_{1}

^{*}

Investors who seek to profit from depreciating currencies may invest in put options. Upon option exercise, the currency is sold at a high price, and then purchased at the lower future currency value, resulting in a gain for the put buyer. A series of such transactions yields a stream of income for the put investor. Alternatively, the investor could short sell the currency, reaping gains from the difference between the high short sale price and the low future purchase price. This paper derives the theoretical formulations for combined short sale and put s purchase strategies for the US dollar, the Euro, the Australian dollar and the New Zealand dollar, and the Mexican peso. Utility functions are based upon an assumption of declining risk aversio with negative rescale factors and positive threshold factors in a hyperbolic cosine distribution. This distribution intersects with the cosine distribution of short sale prices on the U. S. dollar, the lognormal distribution of short sale prices on the Euro, the Weiner process for shorts on the Australian dollar and the New Zealand dollar, and the Laplace currency distribution for peso shorts. Similar utility functions intersect with Levy-Khintchine jump processes to provide put option prices for each type of foreign currency.

Currency put options are only profitable with declining currency values. The Gain on a Currency Put = (Exercise Price − Spot Exchange Rate) − Put Purchase Price. This outcome suggests that an investor, forecasting future currency depreciation, purchases a put option at the purchase price, sells the underlying currency at a high exercise price and buys back the currency at the lower future spot exchange rate, earning a gain. Conversely, an increase in currency values will result in a loss. If put buyers exercise, as Exercise Price < Spot Exchange Rate, (Exercise Price − Spot Exchange Rate) − Put Purchase Price < 0), therefore, the option will expire, with Loss = (0 − Put Purchase Price).

There is a paucity of literature on currency put valuation and trading strategies. The existing literature explores currency call options which profit from increasing foreign currency prices. Watt [

The purpose of this paper is to present a model of valuation of currency put options, based upon the premise that put options are a unique entity. Unlike currency calls, the market for currency puts has multiple participants. Short sellers attempt to gain from depreciating currencies. They borrow currencies, sell at high prices, and repay with cheap currency. Repeated short selling devalues currencies, raising concerns by regulatory authorities who view the devaluation as a loss of national wealth. Accordingly, they restrict short selling. This paper contributes to the literature in three ways. It is the first study to propose combined short selling and put buying strategies. An informed investor short sells currency in the foreign currency market, earns the modest gains, then ceases short selling upon restrictions imposed by regulatory authorities. Yet, he or she still has unsatiated demand for depreciating currencies, which may be fulfilled in the unregulated options market. Therefore, the trader purchases put options, devaluing currencies further. Secondly, the Miller model [

The remainder of this paper is organized as follows. Section 2 is a Review of Literature, Section 3 is a presentation of the joint short sale and put option formulations followed by valuation of put options, and Section 4 provides Conclusions.

A currency raid begins with signs of weakness in macroeconomic fundamentals. Speculators, often hedge funds, short sell the currency, being joined by other speculators, who aggressively drive the value of the currency to a minimum level. Morris [

A few special cases of central bank intervention may be noted. Political pressure from the United States to overcome the persistent depreciation of the US dollar has led to the Japanese central bank purchasing large quantities of US dollars, in a bid to support the value of the US dollar in the post-2010 period. Other central banks maintain their foreign reserves in US dollars due to the United States’s economic size, developed financial markets, and historical practice [

The first set of put currency option models are based on the [

C t = S t B t , T ∗ N ( d 1 ) + σ T − t − X B t , T N ( d 1 ) (1)

where,

d 1 = ln [ S t B t , T ∗ / X B t , T ] − 1 / 2 σ 2 ( T − t ) σ ( T − t ) (2)

Put-Call Parity yields the value of a put option,

P t = X B t , T N ( − d 1 ) − S t B t , T ∗ N ( − d 1 ) / σ ( T − t ) (3)

C_{t} = Value of Call Option in Foreign Currency at time t, S_{t} = Spot Echange Rate at time t, (T − t) = Time to maturity, B t , T ∗ = Price of a Foreign Currency Bond at time t, N(d_{1}) =Cumulative Normal Density Function, σ = Volatility of the Spot Currency Price, X = Exercise Price of the Call Option, B_{t}_{,T} = Price of a Domestic Currency Bond at time t, P_{t} = Value of Put Option in Foreign Currency at time t.

This formulation assumes that the investor owns a domestic bond and a call option and sells a foreign bond and a put option. The only differences from the Black-Scholes model are that 1) the stock price is replaced with the spot exchange rate, as the underlying asset is foreign currency, not stock, 2) domestic and foreign bonds are included, and, 3) time to maturity is computed as (T − t), instead of t. Similarities outweigh differences, including, 1) the assumption of a geometric Brownian motion for the movement of currency prices over time. However, currencies have values set by government policy, which have been shown to depart from a geometric Brownian motion [

The [

P = C + ( X − F ) e − r t , where F = forward rate (4)

This substitution may be an improvement over the Black-Scholes-based formulation in that the forward rate approximates the return on a put option more closely than bond returns. Returns on riskless bonds are typically lower than those on risky puts. The forward rate, on the other hand, is based on market-based judgements of future exchange rates, and may therefore, be more realistic in predicting returns on put options than risk-free bonds. However, both the [

The above models omit the inclusion of jumps in currency options. A series of studies [

Both short selling and put buying strategies are risky. Profits are only earned if currency prices depreciate, with greater payoffs as currency depreciation continues and reaches a minimum level. Not all traders wish to maximize profits. Risk-averse investors will be satisfied with minimal profits, exiting both short selling and put buying if they suspect that short sale borrowing rates will eliminate profits, or that put premiums will rise to overwhelm put gain. Other investors may trade in a few rounds of short selling or put buying, achieving satiation of payoffs with higher short sale or put buy gains. Risk-takers will pursue maximum gain, trading to the last round of either strategy. In other words, the attitude to risk of each type of trader governs the investment choices made and the prices they are willing to pay. One type of attitude is absolute risk aversion. Absolute risk aversion is the innate attitude to risk based on values, beliefs and personal predispositions. Such attitudes are only likely to change under exceptional circumstances. For example, both Pratt [

Certain utility functions of investor preferences omit changes in risk aversion altogether. In both [

Pratt [

− [ p θ ( θ − 2 Π b ) + Π 2 b ] m 1 / 2 < [ p θ ( θ 2 − 3 θ Π b + 3 θ Π 2 b ) − Π 3 b ] m 2 / 6 (5)

Or, (Small gain × absolute risk aversion) < (Large gain × change in absolute risk aversion). Therefore, for a risk-averse individual, only a substantial increase in gain would permit acceptance of the riskiest put or short sale choices.

Three types of investor utility functions will be considered with investors first embarking on short selling. Then, they may or may not engage in put buying. The put strategy will lead to a put distribution, which when weighted by put prices will yield the final expression for the put value.

Case 1: The Risk-Averse Investor. The risk-averse investor has a coefficient of absolute risk aversion, m_{1} > 0, or will only accept an investment if certain that the payoff will not result in loss, which could occur if short sale prices or put prices trend upwards. Therefore, this strategy may be confined to 2 rounds of short selling, before foreign governments realize that they need to support the US dollar, and start purchasing it for their reserves. The investor’s utility function may be modeled by an hyperbolic cosine distribution, shown in

F = ∑ x 2 n / ( 2 n ) ! + ∫ a + ( € ) cos ( 2 Π € ( x + t ) ) + a − ( € ) cos ( 2 Π € ( x − t ) ) + b + ( € ) sin ( 2 Π € ( x + t ) ) + b − ( € ) sin ( 2 Π € ( x − t ) ) d € + d / d x [ ( x − μ ) 3 / σ ] (6)

First term = Taylor series expansion of a hyperbolic cosine distribution, Remaining terms = Cosine distribution of short sale prices, and Skewness of short sale prices.

Taking derivatives of Equation (6) to explain the change in risk aversion and change in short sale prices to point S.

For the hyperbolic cosine distribution, the derivative is given by inserting Ito’s multiplication rule into the Taylor series in Equation (6). For the cosine series, the derivative is given by applying Euler’s formula. For the skewness, we find the derivative of a Laplace transform,

[ ( d G / d x ) a + ( d G / d t ) + 0.5 ( d 2 / d x 2 ) b b ] d t + ( d G / d x ) b d z + [ ( e − i a x + e − i a x ) ] / 2 + s ∗ L { ( x − μ ) 3 / σ } (7)

The necessary condition for the investor to achieve maximum gain is described by the following linear programming model.

Maximize

( S − P ) ∗ [ ( d G / d x ) a + ( d G / d t ) + 0.5 ( d 2 / d x 2 ) b b ] d t + ( d G / d x ) b d z − θ [ ( e i a x + e i a x ) ] / 2 − s ∗ L { ( x − μ ) 3 / σ } (8)

where (S-P) is the gain on the short sale transaction and θ is the Lagrange multiplier that relates the cosine distribution of short sale prices and their skewness to the satisfaction of the need by the risk-averse consumer for depreciating currency.

The sufficient condition for the achievement of maximum gain is the second derivative of Equation (8) equated to 0,

( S − P ) ∗ [ ( d 2 G / d x 2 ) a + ( d G / d t ) + ( d 2 / d x 2 ) b 2 ] d t + d 2 G / d x 2 b d z + θ [ ( e i a x + e i a x ) ] / 2 + s ∗ L ′ { ( x − μ ) 3 / σ } = 0 (9)

Since the third derivative is an aberration [

( S − P ) ∗ [ ( d 2 G / d x 2 ) a + d G / d t + d 2 G / d x 2 b d z + θ [ ( e i a x + e i a x ) ] / 2 + s ∗ L ′ { ( x − μ ) 3 / σ } ] = 0 (10)

Case 2: The Moderate Risk-Taker

The moderate risk taker may see more potential for profit in short selling, engaging in three rounds of short selling, after which central bank intervention renders short selling unprofitable. The liquidity of the US Dollar permits more short selling than other currencies, as a large volume of dollars is available to trade. Consequently, we introduce a lower price bound to short selling, after which the risk-taker continues trading in the options market, purchasing puts for about two rounds of trading, before currency prices rise and put profits are exhausted. The investor exercises the put at points P and R, at which his or her utility function of risk preferences, AB, intersects with the put Levy-Khintchine distribution, OS (see Abraham, in press, for a review of Levy-Khintchine jump processes). Therefore,

Taking derivatives of Equation (6) will explain the change in risk aversion and change in short sale prices to point S, For the hyperbolic cosine distribution, the derivative is given by inserting Ito’s multiplication rule into the Taylor series in Equation (6), for the cosine series, it is applying Euler’s formula, and for the skewness, it is finding the derivative of a Laplace transform. A Laplace transform is employed to suppress the considerable skewness of the heavily traded US dollar, which fluctuates with positive and negative skewness.

[ ( d G / d x ) a + 0.5 ( d 2 G / d x 2 ) b b d t + d G / d x b d z + [ ( e − i a x + e − i a x ) ] / 2 + s ∗ L { ( x − μ ) 3 / σ } ] (11)

when the investor is satisfied with the gain from short selling, he or she imposes a lower price bound to the cosine short sale distribution (see Equation (12) and Equation (13)).

The Fourier sine and cosine forms of a boundary condition are,

2 y ( x , 0 ) cos ∫ ( 2 Π € x ) d x = a + a − (12)

and

2 ∫ ∂ y ( u , 0 ) / ∂ t cos ( 2 Π € x ) d x = 2 Π € ( b + b − ) (13)

Substituting t = 0 in the wave equation that satisfies the Fourier transform,

y ( x , t ) = ∫ a + € cos ( 2 Π € ( x + t ) ) + a − € cos ( 2 Π € ( x − t ) ) + b + € sin s ( 2 Π € ( x + t ) ) + b . € ( x − t ) d € (14)

y ( x , 0 ) = ∫ a + € cos ( 2 Π € x ) + a − € cos ( 2 Π € x ) + b + € sin ( 2 Π € x ) + b − sin € x , d € (15)

Multiply (15) by cos(2П?i style='mso-bidi-font-style:normal'>x), and assume all x > 0,

a + a − = a + € cos 2 ( 2 Π € x ) + a − € cos 2 ( 2 Π € x ) + b + cos ( 2 Π € x ) € sin ( 2 Π € x ) + b − cos ( 2 Π € x ) € sin x , d € (16)

By the same token,

b + b − = a + € cos ( 2 Π € x ) + a − € cos ( 2 Π € x ) + b + € sin 2 ( 2 Π € x ) + b − € sin 2 x d € (17)

Solving (16) and (17) will yield the boundary for short selling.

At points P and R in

∫ ( e i θ x t − 1 − i θ x t I | x t < 1 | ) Π ( d x ) (18)

The lowest point of this jump, at which the investor will realize the maximum gain from investing in the put option is the second derivative of Equation (18). Differentiating the investor’s utility hyperbolic cosine distributed utility function below,

= 2 n ( x ) / 2 n ! (19)

At maximum gain, the second derivative of the utility function in Equation (19) = 0. Equating (19) and (20) and including the Laplace transform adjustment for skewness from Equation (12),

= − e i θ x t + i θ x t I | x < 1 | Π ( d x ) + s ∗ L { ( x − μ ) 3 / σ } = 0 (20)

Equation (20) provides a solution for x, the optimal point of the intersection of the utility function and the Levy-Khintchine process. The put price is given by,

P = ( X − S − p p ) ∗ 1 / 2 n x / 2 n ! ( − e i θ x t + i θ x t I | x t < 1 | ) Π ( d x ) + s ∗ L { ( x − μ ) 3 / σ } (21)

Case 3: The Risk-Taker

The trader engages in short selling, until a lower limit on short selling is reached. Therefore, the expressions contained in Equation (11), Equation (12), Equation (13), Equation (14), Equation (15), Equation (16), Equation (17), Equation (18), and Equation (19) will be used in this section. Upon reaching the boundary of short selling, the trader moves to the options market to purchase puts. Being a risk-taker, the trader may wish to continue purchasing put options until the minimum U.S. dollar value is reached. Yet, given the heightened risk, he or she may insist on being paid a premium for taking the risky gamble. We add the [

( X − S ) ∗ [ ( d G / d x ) a + d G / d t + 0.5 ( d 2 x / d x 2 ) b b ] d t + ( d G / d x ) b d z . [ X S ( X − 2 S ) + S 2 ] m 1 / 2 − [ X S ( X 2 − 3 X S + 3 S 2 ) − S 3 ] m 2 / 6 .. (22)

where First 4 terms = Ito’s lemma of the hyperbolic cosine distribution, X − S = Exercise Price of the Put Option − Spot Exchange Rate Last 3 terms = Function for the acceptance of an unlucky gamble, m_{1} = coefficient of absolute risk aversion, m_{2} = change in the coefficient of absolute risk aversion. This utility function in Equation (22) is equated to the premium, Y, that must be paid to the trader for assuming additional risk, The above utility function in Equation (22) intersects the Levy-Khintchine formula for the put distribution, so the maximum gain from the put option investment is achieved at the intersection of the utility function and the second derivative of the Levy-Khintchine process.

( X − S ) ∗ [ ( d G / d x ) + d G / d t + 0.5 ( d 2 / d x 2 ) b b ] d t + d G / d x b d z − [ X S ( X − 2 S ) + S 2 ] m 1 2 − [ X S ( X 2 − 3 X S + 3 S 2 ) − S 3 ] m 2 / 6 − Y = 0 = ( − e i θ t + i θ x I | x < 1 | ) Π + s ∗ L ′ [ ( x − μ ) 3 / σ ] (23)

Solving for x,

x = ( X − S ) ∗ [ ( d G / d x ) a + d G / d t + 0.5 ( d 2 / d x 2 ) b b ] d t + d G / d x b d z − [ X S ( X − 2 S ) + S 2 ] m 1 / 2 − [ X S ( X 2 − 3 X S + 3 S 2 ) − S 3 ] m 2 / 6 − Y / ( − e i θ x + i θ I | x | < 1 ) Π + s ∗ L ′ [ ( x − μ ) 3 / σ ] (24)

Case 1: The Risk-Averse Investor.

The macroeconomic parameters to qualify for the European Union’s exchange rate target of 0% - 1% of normal rates is 0% - 0.5% fluctuation in the inflation rate, 0% - 0.7% change in the long-term interest rate, and 0% - 0.25% change in government debt [

∑ x 2 n / ( 2 n ) ! = [ ( x − μ ) Z 3 / σ ] + e μ + 1 / 2 σ 2 ϕ ( μ + σ 2 − ln k 1 ) / σ + e μ = 1 / 2 σ 2 ϕ ( μ + σ 2 − ln k 2 ) / σ + e μ + 1 / 2 σ 2 ϕ ( μ + σ 2 − ln k 3 ) / σ (25)

where, First term = Taylor series expansion of a hyperbolic cosine distribution, e^{μ} = 1/2σ^{2} quantities = partial expectation of a short sale price, x, conditional upon the spot rate, x < k_{1}, or k_{2}, or k_{3}, where k_{1} is the targeted inflation rate, k_{2} is the targeted long-term interest rate, and k_{3} is the targeted amount of government debt, A necessary condition for maximum short sale gain is the first derivative of Equation (23),

∑ 2 n x / ( 2 n ) ! − 3 [ ( x − μ ) 2 / σ ] − e μ + 1 / 2 σ 2 d / d x ϕ ( μ + σ 2 − ln k 1 ) / σ − e μ = 1 / 2 σ 2 ϕ ( μ + σ 2 − ln k 2 ) / σ − e μ + 1 / 2 σ 2 ϕ ( μ + σ 2 − ln k 3 ) / σ , (26)

Using Ito’s Lemma, where G 1 = ( x + μ 2 − ln k 1 ) / σ , G 2 = ( x + μ 2 − ln k 2 ) / σ , G 3 = ( x + μ 2 − ln k 3 ) / σ ,

∑ / 2 n ! − 3 [ ( x − μ ) 2 / σ ] − e μ + 1 / 2 σ 2 [ ∂ G 1 / ∂ x d x + ∂ G 1 / ∂ t + 1 / 2 ∂ 2 G 1 / ∂ x 2 b 2 d t + ∂ G 2 / ∂ x d x + ∂ G 2 / d t + 1 / 2 ∂ G 2 / ∂ x 2 b 2 d t + ∂ G 3 / ∂ t + 1 / 2 ∂ 2 G 3 / ∂ x 2 b 2 d t ] (27)

Differentiating Equation (25) yields the sufficient condition for short sale gain,

∑ 2 n x / ( 2 n ) ! − 6 [ ( x − μ ) / σ ] + e μ + 1 / 2 σ 2 [ ∂ 2 G 1 / ∂ x 2 d x + ∂ G 1 / ∂ t + 1 / 2 ∂ 2 G 1 / ∂ x 3 b 2 d t + ∂ 2 G 2 / ∂ 2 x d x + ∂ G 2 / ∂ t + 1 / 2 ∂ 3 G 2 / ∂ x 3 b 2 d t + ∂ 2 G 3 / ∂ x 2 d x + ∂ G 3 / ∂ t + 1 / 2 ∂ 3 G 3 / ∂ x 3 b 2 d t ] (28)

After a few rounds of short selling, the risk averse investor is confronted with the sudden appreciation of the Euro, as the central bank forces the Euro into the narrow band. Then, the investor continues trading put options. As shown in

jump, selling the Euro at the exercise price, X, and repurchasing it at the spot rate at point, P, thus realizing a gain. The Euro band, AB, imposes a lower limit below which the Euro may not decline. The investor’s utility function intersects with the Levy-Khintchine distribution of put prices at X.

Thepriceofaputoption = ( X − S − p p ) ∗ ∑ 2 n x / ( 2 n ) ! − 6 [ ( x − μ ) / σ ] + e μ + 1 / 2 σ 2 [ ∂ 2 G 1 / ∂ x 2 d x + ∂ 2 G 1 / ∂ t 2 + 1 / 2 ∂ 3 G 1 / ∂ x 3 b 2 d t + ∂ 2 G 3 / ∂ x 2 d x + ∂ 2 G 3 / ∂ t ] (29)

where, X = Exercise price, S = Spot rate, Pp = Put premium, At the maximum point of the gain, the second derivative the utility function is equated to the Levy-Khintchine distribution, _{ }

The spot rate for maximum put gain for the risk-averse investor = 2 n [ ( e i θ x − i θ x I | x < 1 | ) Π + s ∗ L ′ { ( x − μ ) 3 / σ } ] / [ 2 n ! ] = x (30)

It follows that the put price,

P = ( X − S − p p ) ∗ ( e i θ x − i θ x I | x < 1 | ) Π 2 n ! / 2 n (31)

X = Exercise price of the put option, S = Spot exchange rate of the Euro, pp = Put premium, s × L’ term = Laplace transform adjustment for skewness.

Case 2: The Moderate Risk-Taker

Using Prakash, et al. [

m 2 / 3 m 1 = 1 / ( θ − 1 ) (32)

where, m_{2} = elasticity of absolute risk aversion, m_{1} = coefficient of absolute risk aversion, Right side of Equation (32) = loss from short selling, These risk-takers know that gains may be earned during the narrow window of time during which the Euro declines below the band AB in

1 / ( θ − 1 ) < ( θ − 2 ) / ( θ 2 − 3 θ + 3 ) (33)

Over time, acceptance of risky short selling grows. In

( X − S ) ∗ [ ( d G / d x ) a + d G / d x + 1 / 2 ( d 2 x / d x 2 ) b 2 d t + d G / d x b d z + ( m 2 − 1 ) / 3 m 1 ] = [ e μ + σ 2 ϕ ( μ + σ 2 − ln k 1 ) x / σ ] + [ e μ + σ 2 ϕ ( μ + σ 2 − ln k 2 ) x / σ ] + [ e μ + σ 2 ϕ ( μ + σ 2 − ln k 3 ) x / σ ] + ( e σ 2 + 2 ) ( e σ 2 − 1 ) + ( x 4 σ 2 + 2 x 3 σ 2 + 3 x 2 σ 2 − 6 ) (34)

Last 2 terms in Equation (34) = Measures of skewness and kurtosis, to satisfy the sufficient condition to maximize utility, differentiate Equation (34),

( X − S ) [ ( d 2 G / d x 2 ) + d G / d t + 1 / 2 ( d 3 x / d x 2 ) b b d t + d 2 G / d x 2 b d z + ( m 2 − 1 ) / 3 m 1 ] = e μ + σ 2 ϕ ( μ + σ 2 − ln k 1 ) / σ + e μ + σ 2 ϕ ( μ + σ 2 − ln k 2 ) / σ ( e σ 2 + 2 ) 1 / 2 ( e σ 2 − 1 ) − 1 / 2 + 4 σ 2 x 3 σ 2 + 6 x 2 σ 2 + 6 x σ 2 (35)

In

yield the necessary condition,

= e i i θ x + i θ x I | x | , 1 Π + s L ′ { ( x − μ ) 3 / σ } (36)

The sufficient condition for the maximization of put gain is obtained by differentiating Equation (35) and Equation (36),

Case 3: The Risk-Taker

Put Purchase Euro Transactions for the Risk-Taker.

Risk-Takers have identical functions to moderate risk-takers in _{1}, k_{2}, k_{3}) in the intersection of the utility function with the Levy-Khintchine process. For the necessary condition for maximum utility, we equate Equation (34) to the Levy-Khintchine process to obtain the necessary condition to achieve maximum utility,

The sufficient condition for maximum put gain is the second derivative of Equation (38),

The Australian dollar and the New Zealand dollar are popular reserve currencies in Asia resulting in heavy daily trading, have minimal central bank intervention, are high interest rate currencies, and are countercyclical in movement in that these economies export commodities. Commodity prices fall with recessions depressing currency values. This is counter-cyclical to most currencies, whose central banks intervene to restore currency values during recessions [

Case 1: The Risk-Averse Investor

The risk-averse investor will short sell the Australian dollar and the New Zealand dollar for a few rounds longer than the other currencies, knowing that market forces will not restore currency values and end short sale profits in a timely fashion. However, the lack of liquidity in the options markets indicates that the investor may forego trading in put options, with the few options available being highly risky, and therefore incompatible with the risk-averse inclinations of the investor. We assume that the optimal short sale price will lie at the intersection of the investor’s hyperbolic cosine distributed utility function and the Weiner process of put option prices. Volatility may be limited, given the goal of stability. Therefore, two gradient vectors may capture the skewness risk of increased future currency risk, and the kurtosis risk of outliers. A Weiner process is assumed because of the characteristic of having independent increments. For every time interval in the future, and t, the future increments, W_{t}_{+u} − W_{tu} = 0, u

At point T, the point of intersection of the investor’s utility function, RS, with the Weiner process of short sale prices, OP, both the hyperbolic cosine distribution and the Weiner process are expressed as a Taylor series expansion with

?sub>0 = independent Gaussian variable with mean 0 and standard deviation of 1,

Upon differentiating both sides, the moments > 2 may be omitted, as the gradient vectors capture the volatility in the higher moments,

Equation (42) is the necessary condition for maximum short sale gain (S − P) or (Selling Price ? Purchase Price).

The sufficient condition for maximizing short sale gains is,^{ }

The risk-averse investor foregoes put buying as the limited trading of options in these markets imposes excessive volatility about future put prices.

Case 2: The Moderate Risk-Taker

The moderate risk-taker will expect a payoff for taking the risky short sale for a few more rounds to compensate for the additional risk of uncertain timing of the increase in interest rates.

Adding the penalty for additional risk which is Pratt’s [

Eliminating moments > 3 upon differentiating,

The function for maximum gain on short selling is,

The moderate risk-taker expects that the minimum Australian dollar or New Zealand dollar price,

where, E(M_{x}) = expected minimum exchange rate, m = actual minimum exchange rate, At the boundary, we add Equation (46) to the Taylor series in Equation (44), to obtain the necessary condition for maximum short sale gain,

Differentiating (50), while omitting moments higher than 2,

Setting

Upon differentiating Equation (54), we obtain the sufficient condition for maximum short sale gain,

For the put buy option, only one round of trading may be expected, given the lack of liquidity in the options market for these currencies.

The utility function is equated to the Levy-Khintchine distribution for put prices in Equation (56),

The necessary condition from the differentiation of (56) is,

The sufficient condition for maximum put gain is obtained by differentiating_{ }

Equation (57) as follows,

Case 3: The Risk-Taker.

The risk-taker follows the same choices as the moderate risk-taker for short selling. However, for put buying, the investor relies on personal judgment of the situation, rather than past experience. A utility function that ignores past experience is a continuous time martingale of the form,

Differentiating Equation (59) to obtain the necessary condition for maximum put gain,

Setting t = x, and omitting moments > 2, the sufficient condition for maximum put gain is,

Therefore, combining with the Levi-Khintchine formula, we solve for put gain (X − S) for small and large jumps, respectively, as,

Only the risk-taker is assumed to invest in the Mexican peso, due to its history of external debt default in 1982, and capital flight [

put options on the peso. The point of intersection, T, represents the maximum gain for the put investor, and is represented as the first derivative of the following expression,

The first derivative is,

The sufficient condition for maximum put gain is the first derivative of Equation (65),

This paper has created models of put valuation for four currencies with different characteristics. It has updated the pre-2000 literature on the topic, upholding the position that currencies have unique characteristics, and therefore must be described by different models. We also introduce the concept of multimarket trading into put purchase strategies. Investors are likely to trade in both the foreign currency market and the options market. First, they attempt to optimize gain in the currency market through short selling. However, short selling is restricted, as it includes borrowing. The borrowed currency must be repaid, and the short seller lacks control over the interest rate to be paid. Hence, short sellers take additional profits on declining currency values in the options market. Thus, investments in put options are not restricted to the derivatives market. They are part of a comprehensive currency + derivatives investment strategy that earns gains from all venues of trading in depreciating currencies, which includes both the currency market and the derivatives market.

The Black-Scholes model initiated research in the valuation of stock options. While it was extremely successful in achieving this purpose, the practice of using the same model to value currency options is questionable. Currencies are fundamentally different from securities. Put options on securities do not display jumps, while those on foreign currencies are described by jump processes. This paper models put prices in terms of the Levy-Khintchine formula, which makes allowance for both small and large jumps. Further, currencies are more heterogeneous than stocks. The U.S. dollar’s reserve currency status, the Australian and New Zealand dollar’s lack of government intervention, the Euro’s strict band of fluctuation, and the Mexican peso’s volatility are all significant differences that impact put currency option values, much more than they impact security prices.

This paper views short sales and put option purchase decisions as being governed by investor preferences. Investors are heterogeneous in their beliefs. Therefore, it is reasonable to assume that they differ in perceptions of risk, and that these perceptions must be modeled as intersecting with investment opportunities, be they in short selling or put buying. Floor traders will find these models useful, as they present clients with investment choices. Certain clients will gravitate to safer choices, while others will accept more risk, and pay accordingly. The notion that such perceptions towards risk, termed absolute risk aversion, may change, has also been included in our models. If the reward is sufficiently attractive, risk-averse investors may revise their original investment choices, opting for risky short selling and put buying. Have we supported Miller’s [

Certain limitations of these models must be considered. The paper does not address most emerging market currencies or Organization of Petroleum Exporting (OPEC) currencies. Given the rapid growth of the former, and the recent fluctuations in oil prices, put strategies on these currencies must be examined in future research. Short selling has been presented as a means to correct currency mispricing. What are the limitations of this practice? Can short selling correct mispricing of emerging market currencies? Other theoretical formulations must be developed to address this issue. Finally, the hyperbolic cosine distribution was presented as the utility function of choice for investors in this paper. Future examinations must consider other forms, such as Esscher transformed Geometric Levy processes, Legendre functions, or Lebesque integrals.

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

Abraham, R. (2018) The Valuation of Currency Put Options. Theoretical Economics Letters, 8, 2569-2593. https://doi.org/10.4236/tel.2018.811165