Currency Options and Options Markets Chapter 6 Currency Options and Options Markets Learning objectives  Option payoffs – Payoff profiles and profit/loss diagrams – Option value determinants  Hedging with options – Combinations of options & with underlying positions  Exchange rate volatility – Implied volatility

6.1 What Is an Option? An option is an agreement giving the option holder (buyer or owner) the right to buy or sell an underlying asset at a specified price, on (or perhaps before) a specified date. European: exercisable only at expiration American: exercisable anytime until maturity The option writer (seller) holds the obligation to fulfill the other side of the contract. Just for fun (not in the text ) - Options typically have more value if left unexercised because the time value of the option is related to the time to expiration. - As a consequence, the value of a European option typically is close to the value of a similar American option. (Exceptions can occur for deep-out-of-the-money options when interest rates are high, although these circumstances are rare.)

6.1 What Is an Option? A currency option is a contract giving the option holder the right to buy or sell an underlying currency at a specified price by a specified date. Call option = an option to buy Put option = an option to sell Note that when you buy one currency you are selling another currency.

6.1 What Is an Option? CME pound Dec 1500 call (American) Type of option: a call option to buy pounds Underlying asset: CME December pound sterling futures contract Contract size: £62,500 Expiration date: 3rd week of December Exercise price: $1.50/£ Rule for exercise: an American option exercisable anytime until expiration

6.1 What Is an Option? CME currency option quotations British pound (CME contract £62,500; prices in ¢/£) Strike Calls-Settle Puts-Settle Price Oct Nov Dec Oct Nov Dec 1480 4.89 5.49 6.02 2.90 3.51 4.04 1490 4.34 4.96 5.50 3.34 3.97 4.51 1500 3.83 4.46 5.00 3.83 4.46 5.00 1510 3.37 3.99 4.54 4.37 4.99 5.53 1520 2.95 3.57 4.11 4.94 5.55 6.09 The CME pound Dec 1500 call sells for $0.0500/£

6.2 Option Payoff Profiles A forward obligation A £1 million obligation due in four months Currency exposure Underlying transaction v$/£ s$/£ -£1,000,000

6.2 Option Payoff Profiles A forward hedge Buy £1 million in the forward market at the forward price F1$/£ = $1.50/£. Exposure of forward contract Long pound forward v$/£ +£1,000,000 s$/£ -$1,500,000

6.2 Option Payoff Profiles A currency option hedge A currency option is like one side of a currency forward contract: The option holder gains if pound sterling rises. The option holder does not lose if pound sterling falls. v$/£ Long pound call (option to buy pound sterling) s$/£

6.2 Option Payoff Profiles Payoff profile of a pound call at expiration

6.2 Option Payoff Profiles Profit (loss) on a long call at expiration FX rate at expiration $1.50/£ $1.55/£ $1.60/£ (Premium)*(contract size) -$3,125 –$3,125 –$3,125 (Exercise price)*(contract size) $0 –$93,750 –$93,750 (Spot £ sale)*(contract size) $0 $96,875 $100,000 Net profit –$3,125 +$0 +$3,125

6.2 Option Payoff Profiles Payoff profile of a call option at expiration Long call Short call CallT$/£ -CallT$/£ KT$/£ ST$/£ ST$/£ KT$/£ Out-of-the- money Out-of-the- money In-the- money In-the- money

6.2 Option Payoff Profiles Payoff profile of a put option at expiration PutT$/£ -PutT$/£ Long put Short put KT$/£ ST$/£ ST$/£ KT$/£ In-the- money Out-of-the- money In-the- money Out-of-the- money

6.2 Option Payoff Profiles Puts and calls Call option to buy pounds at KT$/£ An option to sell dollars at KT£/$ DCallT$/£ DPutT£/$ = DST$/£ DST£/$

6.2 Option Payoff Profiles Forwards, puts, and calls A combination of a long call and a short put at the same exercise price and with the same expiration date results in a long forward position at that forward price. Long call Short put Long forward DCallT$/£ + –DPutT$/£ DFT$/£ DST$/£ DST$/£ = DST$/£

6.2 Option Payoff Profiles Put-call parity at expiration: CallTd/f - PutTd/f + Kd/f = FTd/f CallT$/£ -PutT$/£ KT$/£ + + ST$/£ ST$/£ ST$/£ KT$/£ Exercise price Long call Short put FT$/£ = Long forward ST$/£

6.3 Currency Option Values Prior to Expiration Option value = (Intrinsic value) + (Time value) Intrinsic value = Value if exercised immediately Time value = Additional value prior to exercise The time value of a currency option is a function of the following six determinants Exchange rate underlying the option Exercise price or strike price Riskless rate of interest id in currency d Riskless rate of interest if in currency f Volatility in the underlying exchange rate Time to expiration

6.3 Currency Option Values Prior to Expiration Time value and volatility

6.3 Currency Option Values Prior to Expiration The interaction of time and volatility If instantaneous changes are a random walk, then T-period variance is T times the one-period variance sT2 = T s2  sT = (√T) s where s2 = 1-period variance sT2 = T-period variance Estimation of exchange rate volatility Historical volatility Implied volatility

6.4 Hedging with Currency Options Currency option hedges Advantages Disaster hedge insures against unfavorable currency movements. Disadvantages Option premiums reflect option values and so are high when: The option hedge is in-the-money. Exchange rate volatility is high. There is a long time until expiration.

6.4 Hedging with Currency Options Summary of option payoffs CallTd/f Long call PutTd/f Long put Kd/f Kd/f STd/f STd/f -CallTd/f Short call -PutTd/f Short put STd/f STd/f Kd/f Kd/f

6.4 Hedging with Currency Options A currency call option A pound call is an option to buy pounds: The option holder gains if pound sterling rises. The option holder does not lose if pound falls. V$/£ A long pound call is an option to buy pounds sterling at a contractual exercise price. Exercise price $1.50/£ Option premium S$/£ -$0.30/£

6.4 Hedging with Currency Options A call option hedge Call option hedge V$/£ $1.50/£ S$/£ -$0.30/£ -$1.50/£ Option hedged position -$1.80/£ Short exposure

6.4 Hedging with Currency Options A currency put option A pound put is an option to sell pounds The option holder gains if pound sterling falls. The option holder does not lose if pound rises. V$/£ A long pound put is an option to sell pounds sterling at a contractual exercise price. Exercise price $1.50/£ Option premium S$/£ -$0.30/£

6.4 Hedging with Currency Options A put option hedge Long exposure V$/£ Option hedged position +$1.50/£ +$1.20/£ S$/£ -$0.30/£ $1.50/£ Put option hedge

6.4 Hedging with Currency Options Profit (loss) on a short straddle Short straddle Short put Short call

6.5 Exchange Rate Volatility Revisited Continuously compounded returns Continuously compounded changes in exchange rates s (in italics) are related to holding period changes s according to s = ln(1 + s) or es = eln(1+s) = 1 + s Eqn (6.2) Normally distributed returns If continuously compounded returns are normally distributed, then return variance increases linearly with time. For one-period variance σ2, that means σT2 = (T)σ2 or σT = (T)σ Eqn (6.3)

6.5 Exchange Rate Volatility Revisited Volatility estimates Historical volatility is estimated from actual returns: The standard deviation of currency returns over a recent period provides a easy-to-calculate and easy-to-understand estimate of volatility. Time-varying estimates (see Chapter 3) such as realized volatility and conditional volatility (e.g., GARCH) provide more sophisticated estimates of historical volatility.

6.5 Exchange Rate Volatility Revisited Volatility estimates Implied volatility is implied by an option’s value given the other determinants of option value: Currency option value determinants are: (1) price of the underlying asset, (2) exercise price, (3) domestic risk-free rate, (4) foreign risk-free rate, (5) time to expiration, and (6) volatility of the underlying asset. If option value and the first five determinants are known, then the underlying volatility that generates the option value can be found by trial-and-error given an option pricing model.

Appendix 6-A Currency Option Valuation Learning objectives  A primer on continuous compounding  The Black-Scholes option pricing model  The Biger-Hull currency option pricing model

Appendix 6-A Currency Option Valuation A primer on continuous compounding Natural logarithms and exponentials y = logex = ln(x)  ey = x r = holding period return (e.g., annual return) r = continuously compounded return r = ln(1+r) Û er = (1+r) where e ≈ 2.718 Price changes Compounding: Vt+1 = Vt (1+r) = Vt er Discounting: V0 = Vt / (1+r) = Vt (1+r)-t = Vt e -r t

Appendix 6-A Currency Option Valuation The Black-Scholes option pricing model If continuous returns are normally distributed with mean μ and standard deviation σ, then the value of a European call on a non-dividend-paying stock is Call = P N(d1) – e(–iT) K N(d2) P = the current share price K = exercise price of the call option i = risk-free rate (continuously compounded) σ = instantaneous standard deviation of return T = time to expiration expressed as a fraction of one period d1 = [ ln(P/K) + (i + (σ2/2))T ] / (σ√T) d2 = (d1 – σ√T) N(•) = the standard normal cumulative distribution function

Appendix 6-A Currency Option Valuation The Black-Scholes option pricing model Call = P N(d1) – e(–iT) K N(d2) N(d1) = Probability the option expires in the money P N(d1) = E[ PT | PT > K ]

Appendix 6-A Currency Option Valuation The Biger-Hull currency option pricing model Calld/f = e(–i dT) [ FTd/f N(d1) – Kd/f N(d2) ] = e(–i fT) [ S0d/f N(d1) ] – e(–i dT) [ Kd/f N(d2) ] S0d/f = spot exchange rate FTd/f = S0d/f e(+i dT) e(–i fT) = forward exchange rate Kd/f = exercise price on one unit of foreign currency id = domestic risk-free rate (continuously compounded) if = foreign risk-free rate (continuously compounded) σ = instantaneous standard deviation of the FX rate T = time to expiration d1 = [ ln(Sd/f/Kd/f ) + ( (id – if) + (σ2/2) ) T ] / (σ√T) d2 = (d1 – σ√T) N(•) = the standard normal cumulative distribution function

Appendix 6-A Currency Option Valuation The Biger-Hull currency option pricing model Put option values are found from put-call parity Calld/f – Putd/f = e(–i dT) ( FTd/f – Kd/f ) or Putd/f = Calld/f – e(–i fT) S0d/f + e(–i dT) Kd/f for FTd/f = S0d/f e(+i dT) e(–i fT)

Appendix 6-A Currency Option Valuation The Biger-Hull currency option pricing model