CURRENCY OPTION PRICING II

Transcription

1 Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility Black-Scholes Moel for Currency Options Properties of the BS Moel Option Sensitivity Analysis Delta Gamma Vega Theta Rho

2 Calibrating the Binomial Tree Instea of u an, you will usually obtain the volatility, σ, either as Historical volatility Compute the stanar eviation of aily log return (ln(s t / S t- )) Multiply the aily volatility by 5 to get annual volatility Implie volatility In reality, both types of volatility are available from, e.g., Reuters. In fact, OTC options are quote by implie volatility. Euro options quotes, Reuters, /7/003 To construct a binomial tree that is consistent with a given volatility, set u σ t = e, σ t = e, where t = /n is the length of a time step (a subtree) an n is the number of time steps (subtrees) until maturity.

3 Example. Let us construct a one-perio tree that is consistent with a 0% volatility. Using this tree, we will price call an put options. Spot S =.5$/ Strike K =.5$/ Domestic interest rate i $ =.% (continuously compoune) Foreign interest rate i =.% (continuously compoune) Volatility σ = 0% (annualize) Time to maturity = 6 months ( = 0.5) u = exp(..5) =.0773 = exp(-..5) = q = [exp((.0.0).5) ]/(u ) = Risk neutral pricing gives C = [.0846 q + 0 ( q)] exp(-.0.5) = P = [0 q ( q)] exp(-.0.5) =.0435 Spot rate ($/ ) q= q= Call.0846 Put

4 As the number of time steps increases (n an t 0), the binomial moel price converges to the Black-Scholes price. Time step n BS Call Put Graphically: Convergence of the Binomial to BS Moel Call Option # Time Steps Tree BS 4

5 Black-Scholes Moel for Currency Options To price currency options, you can use the Black-Scholes formula on a ivien-paying stock with the ivien yiel replace by the foreign interest rate. Notation Toay is ate t, the maturity of the option is on a future ate T. S /f t or S: Spot rate on ate t, value of currency f in currency F : Forwar rate K : Strike price i : Interest rate on currency (continuously compoune) i f : Interest rate on currency f (continuously compoune) = T t : Time to maturity σ : Volatility of the spot rate (annualize) C : Call P : Put f i i i i C = Se N( ) Ke N( ), P = Ke N( ) Se N( ) () where f f ln( S / K) + i i + σ ln( S / K) + i i σ, () σ σ = σ f This is known as the Garman-Kohlhagen moel 5

6 Note that, in the FX context, you can write the formula in terms of the forwar rate so that the foreign interest rate (or even the spot rate!) oes not appear. Since F ( i i ) = Se, f i i C = e [ FN( ) KN ( )], P = e [ KN ( ) FN( )] (3) where ln( F / K) + σ ln( F / K) σ, (4) σ σ = σ Discounting by the risk-free rate in Equation (3) inicates that the terms in the square brackets are certainty equivalent of the option payoff at maturity. Note: you o have to use the forwar rate that correspons to the maturity of the option. Example. Spot S =.5$/ Strike K =.5$/ Domestic interest rate i $ =.% (continuously compoune) Foreign interest rate i =.% (continuously compoune) (Or you might observe the forwar rate, F =.443$/. Then use (3)-(4)) Volatility σ = 0% Time to maturity = 6 months = [ln(.5/.5) + ( /).5]/(..5) = =..5 = N( ) =.48590, N(- ) = N( ) =.540 N( ) =.44776, N(- ) = N( ) =.544 C =.5 e e =.0939 P =.5 e e =

7 Properties of the Black-Scholes Moel for Currency Options These are also the properties of the BS moel on a ivien-paying stock.. The B-S moel assumes the future spot rate is istribute lognormally. Without this strong assumption we can still put a lower boun on a European call option: C Max(S exp(-i f ) K exp(-i ), 0) To see this, let us first erive the following important result: Result The present value of receiving S T at maturity is S t exp(-i f ). Note: this is NOT S t. The value of foreign interest is subtracte. This can be seen by consiering the following investment strategy (take currency = $, currency f = ): Now: invest $S t exp(-i f ) = exp(-i f ) in a euro eposit. Reinvest the interest continuously in the eposit itself. At maturity: you will receive (=exp(-i f ) exp(i f )) or equivalently $S T. The payoff of a call option at maturity is the larger of S T K or 0. The PV of receiving S T K at maturity is, from the above result, S t exp(-i f ) Kexp(-i ). Since the value of a call option is never negative, we have the above inequality. Graphically: 7

8 Call Spot Rate Call Lower Boun. The prices of forwar ATM call an put options are the same. Graphically, recall that we can create a synthetic forwar by a call an a put. Since the forwar costs nothing, the price of the call an the put must balance. Long Forwar Long Call Short Put Payoff Payoff Payoff F S T /f = + K=F S T /f K=F S T /f Mathematically, since we always have N( ) N( ) = N( ) [ N( )] = N( ) N( ), if K = F in (3), 8

9 9. )] ( ) ( [ )] ( ) ( [ )], ( ) ( [ C N N F e N N F e P N N F e C i i i = = = = Note: Equation (4) also becomes very simple: / = σ, / = = σ. Thus, N( ) = N( ) = N( ) an therefore we can further write ] ) ( [ = = N F e P C i.

10 3. Digital (Binary) Options. Asset-or-nothing (AON) an cash-or-nothing (CON) options are not really exotic. They are the basic builing blocks of the Black-Scholes Moel. By efinition, C = AON(S T > K) K CON(S T > K) P = K CON(S T < K) AON(S T < K) Compare with (). We obtain: AON( S T > K) = Se f i N( ), CON( S T > K) = e i N( ) AON( S T < K) = Se f i N( ), CON( S T < K) = e i N( ) From the expressions for CON, we know that the probability of the call ening up in the money (S T > K) is N( ), the put ening up in the money (S T < K) is N( ). Risk-neutral probabilities of the call an the put ening up ITM Prob. Put ITM N( ) Call ITM N( ) K S T 0

11 Option Sensitivity Analysis Delta The elta of an option (or a portfolio),, is the rate of change in the price of the option (or portfolio) with respect to the spot rate. C f i Delta of a European call: Call = = e N( ) S Recall that 0 < Call <. In the B-S worl, a tighter boun obtains: 0 < Call < exp(-i f ) <. P f i Delta of a European put: Put = = e N( ) S - < -exp(-i f ) < Put < 0. Delta of the European call option in the Example Delta Spot Rate As the figure shows, Call exp(-i f ) (closer to ) as S t (ITM). Call 0 as S t 0 (OTM). Q. What is the limit of the elta of a put as S t or S t 0?

12 Delta of ITM, ATM, an OTM calls plotte against time to maturity Delta Time to Maturity Q. In the above example, recall that S =.5 an F =.443. Why oes the elta of the.5 call converge to 0? Why oes the elta of the.05 call become closer to? The elta of a spot contract is (confirm this). The elta of a forwar contract is exp( i f ). Q3. Show this by first writing own the value of a long forwar contract (this is ifferent from the formula for the forwar rate) an then ifferentiating it with respect to the spot rate. The elta of a futures contract is exp((i i f )). This is slightly ifferent from the forwar contract because of the mark-to-market. The mark-to-market enables you to realize the gain or loss cause by the change in the spot rate immeiately (aily). The elta of a portfolio is the sum of the eltas of its component assets.

13 Example, continue. The elta of the above call option is exp( i f )N( ) = exp(.% 0.5) = The elta of the put option is exp( i f )N( ) = exp(.% 0.5) = Q4. A bank has sol the above put option on million. How can the bank make its position elta neutral using the CME futures contract? The size of the CME euro futures contract is 5,000. What cares shoul be taken about this elta hege? Gamma The gamma of an option/portfolio is the rate of change of the option/portfolio s elta with respect to the spot rate. It is the secon partial erivative of the option/portfolio price with respect to the spot rate. It measures the curvature of the relation between the option/portfolio price an the spot rate. The gamma of a European call option an a put option with the same strike price turns out to be the same: C Γ = S P e = = S i f n( ) Sσ where n( ) is the stanar normal ensity function. Q5. Show this equivalence (not the formula) by the put-call parity. The gammas of a spot, forwar, an futures contract are all zero. You shoul be able to erive this. 3

14 Gamma of the European call in the example 6 Gamma Spot Rate Graphically, this is the slope of the graph of the elta. Confirm this. Delta neutrality provies protection against relatively small spot rate movements. Delta-Gamma neutrality provies protection against larger spot price movements. Traers typically make their position elta-neutral at least once a ay. Gamma an vega (see below) are more ifficult to zero away, because they cannot be altere by traing the spot, forwar, or futures contract. Fortunately, as time elapses, options ten to become eep OTM or ITM. Such options have negligible gamma an vegas. 4

15 Vega The vega of an option/portfolio, υ, is the rate of change of the option/portfolio value with respect to the volatility of the spot rate. The vega of a regular European or American option is always positive. Intuitively, the insurance (time) value of an option increases with volatility. The vegas of a European call an a put are the same. Q6. Show this using put-call parity. Vega of the European call in the example Vega Spot Rate 5

16 Theta The theta of an option/portfolio, Θ, is the rate of change of the option/portfolio value with respect to the passage of time. Theta is usually negative for an option, because the passage of time ecreases the time value. For an ITM call option on a currency with a relatively high interest rate, theta can be positive, because the passage of time shortens the time until the receipt of the foreign interest if exercise in the money (see the picture below). Theta is not the same type of hege measure as others Greeks, because there is no uncertainty about the passage of time. Theta of the European call in the example Theta Spot Rate 6

17 Rho The rho of an option/portfolio is the rate of change of the option/portoflio value with respect to the interest rate. For currency options, there are two rhos, one corresponing to the change in the omestic interest rate an the other corresponing to the foreign interest rate. The rho of a European call option with respect to the omestic interest rate is positive. It is negative for a put. The rho of a European call option with respect to the foreign interest rate is negative. It is positive for a put. Q7. Explain the above two points. Rho of the European call w.r.t. the omestic interest rate in the example Rho Spot Rate 7

18 Suggeste solutions to questions Q. Put 0 as S t (OTM). Put -exp(-i f ) (closer to -) as S t 0 (ITM). Q. The elta of the.5 call converges to 0 because, as 0, it becomes progressively sure that the option will en up OTM. On the other han, the elta of the.05 call becomes closer to because it becomes very likely that the option will en up ITM. Q3. The value of a forwar contract is f = S exp( i f ) K exp( i ) Thus, the elta of a forwar contract is f/ S = exp( i f ). Q4. The elta of the futures contract is exp((.%.%) 0.5) = Let x be the amount of futures contracts that the bank shoul hol long. We set ( million) x = 0. x = 0.5 million x / 5,000 = Thus, the bank shoul sell four CME euro futures contracts. As in the Dozier case, this is not a perfect hege. The bank must rebalance ynamically to remain elta-hege. There is a size mismatch. Q5. By the put-call parity, we have C P = S exp( i f ) K exp( i ). (*) Differentiating with respect to S gives C P = e S S That is, the ifference between the eltas of the call an the put equals the foreign iscount factor. Further ifferentiation yiels f i C P = 0. S S This shows the equivalence of the two eltas.. 8

19 Q6. The put-call parity relation (*) above oes not involve σ. Thus, ifferentiating with C P respect to σ gives = 0. σ σ Q7. Equations (3) an (4) can be consiere a B-S moel on a forwar contract. A higher omestic interest rate or a lower foreign interest rate will increase the forwar rate an therefore makes the call option more ITM, an the put more OTM. 9