A currency option is the right - but not the obligation - to buy (in the case of a call) or sell (in the case of a put) a set amount of one currency for another at a predetermined price at a predetermined time in the future.
The two parties to a currency option contract are the option buyer and the option seller/writer. The option buyer may, for an agreed upon price called the premium, purchase from the option writer a commitment that the option writer will sell (or purchase) a specified amount of a foreign currency upon demand. The option extends only until the expiration date. The rate at which one currency can be purchased or sold is one of the terms of the option and is called the exercise price or strike price. The total description of a currency option includes the underlying currencies, the contract size, the expiration date, the exercise price and another important detail: that is whether the option is an option to purchase the underlying currency - a call - or an option to sell the underlying currency - a put. There are two types of option expirations - American-style and European-style. American-style options can be exercised on any business day prior to the expiration date. European-style options can be exercised at expiration only. Currency options may be quoted in one of two ways: American-terms, in which a currency is quoted in terms of the U.S. dollar per unit of foreign currency; and European-terms (inverse terms), in which the dollar is quoted in terms of units of foreign currency per dollar. The same logic can be applied to currency pairs in which the U.S. dollar is not one of the currencies. Either currency can be expressed in terms of the other. Prices for currency options - called premiums - are calculated using several complex statistical models. The premium quoted for a particular option at a particular time represents a consensus of the option's current value which is comprised of two elements: intrinsic value and time value. Intrinsic value is simply the difference between the spot price and the strike price. A put option will have intrinsic value only when the spot price is below the strike price. A call option will have intrinsic value only when the spot price is above the strike price. Options which have intrinsic value are said to be "in-the-money."
Time value is more complex. When the price of a call or put option is greater than its intrinsic value, it is because it has time value. Time value is determined by five variables: the spot or underlying price, the expected volatility of the underlying currency, the exercise price, time to expiration, and the difference in the "risk-free" rate of interest that can be earned by the two currencies. Time value falls toward zero as the expiration date approaches. An option is said to be "out-of-the-money" if its price is comprised only of time value. A variety of complex option pricing models such as Black-Scholes and Cox-Rubinstein have been developed to determine option pricing. Another commonly used model for currency option valuation is the Garmen-Kohlhagen model. There are many texts available which cover the specifics of option pricing models in detail. Interest rate differentials between nations and temporary supply/demand imbalances can also have an effect on option premiums. In the final analysis, option prices (premiums) must be low enough to induce potential buyers to buy and high enough to induce potential option writers to sell. These prices result from the interaction of buyers and sellers through the market.
Options on currencies provide foreign exchange risk managers and traders with a wide new array of capabilities for controlling the risks inherent in foreign exchange exposure. Currency options should be considered for their potential to meet needs which cannot be adequately addressed by forward contracts or futures. However, the relationship of the forward, futures and options markets is close enough to allow opportunity for intermarket hedging and arbitrage. The following pages will illustrate just a few strategies available to users of currency options. Readers should keep in mind that this brief explanation can only provide a thumbnail sketch of the possibilities presented by options. A list of relevant publications which cover the topic in greater detail is provided by Jet Investments. To simplify the example sections, all options are held through expiration and examples do not take into account brokerage fees, transaction costs, margin requirements or tax implications. These factors should be examined when considering an option transaction and discussed with your broker or investment advisor.
Since a call option entitles the holder to purchase units of the underlying currency at the option exercise price, it follows that the option holder will realize a profit if the value of the option at expiration is greater than the premium paid to acquire the option. That is, if the spot market price of the currency is above the option exercise price plus the initial option premium. Call options can be used to protect against currency appreciation. As an example, Surinam companies purchasing goods from foreign companies may incur substantial risk if, before completion of an agreed upon transaction, the currency of the seller's nation should rise in value relative to the Surinam Guilder. Such a rise means more Surinam Guilders than originally contemplated would be required to purchase the goods, resulting in reduced profits or even potential losses. Buying call options on the seller's currency, in this case, can provide protection or "insurance" against the risk of a rising currency. In return for the premium paid for a call option, the buyer can be assured that the specified currency can be purchased at the predetermined price. Yet, the holder of such an option has no obligation to purchase at the price guaranteed by the option. Instead, the opportunity to reduce the cost by spending fewer Surinam Guilders for the goods is preserved should the currency decline in value relative to the Surinam Guilder ; the option would be allowed to expire without further consequence to its holder. Example: In June, an Surinamese importer wishes to purchase American goods for resale. A price of 12,500,000 US$ is negotiated for delivery in September. The Surinamese importer observes that the current spot exchange rate is Sf.1530 per US$. A decision is made to protect against an adverse movement of the Surinam Guilder by purchasing 200 OTC call options (US$ 62,500/contract) with a strike price of Sf.1500, for expiration in September. If the premium for the option was 1.36 cents, the importer would pay a total of $170,000, or about 2% of the value of the underlying, ($.0136 x 62,500 US$ = $850 x 200 contracts). In exchange for the premium, the importer is guaranteed US$ at a price of Sf.1500, even if the US$ appreciates above Sf.1500 before expiration. If the US$ declines in value below the exercise price of Sf.1500, the original purchase price would actually require fewer U.S. dollars than was initially required and the option would not be exercised.
A currency put option guarantees its owner the right, but not the obligation, to sell currency at a pre-agreed exercise price within a specified time period. It follows that the option holder will realize a profit if the spot market price of the currency is below the option exercise price by an amount greater than the premium paid. Put options can be used to protect or "insure" against currency price declines. As an example, a Surinamese exporter of goods to the U.S. may well find it advantageous to purchase put options on U.S. dollars in connection with a specific sale of goods. The Surinamese seller of goods will be at risk for any decline in the value of U.S. dollars relative to the Surinam Guilder. Ideally, the seller would like to eliminate this risk and retain the ability to receive more Surinam Guilders for the merchandise if the U.S. dollar should rise in value. Example: During the month of September a Surinamese company agrees to sell 115,050,000 Sf worth of merchandise to an American buyer for delivery in December. At the time of the agreement, 1 U.S. dollar can be exchanged for 1530 Sf. Thus, a contract is agreed upon to sell the goods for 75,196.07 U.S. dollars (115,050,000 DEM / 1530 Sf). In order to protect against the risk of a decline in the U.S. dollar relative to the Surinam Guilder, the seller decides to purchase 17.5 ($75,196/$10,000 per contract) December 1530 customized put options on the U.S. dollar. The strike price is in Surinamese Guilders per dollar, while the underlying contract is U.S. dollars. The premium for the put option is 2.45% or $1,225 per contract, for a total premium payment of $42,875 ($10,000 per contract x .0245 x 17.5 contracts). For the cost of the premium, the exporter is guaranteed that the 75,196.07 U.S.dollars received from the sale of merchandise can be exchanged for a minimum of 1530 Sf per U.S. dollar, until the option expires in December. If the U.S. dollar should increase in value above the current rate the option contract would not be necessary since more Surinamese Guilders would be realized upon the sale of the goods thus benefiting from a favorable currency move.
Since the sale of call options results in the payment of a premium to the seller of the option, it is possible to sell call options in order to achieve a potentially attractive rate of return. Of course, since the writing of call options on currencies places an obligation on the option writer, considerable risk accompanies such a strategy. Writing calls on a currency can be done on a "covered" basis by holders of currency positions. It should be recognized that the writing of call options limits the rewards available in the event of a sharply rising currency. This could be considered an "opportunity cost." The strategy of selling call options against assets already held is called "covered" call writing. It should not be confused with the much more speculative "naked" call writing. The writer of an option is obligated, if the option is exercised, to perform according to the terms of the option contract: to deliver the required number of units of the underlying currency at the option exercise price if the option is a call, or to purchase the required number of units at the option exercise price if the option is a put. The investor considering writing options should clearly understand that the holder of an American-style option can exercise his rights under the option at any time. Moreover, once a writer has been assigned a notice of exercise, he may no longer liquidate his option position by an offsetting purchase. In general, covered writing strategies afford the investor additional income and a measure of "downside protection" - a cushion against a decline in value of the underlying assets. In exchange for these benefits, the writer forgoes the opportunity to profit should the underlying asset increase in value beyond the exercise price of the option written. Example: With the US$ exchange rate at Sf.1530, an Surinamese investor has 62,500 Sf in interest bearing investments yielding 7%. By writing a three month out-of-the-money Sf /US$ call, with a strike price of Sf1530, at a premium of Sf750, the investor could increase the rate of return on his investment by 7.73% annually - a total yield of 14.73% annually (interest plus option premium), calculated as follows: The option premium for a 3 month option represents 1.93% of the underlying value [$750 / (62,500 x $.6212)], or 7.73% on an annualized basis (assuming a similar option premium can be collected each quarter). The 7.73% annualized return, plus the original investment yield of 7% earns 14.73% annualized. An option position known as a "spread" is created by the purchase of one option with a given exercise and expiration, and the sale of the same class of option (a call or a put) with a different exercise price and/or expiration. Although some investors engage in complex spread trading strategies, which are not covered in these pages, a number of relatively simple spread techniques exist that can expand an investor's opportunities once the basic purpose of a spread is understood. The purpose of a spread transaction is to establish a position in the options market that has clearly defined risk and reward parameters. A spread can be designed to match the investor's currency value expectations and tolerance for risk by selecting specific options to be bought and sold. Thus, as with other option strategies, the spread trader should have an opinion as to the probable direction of exchange rates and choose a spread position accordingly. Spread strategies are not solely for a directional outlook. There are many varieties of spread strategies, using various option instruments in differing proportions, that an investor may employ to protect against foreign exchange exposure or to capitalize on just about any forecasted market scenario. Just as spread strategies pose a large variety of investment alternatives to the investor, there are limitless other ways options can be combined to exploit market opportunities. Straddles, strangles, combos, 3-ways, etc., are just some of the ways options can be used as flexible tools to meet the objectives of the investor. The countless strategies that can be employed using options are well beyond the scope of this introductory manual but we do encourage the reader to take advantage of the wealth of information on the subject.
S R G SURINAM GUILDER SPOT USD/SRG Price of SRG Crncy 691.5 USD/SRG Strike: 691.5 100.000% (USD)Rate: 5.505%Mmoney Market Forward USD/SRG Exercise Type: E European (SRG)Rate: 5.505%Mmoney Market Price: 691.5 Put or Call: C Call Time to Expiration: 90 11:03 Trade: 9/ 9/99 10:56 Expiration: 12/ 8/99 22:00 Settle Date: 9/13/99 Exercise Delay: 2 Option Valuation and Risk Parameters Notes USD Value Percent Time Value: 28.334843 Price: 28.33486 1.959% Theta: .151823 Option to BUY 10MM SRG Volatility: 10.000% Type:1User entered Delta: 0.50312 Spot hedge: 503123.81983 Gamma: 5465.72852 Vega: 0.00000 - The Italic entries represent INPUT-fields. The screen will differ slightly for each option type. 1) Standard options 10) Spread options 2) Warrant options 11) Chooser options 3) Cross-currency options 12) Compound options 4) Executive options 13) Power options 5) Enhanced options 14) Digital barrier options 6) Barrier options 15) Ratchet [cliquet] options 7) Asian [average] options 16) Range accrual options 8) Digital [binary] options 17) Two color rainbow options 9) Lookback options
1= Default Models American call options: The Roll-Geske Model American put options: 100 Step Binomial Model European call options: The Black-Scholes Model European put options: The Black-Scholes Model 2= Black-Scholes (European exercise is assumed) 3= Binomial (Cox-Ross-Rubenstein) 4= Modified Roll (Roll-Geske Model for calls on stock with dividends) 5= Square Root Constant Elasticity of Variance (CEV) 6= Enhanced Discrete Dividends (a binomial tree for discrete dividends where the stochastic variable is the full stock price. NOTE: American style options can be exercised at any time up to expiration. European style options can be exercised only at expiration. As is customary with short dated equity option models such as the Black-Scholes model and the Binomial model, Jet Investments makes the following assumptions: Trading in the markets for stocks, options and bonds is frictionless. The risk free interest rate is constant over the life of the option. Use the Modified Roll model for calls with dividends. Use the 100 Step Binomial model for puts. For American options, the principal model used is the Binomial Tree model of Cox, Ross and Rubenstein, where: stock prices S(t)follow a multiplicative binomial process such that for all times t s(t)*exp(u) with probability q S(t+1) = S(t)*exp(v) with probability 1-q Consistent with the approach of Jarrow and Rudd we choose:
u = (r-.5*sigma**2)*t/I + sigma*sqrt(t/I) v = (r-.5*sigma**2)*t/I - sigma*sqrt(t/I) q = .5 where I is the granularity of the walk (i.e. the number of steps) and t is the time to expiration. In the Binomial Tree model, u and v are arbitrary but constrain q to equal: (r-d)/(u-d) to avoid riskless profits in the binomial walk. NOTE: Jarrow Rudd parameters guarantee that the binomial walk value for an American call on an equity with no dividends during the life of the option converges to the Black-Scholes value. At any node, N in the tree with successor nodes Nu (upward stock price changes) and Nd (downward price changes) the call option value opt(N) satisfies:
opt(N)=max[.5*(opt(Nu + Nd)*exp(-r*t/I),S(N) -E] If a dividend occurs at the time period corresponding to N S(N)-E in the above formula it is replaced by S(N)-E-div. In general, when a dividend occurs its value is subtracted from the stock price process. European options are evaluated via the standard Black-Scholes formula below. Call price = exp(-r*t)*[S*exp(r*t)*N(x)-E*N(x-v*sqrt(t))] Put price = Call price - S + E*exp(-r*t) where: S = Underlying Equity Price E = Strike Price v = percent volatility of Equity Price, annualized (expressed as decimal) t = time to expiration in years r = short term interest rate (decimal) x = log(S*exp(r*t)/E)/v*sqrt(t) + .5*v*sqrt(t) N() = the cumulative normal distribution function.
Source: Jarrow, Robert and Rudd, Andrew, OPTION PRICING, Richard D. Irwin, Inc., Homewood,Ill., 1983. For example: Suppose INTEL Corp. trades at 44 1/2. What is the value of a ninety day call on this stock at a strike of 45 if we assume a 25% price volatility and a short rate of 9.2%? S = 44.5 E = 45. t = .2465 r = .092 v = .25 x = .1537 N(x) = .56108 x - v*sqrt(t) = .02998 N(x-v*sqrt(t)) = .51196 exp(-r*t) = .97758 Call Price = .97758*(44.5*1.0227*.56108-45.*.51196) = 2.45 If the option market perfectly obeyed the Black-Scholes model we would expect the same implied volatilities for both calls and puts over all strikes and maturities for a given underlying security. However, as one may observe from the CALL and PUT screens for equities, this is not always the case. Below, we cite historical evidence for what appear to be systematic biases in the model.
The warrant pricing models available use an option-theoretic approach. This approach says that the value of the warrant is determined by specifying a stochastic process, such as a random variable which changes through time. For clarity we temporarily assume that the underlying stock pays no dividends. As a starting point, recall that the Black-Scholes formula begins by specifying how the stock price moves through time. More accurately, what is specified is the return - the change in value divided by starting value (dS/S)
(1a) says that the return over a short time period, dt, has an expected component ( the m * dt term), and a random component (the sigma * dZ term). Further, Black-Scholes take m and sigma to be constant, and dZ to be Gaussian (i.e. normal), which is equivalent to saying that S = S(t) follows a lognormal process. Let V denote the value of the firm, n_S the number of outstanding shares of common stock. Then
We could rewrite (1a) in terms of V and state that V follows a lognormal process. Now, suppose that the firm issues warrants, say n_W of them. We extend (1a,1b) as follows:
Note that (2a) further generalizes (1) so that the coefficient sigma is now (potentially) a function of the value of the firm V. We will use this generalization in the CEV model. Finally, consider the additional parameters, the exchange rate, E and the foreign riskfree rate, q. E = amount of currency of warrant per one unit of currency of stock (2c) r = riskfree rate in currency of warrant (continuous compounding) q = riskfree rate in currency of underlying (continuous compounding) Using the notation of (2a) - (2c) we write the warrant price as
Note that (3) is not a formula, it simply lists the items on which the warrant price depends. The omission of m is actually a substitution by r, the riskfree rate, and follows from options pricing theory. X is the exercise price, denominated in the currency of the underlying. In order to fully specify a model we must pin down a choice for sigma. The warrant price depends on a lognormal process for V. This choice for sigma extends the Black-Scholes (for European) or the Cox-Ross-Rubinstein binomial tree method (for American). For example, for the single currency case, the price of a European warrant is given by
It is important to note that W appears on both sides of (4), so that this represents an implicit equation for W, which is solved for using numerical techniques. Note also that V appears in (4) since V = n_S*S+n_W*W, by (2b). We can see that (4) is an extension of Black-Scholes by setting n_W = 0, in which case the right-hand-side of (4) reduces to the classic formula for the premium of a call option. For the two currency case, proceed as follows. First, consider a related warrant which is identical to the given warrant except that it is denominated in the same currency as the underlying. We can use the single currency formula above to compute its value in the currency of the underlying. We must be careful to replace the riskfree rate, r, in (4) by q, the riskfree rate in the underlying currency. Once the warrant value is determined in this currency, use the current spot F/X rate, E, to convert the price to the warrant currency.
This case is the square root CEV model, where significantly the volatility sigma(V) is not constant but depends on the value of the firm. The key idea is that as the value of the firm, V, becomes lower, the volatility of the return from V increases. The increased volatility can be thought of in several ways. The ratio of operating revenues to fixed expenses has presumably deteriorated, making V a riskier investment. Similarly, the inherent leverage has increased, increasing the return volatility. Conversely, as V rises in value, the opposite occurs. More importantly, empirical research has found such a relationship between sigma and V, (i.e. that sigma increases if V decreases) and also that the CEV based-model appears to perform better than the sigma = constant choice above. (NOTE: In all cases the models are refined to take dilution into account, as we see for example in (4) above.) We do not include any formulas here, but the interested reader can refer to Lauterbach and Schultz, esp. pp. 1200-1201. (see references). So far, in discussing both models, we have made a simplifying assumption, for the sake of staying focused on the essential points. Namely, that the underlying stock pays no dividends during the life of the warrant. There are a number of modeling choices when dividends are allowed. The crucial issue comes in dealing with American warrants. For standard call options, it is well known that rational early exercise will only occur just before an ex-dividend date, if at all. In general, early exercise is justified by either a relatively large dividend or a short time between ex-dividend and option expiration. A rule of thumb states that typical dividends are not large enough to justify early exercise other than on the last ex-dividend date before expiration. This rule leads to the (modified) Roll call option pricing method, which is used in the Jet Investments equity options screens. Besides treating dividends as discrete events, another technique is to approximate dividends with a dividend rate (resulting in dividends proportional to the value of the firm.) Over the long term, a reasonable economic argument can be made that reduced firm value will coincide with reduced dividend pay outs. With the Jet Investments models, both the proportional dividend method and the Roll method are provided, with the latter being labeled explicitly in the warrant screens. This information has been prepared to give an introduction to the basics of trading options on currencies. Numerous books and widely published research are available regarding the trading of currency options which can be useful to the interested investor. A list of relevant publications is contained in the appendix. Furthermore. Jet Investments can always be contacted for detailed information and ‘tailor-made’ advice.
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