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Currency Option Pricing Ii Jones Graduate School Masa Watanabe Rice University INTERNATIONAL FINANCE MGMT 657 CURRENCY OPTION PRICING II ¾ Calibrating the Binomial Tree to Volatility ¾ Black-Scholes Model for Currency Options Properties of the BS Model ¾ Option Sensitivity Analysis Delta Gamma Vega Theta Rho International Finance Fall 2003 CURRENCY OPTION PRICING II Calibrating the Binomial Tree Instead of u and d, you will usually obtain the volatility, σ , either as Historical volatility à Compute the standard deviation of daily log return (ln(St/ St-1)) à Multiply the daily volatility by √252 to get annual volatility Implied volatility ¾ In reality, both types of volatility are available from, e.g., Reuters. ¾ In fact, OTC options are quoted by implied volatility. Euro options quotes, Reuters, 11/27/2003 ¾ To construct a binomial tree that is consistent with a given volatility, set u = eσ dt , d = e−σ dt , where dt = τ/n is the length of a time step (a subtree) and n is the number of time steps (subtrees) until maturity. 2 International Finance Fall 2003 CURRENCY OPTION PRICING II Example. Let us construct a one-period tree that is consistent with a 10% volatility. Using this tree, we will price call and put options. Spot S = 1.15$/€ Strike K = 1.15$/€ Domestic interest rate i$ = 1.2% (continuously compounded) Foreign interest rate i€ = 2.2% (continuously compounded) Volatility σ = 10% (annualized) Time to maturity = 6 months (τ = 0.5) u = exp(.1×√.5) = 1.0773 d = exp(-.1×√.5) = 0.93173 q = [exp((.012 – .022)×.5) – d]/(u – d) = 0.44709 Risk neutral pricing gives C = [.08426×q + 0×(1 – q)]×exp(-.012×.5) = .03745 P = [0×q + .07851×(1 – q)]×exp(-.012×.5) = .04315 Spot rate ($/€) q=.44709 1.2343 1.15 1-q=.55291 1.0715 Call Put .08426 0 .03745 .04315 0 .07851 3 International Finance Fall 2003 CURRENCY OPTION PRICING II ¾ As the number of time steps increases (n → ∞ and dt → 0), the binomial model price converges to the Black-Scholes price. Time step n 1 5 10 50 100 500 BS Call 0.0374 0.0310 0.0286 0.0292 0.0293 0.0294 0.0294 Put 0.0431 0.0367 0.0343 0.0349 0.0350 0.0351 0.0351 ¾ Graphically: Convergence of the Binomial to BS Model Call Option 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.025 0.024 2 6 10 14 18 22 26 30 34 38 42 46 50 # Time Steps Tree BS 4 International Finance Fall 2003 CURRENCY OPTION PRICING II Black-Scholes Model for Currency Options To price currency options, you can use the Black-Scholes formula on a dividend-paying stock with the dividend yield replaced by the foreign interest rate.1 Notation Today is date t, the maturity of the option is on a future date T. d/f St or S: Spot rate on date t, value of currency f in currency d F : Forward rate K : Strike price id : Interest rate on currency d (continuously compounded) if : Interest rate on currency f (continuously compounded) τ = T – t : Time to maturity σ : Volatility of the spot rate (annualized) C : Call P : Put −i f τ −idτ −idτ −i f τ C = Se N(d1 ) − Ke N(d2 ), P = Ke N(−d2 ) − Se N(−d1 ) (1) where 1 1 ln(S / K) + i d − i f + σ 2 τ ln(S / K) + i d − i f − σ 2 τ 2 2 d1 ≡ , d2 ≡ (2) σ τ σ τ = d1 − σ τ 1 This is known as the Garman-Kohlhagen model 5 International Finance Fall 2003 CURRENCY OPTION PRICING II Note that, in the FX context, you can write the formula in terms of the forward rate so that the foreign interest rate (or even the spot rate!) does not appear. d f Since F = Se(i −i )τ , −idτ −idτ C = e [FN(d1 ) − KN(d2 )], P = e [KN(−d2 ) − FN(−d1 )] (3) where 1 1 ln(F / K) + σ 2τ ln(F / K) − σ 2τ 2 2 d1 ≡ , d2 ≡ (4) σ τ σ τ = d1 − σ τ ¾ Discounting by the risk-free rate in Equation (3) indicates that the terms in the square brackets are certainty equivalent of the option payoff at maturity. ¾ Note: you do have to use the forward rate that corresponds to the maturity of the option. Example. Spot S = 1.15$/€ Strike K = 1.15$/€ Domestic interest rate i$ = 1.2% (continuously compounded) Foreign interest rate i€ = 2.2% (continuously compounded) (Or you might observe the forward rate, F = 1.1443$/€. Then use (3)-(4)) Volatility σ = 10% Time to maturity = 6 months 2 d1 = [ln(1.15/1.15) + (.012 – .022 + .1 /2)×.5]/(.1×√.5) = -.035355 d2 = d1 – .1×√.5 = -.10607 N(d1) = .48590, N(-d1) = 1 – N(d1) = .51410 N(d2) = .44776, N(-d2) = 1 – N(d2) = .54224 C = 1.15×e-.22×.5×.48590 – 1.15×e-.12×.5×.44776 = .02939 P = 1.15×e-.12×.5×.54224 – 1.15×e-.22×.5×.51410 = .03509 6 International Finance Fall 2003 CURRENCY OPTION PRICING II Properties of the Black-Scholes Model for Currency Options These are also the properties of the BS model on a dividend-paying stock. 1. The B-S model assumes the future spot rate is distributed lognormally. Without this strong assumption we can still put a lower bound on a European call option: C ≥ Max(S exp(-ifτ) – K exp(-idτ), 0) To see this, let us first derive the following important result: Result 1 f The present value of receiving ST at maturity is Stexp(-i τ). à Note: this is NOT St. The value of foreign interest is subtracted. This can be seen by considering the following investment strategy (take currency d = $, currency f = €): f f à Now: invest $Stexp(-i τ) = €exp(-i τ) in a euro deposit. Reinvest the interest continuously in the deposit itself. à At maturity: you will receive €1 (=exp(-ifτ)×exp(ifτ)) or equivalently $ST. The payoff of a call option at maturity is the larger of ST – K or 0. The PV of receiving ST – K at maturity is, from the above result, f d Stexp(-i τ) – Kexp(-i τ). Since the value of a call option is never negative, we have the above inequality. Graphically: 7 International Finance Fall 2003 CURRENCY OPTION PRICING II Call 0.3 0.25 0.2 0.15 0.1 0.05 0 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 -0.05 Spot Rate Call Lower Bound 2. The prices of forward ATM call and put options are the same. Graphically, recall that we can create a synthetic forward by a call and a put. à Since the forward costs nothing, the price of the call and the put must balance. Long Forward Long Call Short Put Payoff Payoff Payoff F K=F d/f d/f d/f ST = K=F ST + ST Mathematically, since we always have N(–d2) – N(–d1) = 1 – N(d2) – [1 – N(d1)] = N(d1) – N(d2), if K = F in (3), 8 International Finance Fall 2003 CURRENCY OPTION PRICING II −idτ −idτ C = e F[N(d1 ) − N(d2 )], P = e F[N(−d2 ) − N(−d1 )] −idτ = e F[N(d1 ) − N(d2 )] = C. Note: Equation (4) also becomes very simple: d1 = σ τ / 2 , d2 = −σ τ / 2 = −d1 . Thus, N(d2) = 1 – N(–d2) = 1 – N(d1) and therefore we can further write −idτ C = P = e F[2N(d1 ) −1]. 9 International Finance Fall 2003 CURRENCY OPTION PRICING II 3. Digital (Binary) Options. Asset-or-nothing (AON) and cash-or-nothing (CON) options are not really exotic. They are the basic building blocks of the Black-Scholes Model. By definition, C = AON(ST > K) – K·CON(ST > K) P = K·CON(ST < K) – AON(ST < K) Compare with (1). We obtain: −i f τ −idτ AON(ST > K) = Se N(d1 ), CON(ST > K) = e N(d2 ) −i f τ −idτ AON(ST < K) = Se N(−d1 ), CON(ST < K) = e N(−d2 ) From the expressions for CON, we know that the probability of the call ending up in the money (ST > K) is N(d2), the put ending up in the money (ST < K) is N(–d2). Risk-neutral probabilities of the call and the put ending up ITM Prob. Put ITM Call ITM N(–d2) N(d2) K ST 10 International Finance Fall 2003 CURRENCY OPTION PRICING II Option Sensitivity Analysis Delta The delta 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 ¾ Delta of a European call: ∆ = = e−i τ N(d ) Call ∂S 1 Recall that 0 < ∆Call < 1. In the B-S world, a tighter bound obtains: f 0 < ∆Call < exp(-i τ) < 1. ∂P f ¾ Delta of a European put: ∆ = = −e−i τ N(−d ) Put ∂S 1 f -1 < -exp(-i τ) < ∆Put < 0. Delta of the European call option in the Example Delta 1.2 1 0.8 0.6 0.4 0.2 0 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Spot Rate As the figure shows, f ∆Call → exp(-i τ) (closer to 1) as St → ∞ (ITM). ∆Call → 0 as St → 0 (OTM). Q1. What is the limit of the delta of a put as St → ∞ or St → 0? 11 International Finance Fall 2003 CURRENCY OPTION PRICING II Delta of ITM, ATM, and OTM calls plotted against time to maturity Delta 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.2 Time to Maturity 1.05 1.15 1.25 Q2.
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