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Creating and Pricing Hybrid Foreign Currency OptionsAuthor(s): Eric Briys and Michel CrouhySource: Financial Management, Vol. 17, No. 4 (Winter, 1988), pp. 59-65Published by: Wiley on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665767 .

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Creating and Pricing Hybrid Foreign Currency Options

Eric Briys and Michel Crouhy

Eric Briys is Associate Professor of Finance and Michel Crouhy is Professor of Finance at Centre HEC-ISA, Jouy en Josas, France.

M For more than ten years, foreign currency markets have been characterized by wide price changes. This high volatility of exchange rates has exposed corporate treasurers and international investors to a significant level of currency risk. Currency options markets have developed to provide new means of dealing with this growing risk. The purchaser of a foreign currency call (put) option has the right, but not the obligation, to buy (sell) a given amount of a foreign currency at a predetermined ("strike") price at anytime on or before maturity date for an American option, or only at maturity for a European option.

Currency options have been the subject of consid- erable professional and academic interest as well.1 In-

deed, these are traded on several security exchanges throughout the world. A sizable over-the-counter market has also developed, offering a variety of specialized currency options. Among such specialized options are the so-called hybrid foreign currency options. These hybrid foreign currency options are based on some use of the put/call parity relation. For example, the exporting corporation buys a put and simultaneously sells a call from the bank so that the overall cost is less than the cost of a straight put, but then the exporter loses any potential gain from an appreciation of the foreign currency above the strike price of the call. Currency risk management has thus become a delicate compromise between flexibility, protection, and cost. The achieve- ment of such a trade-off amounts to tailoring an instru- ment which perfectly matches the needs of the investor, conditional on his anticipations.

Banks world-wide have been quite successful in marketing those hybrids, although theoretically they can be replicated by combinations of instruments traded on organized markets. A typical hybrid like the Cylinder

We are grateful to the anonymous referees and Hans Stoll, Associate Editor, for their insightful comments. The usual caveat applies.

IFor a general background in currency options, see Biger and Hull [1], Garman and Kohlhagen [7], and Giddy [8].

59



60 FINANCIAL MANAGEMENT/WINTER 1988

Exhibit 1. Put with Proportional Coverage P K

\

••B

< I

-.8<1

0 K S

P = payoff, S = spot rate at expiration, K = strikeprice, 0 = proportional coverage rate.

Option from Citibank or the Range Forward from Salomon Brothers allows its holder to customize the hedge by selecting the appropriate strike price and a maturity which might not be available on an organized exchange. Moreover, hybrid foreign currency options are mostly of the European type, making them less costly than American options. The potential for a wide variety of specifications has attracted many customers. In that respect, the French hybrid foreign currency option market is interesting. While no organized currency option market exists in France, French banks have been actively competing among themselves to market various instruments like the Range Forward, the Par- ticipating Forward, or the Conditional Forward-just to cite a few.2 All these contracts are designed in such

Exhibit 2. Put with Bounded Payoff

KP

K-B

0 B K S

P = payoff, S = spot rate at expiration, K = strike price, B = lower bound of the spot rate at maturity under which no

additional coverage is provided.

a way that the buyer does not have to pay any up-front cost, and they are referred to as zero premium hybrid foreign currency options.3

I. Tailoring Hybrid Foreign Currency Options

The buyer of a hybrid foreign currency option (e.g., a corporate treasurer of an exporting firm) aims at the best trade-off between protection, flexibility, and cost. On the one hand, corporate treasurers often claim that put contracts are too "expensive." On the other hand, standard zero premium hybrid foreign currency options cost nothing to the buyer to initiate, but the

2See Briys and Crouhy [3,4].

3In a Range Forward contract, the buyer and the bank agree on two

prices, Si and S2, at the inception of the contract. At the maturity of the forward contract, the buyer will purchase the foreign currency either at S1 if the spot price is less than SI, or at S2 if the spot price is greater than S2, or at the spot price if it is between S1 and S2. The two prices, S1 and S2, are set such that no money exchanges hands at the inception of the contract. A Participating Forward contract guar- antees a minimum exchange rate for a forward sale and a maximum exchange rate for a forward purchase. In addition, the seller (buyer) gets a participation in the foreign currency appreciation (deprecia- tion). Obviously, there is a cost to this participation. Since the contract is structured with no up-front payment, the cost of the seller's

upside participation is that the minimum exchange rate guaranteed through a participating forward sale will necessarily be greater than the outright forward price. Analogously, the maximum exchange rate

guaranteed through a participating purchase will necessarily be

greater than the outright forward price. A Conditional Forward Pur- chase contract is similar to an outright forward purchase, except that the long side of the contract has the right to pull out of the forward

purchase by paying a fee to the short side of the contract on the contract maturity date. The contract can also be designed in such a way that there is no cancellation fee, simply by guarantying a buying price lower than the forward price. Similarly, a Conditional Forward Sale contract is equivalent to an outright forward sale, except that the short side of the forward contract has the right to pull out of the agree- ment by paying a fee to the long side of the contract.



BRIYS AND CROUHY/HYBRID FOREIGN CURRENCY OPTIONS 61

Exhibit 3. Put with Disappearing Deductible

P

0 0 K S

P = payoff, S = spot rate at expiration, K = strike price, D = deductible.

residual flexibility of these instruments is slim when they are compared to forward contracts. Three ways of dealing with this issue are considered.

The first way to lower the cost of a put is to slightly reduce the level of protection, as in the "put with proportional coverage" shown in Exhibit 1. If the contract expires in the money, the payoff is limited to a percent- age P of what it would be for an ordinary put:

P = P(K-S) 0 < 0 < 1. (1)

When 13 = 1, the contract becomes identical to an ordinary put. P can be understood as a coinsurance rate and as a means of reducing the cost of the premium. Obviously, an equivalent payoff can be obtained by holding 3 puts.

The second way to reduce the cost of the protection of a put is to bound the payoff of the put from above as shown in Exhibit 2. If at maturity the spot rate lies between the strike price K and the limit B, the payoff is that of a regular put, i.e., K - S. If the spot rate falls below B, the payoff is bounded at K - B. Such an instru- ment may be of interest to a corporate treasurer whose view on the currency is one of a moderate decline.4

The last case considered here corresponds to the so- called "put with disappearing deductible" as shown in

Exhibit 3. In that case, the protection is not effective as long as the spot rate at maturity lies above D. For a spot rate belowD, the option is exercised and the payoff corresponds to what would be obtained with a straight put with a strike price K The payoff profile exhibits a dis- continuity at D.

These basic protection schemes are obviously not mutually exclusive and can be easily combined. A tailored hedge can thus be set up to provide the exporter with the desired trade-off between protection, flexibility, and cost.

II. A General Pricing Formula Instead of pricing each of the above contracts in-

dividually, a general pricing formula for put contracts5 is derived. It can then be adapted to the exact specifications of the hybrid to be priced. Clearly, as long as the payoff function of a put contract is given as a stepwise decreasing linear function, it is straightforward to sepa- rate it into decreasing continuous segments, as shown in Exhibit 4.6

The payoff at maturity is equal to:

0 if St <

Kj.1 Pi(S,)

= Lj + tj(St - K) if K St (2) 0 if St >Kj

where j is thejth segment of the overall payoff profile and St is the foreign currency price at maturity date t. An ordinary put is the case with olj = -1, Lj =

0, Kj-1 = 0, andj = I with Ki = K

Following the risk neutrality approach proposed by Cox and Ross [5] to value options, the current premium for the above generalized payoff segment is equal to its discounted expected value,

pi = e-rd t

(Lj - aojKj + otSt)L'(St)dSt , (3)

KjI

with the following notation:

t = time to maturity, St = the foreign currency spot price at

4Note that this payoff is similar to that of a spread.

5Since we are only considering European contracts, the extension to call profiles can easily be shown by applying the well-known put/call parity condition first proposed by Stoll [9].

6See Cox and Rubinstein [6], pp. 371-375.




Exhibit 4. Generalized Payoff Segment

P

0 Lj

0 Kj-1 Kj S

P = payoff, S = spot rate at expiration,

Kj i-K = spot rate range for the payoff function, Lj = payoff at maturity date for S = Kj, oa0 = slope of the payoff function.

maturity date t, rd = the domestic riskless interest rate,

L'(St) = the "risk neutral" lognormal density function of the foreign currency price at the maturity of the option (see Appendix).

Integrating this expression yields7

P (So, t) =(Lj -

otjKj)erd t[N(d-) - N(dJ )I

+ tSoe-rf't[N(d' -N(d)], (4)

with So being the current spot exchange rate, rf being the foreign riskless interest rate, and where

-1

_ln(So/Kj_)

+ (rd - rf- u2/2)t

d2 ,

(5)

In(So/Kj) + (rd - f r U2/2)t

d2 (6)

-1 j-1 d, =d2 +o X/• (7)

d d = d2 + -- (8)

The total premium for an hybrid put contract is therefore

P = Pj(So,t) . (9)

We can verify that (4) collapses to the Garman-Kohl-

hagen pricing formula when tj = -1, Lj = O, Kj-j = 0, andj = 1 with Ki = K, which corresponds to the ordinary put

P1(So,t) = Ke-rdtN(-d2) - Soe-rftN(-dl) , (10)

where

In(So/K) + (rd - rf- o2/2)t d2 = , (11)

d1 = d2 + t(Ti (12)

Using the above general pricing formula, the premium on the put with the "disappearing deductible," and with

Kj = D, Kj1 = 0, Lj = K - D, j = 1, and ao = -1 is equal to

Pj(So,t) = Ke-rdtN(-d2)-Soe-rftN(-d) , (13)

where

In(So/D) + (rd - rf- o2/2)t

d2 = (14)

d,= d2 + T N/

The pricing of the other schemes is straightforward and follows the same procedure.

Exhibit 5 graphs the premium of the put with "dis-

appearing deductible" as a function of the current spot price for 3 months, 2 weeks, and 5 days to maturity. This can be compared with the graph of a straight European

7See the Appendix for a sketch of the proof.

8This example is derived for the $US against the French Franc as the domestic currency. The domestic and foreign interest rates are 8.375% and 6.875% respectively, the volatility is 18%, the strike of the straight put is K = 5.75 FF, the deductible is (K -D) = 0.75%.




Exhibit 5. Graphs of the Premia for the Put with Disappearing Deductible

1.7 - 1.7

1.6

1.5

1.4

1.3

1.2

1.1

2. 0.9

S 0.8

0.7

0.6

0.5

0.4

0.3

0.2 0.1

- 4 4.4 4.5 5.2 5.6 6 6.4 EXCHANGE RATE

3 G-K(3 months) + 3 months A 2 weeks X 5 days

put option at maturity.8 When the residual life of the option becomes less than two weeks, continuous hedg- ing of the bank's position requires special attention and might become costly. Exhibit 6 shows that the delta of the hybrid becomes quite substantial around the criti- cal value D, and is very sensitive to changes in the spot rate. Indeed, one can notice for the option with 5 days to maturity that when the currency price goes from 5.0 to 5.1 FF per $US, the hedge ratio changes from -3.33 to -1.96 and the gamma "jumps" from 4.31 to 18.3.9

III. Conclusion This paper stresses the wide applicability of the op-

tion pricing theory when it comes to protecting or covering financial assets or a foreign currency position. The current trend in financial institutions is to propose

tailored conditional profiles which are well suited to match the needs and expectations of customers. The three types of put contracts which have been described above fulfill this goal. Again, constructing payoff profiles from known instruments is a standard exercise.10 This construction can, however, become an intricate task implying significant transaction costs. This ex- plains why banks have been so successful in marketing these "jigsaw" products to their customers.

References 1. N. Biger and J. Hull, "The Valuation of Currency Options,"

Financial Management (Spring 1983), pp. 24-28. 2. D.T. Breeden and R.H. Litzenberger, "Prices of State-Contin-

gent Claims Implicit in Options Prices," Journal of Business

(March 1978), pp. 621-651. 3. E. Briys and M. Crouhy, "Les Options de Change

' Prime Z6ro," Revue Banque (November 1987), pp. 1046-1050.

9Delta = 3P/lS is the hedge ratio, and gamma = d2PI/S2 is the sensitivity of the hedge ratio to the foreign currency price. 10See Breeden and Litzenberger [2].




Exhibit 6. Delta's and Gamma's of the Hybrid with Disappearing Deductible Time to maturity

3 months 1 month 2 weeks 5 days S Delta Gamma Delta Gamma Delta Gamma Delta Gamma

4.5 -1.21 -0.44 -1.14 -1.13 -1.02 -0.37 -1.00 0.00 4.6 -1.25 -0.27 -1.28 -1.71 -1.10 -1.44 -1.00 -0.05 4.7 -1.26 -0.01 -1.46 -1.90 -1.34 -3.45 -1.04 -1.13 4.8 -1.25 0.32 -1.63 -1.31 -1.77 -4.73 -1.42 -7.93 4.9 -1.20 0.67 -1.70 0.05 -2.17 -2.59 -2.65 -14.10 5.0 -1.11 0.99 -1.61 1.72 -2.18 2.57 -3.33 4.31 5.1 -1.00 1.23 -1.37 3.01 -1.69 6.55 1.96 18.30 5.2 -0.87 1.37 -1.04 3.47 -1.00 6.62 -0.51 8.92 5.3 -0.73 1.40 -0.70 3.14 -0.45 4.20 -0.06 1.53 5.4 -0.59 1.34 -0.42 2.37 -0.16 1.87 0.00 0.11 5.5 -0.46 1.20 -0.23 1.53 -0.04 0.61 0.00 0.00

Where delta = aP/aS is the hedge ratio, and gamma = 32P/lS2 is the sensitivity of the hedge ratio to the foreign currency price.

4. -, "Les Hybrides Pour la Gestion du Risque de Change," Revue Banque (February 1988), pp. 161-165.

5. J.C. Cox and S.A. Ross, "The Valuation of Options for Alter- native Stochastic Processes," Journal of Financial Economics (March 1976), pp. 145-166.

6. J.C. Cox and M. Rubinstein, Options Markets, Englewood, NJ, Prentice-Hall, 1985.

7. M.B. Garman and S.W. Kohlhagen, "Foreign Currency Option Value," Journal of International Money and Finance (March 1983), pp. 231-237.

8. I.H. Giddy, "Foreign Exchange Options," The Journal of Futures Markets (March 1983), pp. 143-166.

9. H. Stoll, "The Relationship Between Put and Call Option Prices," Journal of Finance (September 1969), pp. 801-824.

Appendix Equation (3) can be decomposed as the sum of two

integrals:

Kj Kj-1

PJ = e-rdt I ajcStL'(St )dSt

+ e-rd (L - ctjK)L'(S,)dSt Kj-1 I

=A-B

We first evaluate the first integral, A. Define the new variable u such that eu = St, which implies that eudu = dSt.

If St is the spot price of the foreign currency at ma-

turity date t, being lognormally distributed, then it follows that

E(St) = e ([ + O.5?2)t

(A 2)

where [L and 02 are the "risk neutral" distribution mean and variance per unit of time. If we assume that the covered interest arbitrage relationship holds, then

Ft erdt

So erft (A3)

where So is the current spot rate and Ft is the t-period forward rate. By the assumption of risk neutrality, the forward rate is equal to the expected value of the spot rate at expiration, i.e., Ft = E(St), which when combined with (A2) and (A3) yields:

E(St) = Soe(rd- rf)t = e( + 0.52)t .

(A4)

It then follows that the parameters of the risk neutral density function are given by

([ + 0.5o2)t = In(So) + (rd - rf)t . (A5)

We can then rewrite A, as

oxje-rdt

Kj

A = eue-(u- t2/2a2du . (A6)

2,ro2 Kj_ I




But since u - (u - pt)2/2o2t = -[u - (jx + o2)t]2/2 o2t + pt + o2t/2, A thus becomes

Oc e-rdte( ( + 0.5u2)t

Kg A = - t( + 5

el[u - (pL + g2)t]2/2a2tdu .(A7)

Making the new change of variable: u - (1 + a2)t/l F- = y and using (A4), it follows that

,j•.S

e'frt [ln(Kj) -

(4L +

r2)t]/

A = I ey2/2dy , (A8)

[ln(Kj-1) - ([ + Or2t)]/r

that is

A = otSoe-rft Prob (-d j-l'< y <5 -dl',

(A9)

where y is the standard normal variable, and where

-d

1 In (Kj_)

-

([L +

ro2)t -d =, (AIO)

In(Kj) - ( + U2)t

-da =

. (All) 0VFc

Denoting byN(.) the univariate cumulative normal distribution, it follows

A = ct1Soe-rt [N(-d) - N(-d/j-) , (A12)

but

(pL + U2)t = (p + r2/2)t + C2t/2 = In(So) + (rd -rf)t + C2t/2 ; (A13)

thus

A = aotSoerft[N(dl) -N(d')], (A14)

where we have rewritten

j- In(So/Kj_l)

+ (rd - rf + &2/2)t

d -

(A15)

In(SO/K1) + (rd - rf + o2/2)t d - , (A16)

Q.E.D.

The evaluation of integral B would follow the same line of argument.

CALL FOR PAPERS

The Administrative Science Association of Canada will hold its annual finance conference in Montreal, Quebec, June 1-4, 1989. The purpose of the meeting is to provide a forum where a wide range of theoretical and empirical issues in all areas of financial economics are presented and debated by academic scholars, prac- ticing managers, and government bodies. Monetary awards (sponsored by Bell Canada, Caisse de D6p6t et Placement du Quebec, and Canadian Shareholders Association) will be presented for outstanding research papers in Corporate Finance, Investments, and Professional Investment Analysis. To participate, please send four (4) copies of your completed paper and/or indicate whether you are interested in organizing a panel dis- cussion, or serving as a chairperson/discussant. Send your paper by January 30, 1989 to:

Abol Jalilvand 1989 Finance Program Chairman, ASAC

Concordia University Faculty of Commerce and Administration

1455 de Maisonneuve Blvd. West Montreal, Quebec H3G 1M8, CANADA



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