# Creating and Pricing Hybrid Foreign Currency Options

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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 .Accessed: 11/06/2014 09:02

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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 ma- turity 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 mar- ket 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 export- ing 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 mar- keting 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

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60 FINANCIAL MANAGEMENT/WINTER 1988

Exhibit 1. Put with Proportional Coverage P K

\

B

< I

-.8

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 pro- portional 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 or- dinary 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 be- tween 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 cor- responds 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 tai- lored hedge can thus be set up to provide the exporter with the desired trade-off between protection, flexi- bility, 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 specifica- tions 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.

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62 FINANCIAL MANAGEMENT/WINTER 1988

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 ordi- nary 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%.

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BRIYS AND CROUHY/HYBRID FOREIGN CURRENCY OPTIONS 63

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 pro- files 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 sen- sitivity of the hedge ratio to the foreign currency price. 10See Breeden and Litzenberger [2].

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64 FINANCIAL MANAGEMENT/WINTER 1988

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 fol- lows 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 com- bined 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

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BRIYS AND CROUHY/HYBRID FOREIGN CURRENCY OPTIONS 65

But since u...

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