)

with inflation rates, long-term interest rates, and government debt approaching their means, the variation of these variables about their means may be described by the variation of a series about a point (a, b), which is the following Taylor series expansion generalized to three variables for a single period t,

f ( a , b ) + ( x 1 t − a ) f x 1 t ( a , b ) + ( x 2 t − b ) f x 2 t ( a , b ) + ( x 3 t − b ) f x 3 t ( a , b ) + 1 / 2 ! [ ( x 1 t − a ) 2 f x 1 t x 2 t ( a , b ) + 2 ( x 1 t − a ) ( x 2 t − b ) f x 1 x 2 ( a , b ) + ( x 2 t − b ) 2 f x 2 x 2 ( a , b ) ] + 1 / 2 ! [ ( x 2 t − a ) 2 f x 2 x 2 ( a , b ) + 2 ( x 2 t − a ) ( x 3 t − b ) f x 2 x 3 ( a , b ) + ( x 3 t − b ) 2 f x 2 x 3 ( a , b ) ] + 1 / 2 ! [ ( x 1 t − a ) 2 f x 1 x 1 ( a , b ) + 2 ( x 1 t − a ) ( x 3 t − b ) f x 1 x 3 ( a , b ) + ( x 3 t − b ) 2 f x 3 x 3 ( a , b ) ] (6)

with the subscripts representing partial derivatives ( [

T ( x ) = f ( a ) + ( x − a ) T D f ( a ) + 1 / 2 ! ( x − a ) T { D 2 f ( a ) } ( x − a ) (7)

To minimize the deviation of x_{1}, x_{2}, and x_{3} from their means, the Hessian matrix D^{2}f(a) = 0, is obtained through an iterative process in which the gradient of f at point a D(f)a = 10^{−9} in the final iteration of a linear programming model presented in Equation (8), If (5) = (6) or, we may solve the Taylor series expansion (6) by maximizing the value of the Euro as a product of its value and the distribution presented in (5), Maximize,

C ∗ [ − ( x 1 t − μ 1 t ) 2 / 2 2 Π σ 1 t 3 − ( x 2 t − μ 2 t ) 2 / 2 2 Π σ 2 t 3 − ( x 3 t − μ 3 t ) 2 / 2 2 σ 3 t 3 ] (8)

s. t.

x 1 t − μ 1 t ≤ 0.5

x 2 t − μ 2 t ≤ 0.7 ^{ }

x 3 t − μ 3 t < 0.25 ^{ }

x 1 , x 2 , x 3 > 0

For the Japanese yen, Australian dollar, or Canadian dollar, there is no band limiting the movement of exchange rates from the mean. Therefore, foreign currency values will vary with respect to changes in up to 7 macroeconomic variables. However, given the central banks’ adherence to a policy of price stability, inflation rates, and long-term riskless rates may remain within 1 standard deviation of the mean. Accordingly, the movement of exchange rates may be described by a Weiner process with a drift term, ( [_{1t} = change in the inflation rate, x_{2t} = change in the short-term interest rate; Rate on a riskless discount bond of the same maturity as the holding period of foreign currency of <1 year, x_{3t} = change in the long-term interest rate; Rate of a riskless discount bond of the same maturity as the holding period of foreign currency of >1 year, x_{4t} = change in government debt, Pt + ∆t(Pt + ∆tPPp), x_{5t} = change in export prices, x_{6t} =change in import prices, x_{7t} =varying levels of political stability. Beginning at any point in time t, the foreign currency value X_{t}, varies in an infinitesimal time interval, Δt. The expected probability of movement of the foreign currency value from X_{t} to X_{t}_{+Δt}, is in direct proportion to the change in each macroeconomic variable, x_{t} to x ′ t , in an adaptation of the literature [

E ( p ( X t + 3 t ) ) = ∫ p ( x 1 t P t + Δ t ( x 1 t | x t ) ) d x 1 (9)

Multiply by ℙ t , t ′ ( x t | x ′ t ) , and find the sum over all changes in currency values over a time period for all macroeconomic variables,

∫ p ( x 1 t ) ∫ P t + Δ t ( x 1 t | x t ) P t t ′ ( x t | x ′ t ) d x d x 1 − p ( x t ) P t t ′ ( x t | x ′ t ) d x 1 + ∫ p ( x 2 t ) ∫ P t + Δ t ( x 2 t | x t ) P t t ′ ( x t | x ′ t ) d x d x 2 − ∫ p ( x t ) P t t ′ ( x t | x ′ t ) d x t + p ( x 3 t ) ∫ P t + Δ t ( x 3 | x t ) P t t ′ ( x t | x ′ t ) d x d x 3 − ∫ p ( x t ) P t t ′ ∫ t + Δ t ( x 5 t | x t ) + ⋯ + ∫ p ( x 7 t ) ∫ P t + Δ t ( x 7 t | x t ) P t t ′ ( x t | x ′ t ) d x d x 7 − ∫ p ( x t ) / P t t ′ ( x t | x ′ t ) d x t (10)

Assuming that as currency values change from x_{t} to x_{t}, inflation adjusts, so that the inflation rate is described by the higher currency value, x_{t}. The product, ∫ P t + Δ t ( x 1 t | x t ) P t t ′ ( x t | x ′ t ) d x = P t + Δ ′ t ( x 1 | x t ) from Equation (10) transforms to:

∫ p ( x t ) ∫ P t + Δ t ( x t | x ′ t ) − ∫ p ( x t ) P t t ′ ( x t | x ′ t ) d x t + ∫ p ( x t ) ∫ P t + Δ t ( x t | x ′ t ) x t − ∫ p ( x t ) P t t ′ ( x t | x ′ t ) d x t + ∫ p ( x t ) ∫ P t + Δ t ( x t | x ′ t ) − p ( x t ) / P t t ′ ( x t | x ′ t ) d x t + p ( x t ) ∫ P t + Δ t ( x t | x ′ t ) − ∫ p ( x t ) / P t t ′ ( x t | x ′ t ) d x t + p ( x t ) ∫ P t + Δ t ( x t | x ′ t ) − p ( x t ) / P t t ′ ( x t | x ′ t ) (11)

Taking partial derivatives yields the Fokker-Planck equation from [

∂ t p ( x , t ) = − ∂ t p ( x , t ) + ∂ x 2 ( σ ( x , t ) 2 / 2 ⋅ p ( x , t ) ) (12)

where x_{t} is the currency value based on changes in all 7 macroeconomic variables, so that the following maximum likelihood estimator is to be maximized assuming there are n observations of foreign currency values,

Maximize

1 / n log [ − ∂ x μ ( x t ) ⋅ p ( x t ) + ∂ x 2 ( σ x t ) 2 / 2 ⋅ p ( x , t ) + ∂ x 2 ( σ , x t ) 2 / 2 p ( x t ) ] (13)

Subject to

x t = ∂ x / ∂ y [ x 1 t + x 2 t + x 3 t + x 4 t + x 5 t + x 6 t + x 7 t ] (14)

x 1 t ≤ ( x 1 t − μ 1 t ) 2 / σ 1 t (15)

x 2 t ≤ ( x 2 t − μ 2 t ) 2 / σ 2 t (16)

x 3 t ≤ ( x 3 t − μ 3 t ) 2 / σ 3 t (17)

x 4 t ≤ ( x 4 t − μ 4 t ) 2 / σ 4 t (18)

x 5 t ≤ ( x 5 t − μ 5 t ) 2 / σ 5 t (19)

x 6 t ≤ ( x 6 t − μ 6 t ) 2 / σ 6 t (20)

x 7 t ≤ ( x 7 t − μ 7 t ) / σ 7 t (21)

∂ x 2 [ σ ( x t ) 2 / 2 ⋅ p ( x t ) ] = 0 (22)

The constraint, Equation (14) presents the first-order Kuhn-Tucker condition that the exchange rate at time t is the partial derivative of the composite of inflation rates, the short-term interest rate, the long-term interest rate, government debt, export prices, import prices, and political stability. Equation (15) to Equation (21) present each macroeconomic variable as varying within 1 standard deviation of its mean. Equation (22) sets the second derivative Hessian matrix = 0.

For currencies such as the Mexican peso, Turkish lira, and Russian ruble, where there are no restrictions on macroeconomic variables, currency movements may be described by a Laplace distribution. A Laplace distribution is the composite of two exponential distributions, with kurtosis of 3 and 0 skewness. Accordingly, our model will include quantities for excess kurtosis and skewness by including 2 groups of gradient vectors that describe skewness and kurtosis due to the impact on currency values of each macroeconomic variable. Excess kurtosis above 3 is defined in a separate term. The Laplace distribution’s double exponents account for the large jumps not captured in other distributions described in Section 3.1 or Section 3.2. The cumulative distribution function of a Laplace distribution may be described as follows from [

∫ − ∞ x F ( u ) d u = 1 2 + 1 / 2 sgn ( x t − μ t ) ( 1 − exp ( − | x t − μ t | b ) ) (23)

Adding 2 gradient vectors, ∇g_{i}, to account for skewness, and ∇h_{i}, to account for kurtosis,

= 1 2 + 1 / 2 sgn ( x t − μ t ) ( 1 − exp ( − | x t − μ t | b ) ) + ∑ μ t ∇ g i ( x t ) + ∑ ∇ h i ( x t ) + 1 / ∑ σ i t 2 ) ∑ σ i t 4 ⋅ K u r t [ x i t ] − 3 ) (24)

Skewness, as the third moment about the mean is defined in Pearson’s Moment Coefficient of Skewness. A kurtosis of 3 is captured in the gradient vector, with the excess of kurtosis beyond 3 contained in the last term in Equation (24), ∇g_{1}x_{1} = skewness in the inflation rate; ∇g_{2}x_{2} = skewness in the short-term rate; ∇g_{3}x_{3} = skewness in the long-term rate; ∇g_{4}x_{4} = skewness in government debt; ∇g_{5}x_{5} = skewness in export prices; ∇g_{6}x_{6} = skewness in import prices; ∇g_{7}x_{7} = skewness in political stability; ∇h_{1}x_{1} = kurtosis < 3 in the inflation rate; ∇h_{2}x_{2} = kurtosis < 3 in the short-term rate_t; ∇h_{3}x_{3} = kurtosis < 3 in the long-term rate; ∇h_{4}x_{4} = kurtosis < 3 in government debt; ∇h_{5}h_{5} = kurtosis < 3 in export prices; ∇h_{6}x_{6} = kurtosis < 3 in import prices; ∇h_{7}x_{7} = kurtosis < 3 in political stability.

Taking partial derivatives of each side,

∂ x t / ∂ x F ( x t ) + ∇ 2 f ( x t ) = 1 / 2 sgn ( x t − μ t ) 2 / b ( 1 ) + ∑ μ t ∇ 2 g t 2 ( x t ) + 0.5 σ i t + ∑ [ 4 σ i t 2 + ∂ / ∂ x K u r t ( x i t ) ] + ∑ ƛ i ∇ 2 h 12 ( x t ) (25)

Since excess kurtosis =

∂ / [ K u r t [ x i t ] − 3 = ∑ 1 / 2 σ i + 4 σ i t 2 , (26)

We can substitute Equation (26) in Equation (25),

∂ x t / ∂ x F ( x t ) + ∇ 2 f ( x t ) = 1 / 2 sgn ( x t − μ t ) 2 / b ( 1 ) + ∑ μ ∇ 2 g i 2 ( x t ) + ∑ ƛ i ∇ 2 h 12 ( x t ) + ∑ / 2 σ i t + ∑ 4 σ i t 2 + ∑ / 2 σ i t + ∑ 4 σ i t 2 (27)

Collecting like terms,

∂ x t | ∂ x F ( x t ) + ∇ 2 f ( x t ) = 1 / 2 sgn ( x t − μ t ) 2 / b ( 1 ) + ∑ μ t ∇ 2 g i 2 ( x t ) + ∑ ƛ t ∇ 2 h i 2 ( x t ) + ∑ / σ i t + ∑ 8 σ i t 2 (28)

This is a necessary condition for solution. The sufficient condition for solution is that the second derivative of Equation (25) = 0. Given that the gradients are in the form of the second derivative, equation may be differentiated thus,

∂ 2 x / ∂ x 2 + F ( x t ) + ∇ 2 f ( x t ) = sgn ( x t − μ t ) / b + μ t g i 2 ( x t ) + ƛ j ∇ 2 h i 2 ( x t ) + ∑ / σ j t + ∑ 8 σ i t 2 (29)

At the local minimum, x t ∗ , the gradients g_{i} and h_{j} are reduced to 0, so that Equation (29) may be solved for x_{t},

| x t − μ t | / b + ∑ / σ i t + ∑ 8 σ i t 2 (30)

Given the strict band within which macroeconomic variables may vary, call options on the Euro are likely to follow a distribution of small jumps.

preferences. At point M, in _{t}*n, where P_{t} = price of a single put option, and n = number of puts. There is no benefit to waiting for jumps to materialize, so the trader exits the market. In other words, the Coefficient of Absolute Risk Aversion does not increase if there are additional risky opportunities for gain, as the rate of change in risk aversion remains at u'(c).

The Arrow-Pratt Coefficient of Risk Aversion [

A ( c ) = − u ″ ( c ) / u ′ ( c ) = 1 (31)

where, A(c) = an individual’s propensity to avoid risk, u(c) = the utility function of payoffs to an individual for accepting risk, with u'(c) and u''(c), as first and second derivatives. Small jumps in call option prices ensue with variations of the Euro within its narrow band. A risk-seeking trader has the utility function described in curve OF of

An increase in wealth (as promised by the gain from trading calls), results in the desire to increase wealth from further investment in call options with future jumps in euro values. An increase in the utility of wealth is a decrease in absolute risk aversion or A(c) < 0. In expanded form,

∂ A ( c ) / d c = { − u ′ ( c ) u ( c ) − [ u ( c ) ] 2 } / [ u ′ ( c ) 2 ] (32)

c = gain from an investment in call options, = [(Euro Value After the Jump − Exercise Price of the Call Option)] − Call Premium. This model posits that a trader will exercise the option and take the highest gain on a jump at the point at which the rate of change in the utility of wealth (defined in Equation (32)) equals the change in price of a call option along a Levy-Khintchine process (defined in Equation (33) below). This is the point of intersection of the utility function OF with the peak of the first jump, Q. The Levy process approximates the random walk of a call option with successive discontinuous displacements. Traders make requests for call purchases in a Poisson process, with a multitude of discrete requests in a single interval of time. The Levy-Khintchine expression is as follows,

€ [ e i θ x ( t ) ] = exp ( t ( a i θ − 0.5 σ t 2 θ 2 + ∫ ( e i θ x t − 1 − i θ x t I | x t < 1 | ) Π d x ) ) (33)

As the aiθ quantity is a linear drift, and the 1 / 2 σ t 2 θ 2 term is a Brownian motion, both of which are independent of jumps, they will be omitted from further consideration.

The first jump in

∫ ( e i θ x t − 1 − i θ x t I | x t < 1 | ) Π d x

The peak of this jump, at which the trader will realize the maximum gain from investing in the call option is the second derivative of the first jump.

The second derivative is ( e i θ x − i θ x I | x | < t ) Π .

Differentiating Equation (32) below,

∂ / ∂ x [ ∂ A ( c ) / ∂ c ] = [ − u ′ ( c ) u ( c ) − [ u ( c ) ] ] 2 / ( u ′ ( c ) ) 2 (34)

c = gain from an investment in call options, [(Euro Value After the Jump − Exercise Price of the Call Option)] − Call Premium. At the minimum, the second derivative of the utility function = 0,

2 u ″ ( c ) = 0 , or Euro value after the jump − Exercise price of the call = Call premium (35)

when the condition in Equation (35) is satisfied, a trader will make a trade, or exercise the call option, purchase the euro, and sell it for gain. The trader will continue to make similar trades until the gain < call premium, or the cost of purchasing the option is higher than the profit from the trade.

Given that the macroeconomic variables may vary no more than 1 standard deviation from the mean, currency calls on currencies such as the Japanese yen, Australian dollar and Canadian dollar, may experience small jumps (See Section 3.2, for a description of the currency distribution). It is unlikely that very risk-averse traders who sell put options would trade options in these currencies, as even a 1 standard deviation change of macroeconomic variables would be considered to be excessively risky.

In Section 3.2, the optimal foreign currency value, x_{t}, was presented as the solution to Equation (13).

Where, x_{t}, the optimal foreign currency value, is the spot rate, or currency value at a point in time can also be the exercise price on a foreign currency call option as shown in the modification of Equation (35) below.

[ Currency Value After the Jump − x t ] = Gain > Call Premium (36)

when the condition in Equation (36) is satisfied, a trader will make a trade, or exercise the call option, purchase the currency, and sell it for gain. The trader will continue to make similar trades until the gain < call premium, or the cost of purchasing the option is higher than the profit from the trade.

In

Gain on the option = Gain on the currency (37)

Or,

[ Final currency price − Exercise price ] − Call price > Final currency price − Currency purchase price (38)

Currency function + Gain on the option = Levy − Khintchine option function (39)

If x_{t} = value of foreign currency

0.5 sgn ( x t − μ t ) 2 ⋅ b − 1 ∑ μ t ∇ 2 g 2 ( x t ) + ∑ ƛ t ∇ 2 h 2 ( x t ) + ∑ ( σ i t ) − 1 + ∑ 8 σ t 2 + C V − ∑ X − C P = e i θ ? δ − 1 − i θ δ I | x t > 1 | (40)

where, δ = jump size, CV = currency value, Ex = exercise price of the currency call option, CP = call premium.

At the point of earning the gain in the call option, T, in

We will draw upon the currency distributions described in Section 3, as well as

the call option distributions presented in Section 4, to develop expressions for pricing currency calls in this section. This includes combining the currency distribution for euros in Section 3.1, with call option distributions on euros in Section 4.1 to develop the currency call pricing formula in Section 5.1. Likewise, the currency distribution for stable currencies in Section 3.2 will be combined with call option distributions on stable currencies in Section 4.2, and in turn, currency call pricing formulations in Section 5.3. The pattern continues for volatile currencies in Sections 3.3, 4.3, and 5.3. This paper employs the underlying asset pricing concept contained in [

The Value of A Contingent Claim such as a call option = (Gain From The Claim − Price Change Due To State Variables) + (Value of the Contingent Claim*. Distribution of the Contingent Claim),

Price of acall = Present value of the currency − Present value of the exercise price of the option − Price change due to change inmacroeconomy + Call premium ∗ Distribution of the call option

= ( C ∗ e − r d t − Forwardrate e − r d t ) − ( C ∗ [ − ( x 1 t + Δ t − μ 1 t + Δ t ) 2 / 2 2 Π σ 1 t + Δ t 3 − ( x 2 t + Δ t − μ 2 t + Δ t ) 2 / 2 2 Π σ 2 t + Δ t 3 − ( x 3 t + Δ t − μ 3 t + Δ t ) 2 / 2 2 Π σ 3 t + Δ t 3 ] − C ∗ [ − ( x 1 t − μ 1 t ) 2 / 2 2 Π σ 1 t 3 − ( x 2 t − μ 2 t ) 2 / 2 2 Π σ 2 t 3 − ( x 3 t − μ 3 t ) 2 / 2 2 Π σ 3 t 3 ] − [ L 1 ( x 1 t + Δ t − μ 1 t + Δ t − 0.5 )

− L 1 ( x 1 t − μ 1 t − 0.5 ) ] − [ L 2 ( x 2 t + Δ t − μ 2 t + Δ t − 0.7 ] − L 3 ( x 2 t − μ 2 t − 0.7 ) ] − L ( x 3 t + Δ t − μ 3 t + Δ t − 0.25 ) − L 3 ( x 3 t − μ 3 − 0.25 ) + Callpremium ∗ i θ x t − 1 − i θ x I | x < 1 | Π d x (41)

where, First term = Gain on exercise of the call option at the forward rate, or the present value of the gain (C*, currency value-forward rate) earned in the future at the foreign interest rate, r_{d}, over the period, t.

Second and third terms = Objective function of Equation (8) that describes the price change in a currency call option due to change in macroeconomic variables from t to t, Δt. All L terms = Price change due to macroeconomic variables at t and t, Δt. L_{1}, L_{2}, L_{3} = Lagrange multipliers of constraints in Equation (8). Last term = Call premium*Levi-Khintchine call option distribution listed in Equation (33).

According to [

Price of a call option = ( Present value of the currency − Present value of the exercise price of the option ) − ( Price change due to change inmacroeconomy ) + ( Call premium ∗ Distribution of the call option )

while the present value of the currency − present value of exercise price is identical to Section 5.1, other remaining terms need to be adjusted by the expressions in Section 3.2 and Section 4.2, for models with price stability and small jumps,

= [ ( C ∗ e − r d t − Forwardrates e − r d t ) − ( 1 / n log [ − ∂ x [ μ t ⋅ p x t + ∂ x 2 t / ( 2 p x t ) ] − L 1 [ ∂ x / ∂ y [ x 1 t + x 2 t + x 3 t + x 4 t + x 5 t + x 6 t + x 7 t ] − L 3 [ ∂ x / ∂ y ( x 1 t − μ t ) 2 / σ 1 t ] − L 3 [ ∂ x / ∂ y ( x 2 t − μ t ) 2 / σ 2 t ] − L 4 [ ∂ x / ∂ y ( x 3 t − μ t ) 2 / σ 3 t ] − L 5 [ ∂ x / ∂ y ( x 4 t − μ t ) 2 / σ 4 t ] − L 6 [ ∂ x / ∂ y ( x 5 t − μ t ) 2 / σ 5 t ] − L 7 [ ∂ x / ∂ y ( x 6 t − μ t ) 2 / σ 6 t ] − L 8 [ ∂ x 2 / 2 p ( x t ) ] (42)

First term Gain on exercise of the call option at the forward rate, or the present value of the gain (currency value, C-forward rate) earned in the future at the foreign interest rate, r_{d}, over the period, t.

Second term Objective function of Equation (13) that describes the price change in a currency call option due to change in macroeconomic variables, 1/L terms Lagrange multipliers of constraints in Equation (15)-Equation (21), Last term Call premium*Levi-Khintchine call option distribution listed in Equation (33).

We adapt Ito’s Lemma [

Price change due to change in macroeconomic variables = k ( 2 T 2 + 2 T 3 ) (43)

where k = constant, T = ending time period. 2 Equation (30) describes the value of currency in a Laplace distribution with large jumps, Substituting Equation(30) into Ito’s Lemma [

Priceofacalloption = ( C V ∗ e − r d t − Forwardrate e − r d t ) − [ 2 k σ i t 2 ( 1 + σ i t − 1 + 8 σ i t 4 + 8 σ i t 5 ) ] ⋅ b μ t + Callpremium ∗ 1 / 2 sgn ( x t − μ t ) 2 / b + ∑ μ t ∇ 2 g 2 ( x t ) + ∑ ƛ t ∇ 2 h 2 ( x t ) + ∑ ( σ i t ) − 1 + ∑ 8 σ t 2 + C V − E x − C P (44)

CV = currency value, Ex = forward rate, CP = call premium.

This paper has updated the sparse publications on stochastic processes in option pricing, most of which belong to the era of the 1980s and 1990s. Unlike papers that modify stock option pricing models, we recognize that stock and foreign currencies are fundamentally different. A share of stock represents ownership of a business, while foreign currency values are determined by macroeconomic forces and government policy. It follows that the values of stock and foreign currency vary in their movements over time. Stock distributions are typically lognormal with minimal skewness and kurtosis. Foreign currency distributions are discontinuous with jumps and skewness and significant fat-tails, or kurtosis. We create foreign currency distributions that account for skewness and kurtosis with high jump-based distributions such as the Laplace distribution. We improve on jump-diffusion and variance gamma distributions which account for small jumps, with Levy processes that explain large jumps.

From a practitioner standpoint, call options on each of the foreign currencies studied may be used to predict future currency values, so that importers who pay in foreign currencies, may protect their accounts payable from currency appreciation. In other words, if a currency rises in value, an importer who has to make a payment in that currency will experience higher expenses, unless they purchase a call option with low exercise price, that permits them to purchase the foreign currency at cheaper rates.

Future research should extend theoretical formulation beyond the three groups of currencies examined. Other currencies may include the Swiss franc, Australian dollar, Hong Kong dollar, Renminbi, baht, krona, and Singapore dollar. Our approach of first developing the formulation of the currency distribution, followed by the option preserves the definition of an option as an instrument whose value is derived upon the foreign currency, so that is recommended over approaches that do not consider the distribution of the underlying foreign currency. Future research should also explore the use of risk preferences to identify option distributions. Trader demand for call options is governed by risk-seeking. Risk-averse traders will seek modest gains, exiting when the risk of investment exceeds the coefficient of risk-aversion. Risk-takers with low coefficients of risk-aversion will trade longer, until higher gains are achieved. Call options assume a market of appreciating foreign currencies. How would traders react to a market of depreciating foreign currencies? Would they short sell the currency, exiting short selling due to regulatory restrictions, satisfying their demand for declining currency with put option purchases? How would those puts be priced? Future research must be directed to answering these questions.

The authors declare no conflicts of interest regarding the publication of this paper.

Abraham, R. (2018) Pricing Currency Call Options. Theoretical Economics Letters, 8, 2271-2289. https://doi.org/10.4236/tel.2018.811148