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单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chapter 3,Derivative Securities for Currency Risk Management,Currency Options and Options Markets,圣经故事。在圣经创世记第29章曾经提到过，大约在公元前1700年，雅克布用七年的劳动购买了一个准许他与拉班的女儿拉结结婚的期权。但是后来，拉班违约了，他强迫雅克布与自己的大女儿利亚结了婚。雅克布照办了，但是，他深爱的仍然是拉结。于是，他购买了另一个期权，即再劳动七年以换得与拉结结婚。这一次，拉班没有食言。最后，雅克布娶了两个老婆，生了12个儿子。,圣经故事、橄榄压榨机与荷兰郁金香,橄榄压榨机故事。古希腊的数学家和哲学家泰利斯在橄榄丰收之前利用期权获得了低价使用橄榄压榨机的权利。,据说，他是第一个利用期权交易致富的人。泰利斯生活在公元前580年左右古希腊的米利塔斯市，位于今天土耳其的西南海岸。泰利斯运用自己的天文知识在冬季就预测到橄榄在来年春天将获得丰收。虽然没有什么钱，然而他用自己所有的积蓄在冬季淡季就以低价取得了春季旺季所有压榨机的使用权。,当然，他支付的价格也很低，因为当时没有人认为有必要为了这些压榨机来竞价。当春天橄榄获得大丰收时，每个人都想找到压榨机。这时，泰利斯执行他的权利，将压榨机以高价出租，结果赚了一大笔钱。,圣经故事、橄榄压榨机与荷兰郁金香,荷兰郁金香故事。在17世纪30年代的“荷兰郁金香热”时期，郁金香的一些品种堪称欧洲最为昂贵的稀世花卉。,1635年，那些珍贵品种的郁金香球茎供不应求，加上投机炒作，致使价格飞涨20倍，成为最早有记载的泡沫经济。,这股投机狂潮却开启了期权交易的大门。郁金香交易商向种植者收取一笔费用，授予种植者按约定最高价格向该交易商出售郁金香球茎的权利（卖权）。,同时，郁金香交易商通过支付给种植者一定数额的费用，以获取以约定的最低价格购买球茎的权利（买权）。,这种交易对于降低郁金香交易商和种植者的风险十分有用。,圣经故事、橄榄压榨机与荷兰郁金香,Chapter Overview,What is an option,Option payoff profiles,Combinations of options,The determinants of currency option values,Hedging with currency options,A forward obligation,Suppose a U.S. company has a forward obligation of 1 million due at time T in four months. Current spot and forward rates are S,0,$/,= F,T,$/,= $1.45/.,The expected amount due on this forward obligation is,ECF,T,$, = (ECF,T,)(ES,T,$/,) = (1,000,000)($1.45/) = $1,450,000.,If the actual exchange rate is $1.50/, then this 1 million obligation will cost,CF,T,$,= (CF,T,)(S,T,$/,) = (1,000,000)($1.50/)= $1,500,000.,In this case, the U.S. company has an unexpected loss of $50,000.,Underlying transaction,-1,000,000,Currency exposure,D,V,$/,D,S,$/,A forward hedge,This forward exposure can be hedged by buying pound sterling in the forward market, which in this case means simultaneously selling dollars forward.,Buy 1 million in the forward market at the forward price,F,1,$/,= $1.45/,The cash flow time line and the payoff pro the forward contract are,shown on the slide based on the forward rate of exchange is F,T,$/,= $1.45/.,If the actual exchange rate is S,T,$/,= $1.50/, then purchasing 1,000,000 at the forward price of F,T,$/,= $1.45/ will save you $50,000 and offset your loss on the underlying exposure.,Conversely, if the pound falls to $1.40/, you will gain $50,000 on the underlying obligation but lose $50,000 on the forward contract.,A forward hedge,Wouldnt it be nice,to own an insurance policy against a rise in the exchange rate without a corresponding loss,if exchange rates fall?,Long pound forward,+1,000,000,-$1,450,000,Exposure of forward contract,D,V,$/,D,S,$/,An option hedge,A currency option is like one-half of a forward contract,the option holder gains if pound sterling rises,the option holder does not lose if pound sterling falls,Long pound call,(option to buy,pound sterling),D,S,$/,D,V,$/,An option hedge,Options are used for two purposes:,Hedging,Speculation,Hedging is by far the more common use by corporate financial managers.,In this example,a call option,on pound sterling,acts as an insurance policy,(a hedge) against a rise in the value of the pound.,If the actual exchange rate rises to S,T,$/,= $1.50/ at expiration, then the option provides a payoff of (1,000,000)($0.05/) = $50,000.,If the actual exchange rate falls,to S,T,$/,= $1.40/, then the option is,out-of-the-money and is not exercised,.,Of course,this insurance policy does not come free.,The cost of the option is called the option premium.,The option holder pays the option premium when the option is purchased.,Options Contracts: Preliminaries,A,foreign currency option,is a contract giving the option purchaser (the buyer),the right,but not the obligation, to buy or sell a given amount of foreign exchange at a fixed price per unit for a specified time period (until the maturity date).,The buyer,of an option is termed,the,holder,while,the seller,of the option is referred to as,the,writer,or,grantor,.,Every option has three,different price elements,:,The,exercise,or,strike,price, the exchange rate at which the foreign currency can be purchased (call) or sold (put),The,premium, the cost, price, or value of the option itself,The underlying or actual spot exchange rate,in the market,Options Contracts: Preliminaries,An option gives the holder the right,but not the obligation, to buy or sell a given quantity of an asset in the future, at prices agreed upon today.,Calls vs. Puts,Call options gives the holder the right, but not the obligation, to,buy,a given quantity of some asset at some time in the future, at prices agreed upon today.,Put options gives the holder the right, but not the obligation, to,sell,a given quantity of some asset at some time in the future, at prices agreed upon today.,Options Contracts: Preliminaries,European vs. American options,European options can only be exercised on the expiration date.,American options can be exercised at any time up to and including the expiration date.,Since this option to exercise early generally has value, American options are usually worth more than European options, other things equal.,Options Contracts: Preliminaries,Intrinsic Value,The difference between the exercise price of the option and the spot price of the underlying asset.,Speculative Value,The difference between the option premium and the intrinsic value of the option.,Option Premium,=,Intrinsic Value,Speculative Value,+,Currency Options Markets,PHLX(费城证券交易所),HKFE（香港期货交易所）,20-hour trading day.,Options on the over-the-counter,(OTC) market can be tailored to the specific needs of the firm but can expose the firm to,counterparty risk,.,Options on organized exchanges,are standardized, but counterparty risk is substantially reduced.,OTC volume is much bigger than exchange volume.,Trading is in seven major currencies plus the euro against the U.S. dollar.,PHLX Currency Option Specifications,Currency,Contract Size,Australian dollar,AD50,000,British pound,31,250,Canadian dollar,CD50,000,Euro,62,500,Japanese yen,6,250,000,Swiss franc,SF62,500,Foreign Currency Options,Status of an option,a.In-the-money,Call:Spot strike,Put:Spot strike,b.Out-of-the-money,Call:Spot strike,c.At-the-money,Spot = the strike,Currency Futures Options,Are,an option on a currency futures contract,.,Exercise of a currency futures option results in a long futures position for the holder of a call or the writer of a put.,Exercise of a currency futures option results in a short futures position for the seller of a call or the buyer of a put.,Firms may purchase currency call options to,They may purchase currency call options,to hedge future payables;,to hedge potential expenses when bidding on projects; and,to hedge potential costs when attempting to acquire other firms.,Speculators may purchase call options on a currency that they expect to appreciate.,Profit =selling (spot) price option premium buying (strike) price,They may also sell (write) call options on a currency that they expect to depreciate.,Profit = option premium buying (spot) price+ selling (strike) price,The functions of Call Option,The purchaser of a call option will break even when,selling price =buying (strike) price + option premium,The seller (writer) of a call option will break even when,buying price = selling (strike) price+ option premium,Breakeven on Call Option,E,S,T,Profit,loss,c,0,E,+,c,0,Long call,E,S,T,Profit,loss,c,0,E,+,c,0,short 1 call,Firms may purchase currency put options to hedge,future receivables,.,Speculators may purchase put options on a currency that they expect to depreciate.,Profit =selling (strike) price buying price option premium,They may also sell (write) put options on a currency that they expect to appreciate.,Profit = option premium + selling price buying (strike) price,The functions of Put Option,E,S,T,Profit,loss, p,0,E,p,0,long put,E,p,0,E,S,T,Profit,p,0,E,p,0,short put,E + p,0,Breakeven on Put Option,The purchaser of a put option will break even when,buy price =selling (strike) price - option premium,The seller (writer) of a put option will break even when,selling price = buying (strike) price-option premium,Payoff pro a call option at expiration,Long call,Short call,S,T,$/,Call,T,$/,K,T,$/,S,T,$/,-Call,T,$/,K,T,$/,In-the-,money,Out-of-the-,money,Out-of-the-,money,In-the-,money,Currency options are a zero-sum game;,gains on one side are offset by losses on the other.,Payoff pro a put option at expiration,Short put,K,T,$/,Long put,S,T,$/,Put,T,$/,S,T,$/,-Put,T,$/,K,T,$/,In-the-,money,Out-of-the-,money,Out-of-the-,money,In-the-,money,As in the previous slide, options are a zero-sum game;,gains on one side are offset by losses on the other.,Forwards, puts, and callsA Forward by Any Other Name,A combination of a,long call,and a,short put,at the same exercise price and with the same expiration date results in a,long forward,position at that forward price,D,S,T,$/,D,Call,T,$/,D,F,T,$/,-,D,Put,T,$/,D,S,T,$/,D,S,T,$/,Long call,Short put,Long forward,+,=,Forwards, puts, and calls A Forward by Any Other Name,Conversely, a,short call,and a,long put,with the same exercise price and expiration date is equivalent to a,short forward,position.,D,S,T,$/,-D,Call,T,$/,-D,F,T,$/,D,Put,T,$/,D,S,T,$/,D,S,T,$/,Short call,Long put,Short forward,+,=,A Forward by Any Other Name,: Put-call parity,Long forward,=,F,T,$/,S,T,$/,-Put,T,$/,S,T,$/,Short put,+,S,T,$/,Call,T,$/,Long call,K,T,$/,S,T,$/,+,Exercise price,K,T,$/,Put-call parity relates call and put values to the value of a forward contract.,When we want to talk about the value (rather than changes in the value) of a long call and a short put, we need to adjust for the exercise price.,The general case is called “put-call parity” and relates the value of a long call, a short put, the exercise price, and the forward price at expiration:,Call,T,d/f,-,Put,T,d/f,+,K,d/f,=,F,T,d/f,Combinations of options,-A Straddle option,One possible speculative strategy for volatile currencies is to,purchase both a put option and a call option at the same exercise price.,This is called a,straddle,.,Long straddle=a long call and a long put on the same underlying asset and with,the same exercise price.,Short straddle=a short call and a short put on the same underlying asset and with,the same exercise price,.,K,T,S,T,S,T,K,T,V,T,V,T,Long straddle,Short straddle,Combinations of options,-A Straddle option,Heres an example, In early 1995, a rogue trader named Nick Lesson droved the United Kingdoms Barings Bank into bankruptcy through unauthorized speculation in Nikkei stock index futures on the Singapore and Osaka stock exchanges.,Leeson,sold option straddles on the Nikkei index at a time when volatility on the index was low,.,Leeson formed,a short straddle by simultaneously selling calls and put on the Nikkei index,.,Including the proceeds from the sale of the call and the put, the profit/loss diagram on the short straddle position at expiration looks like this:,profit/loss on a short straddle,S,T,Nikkei,K,T,Nikkei,V,T,Nikkei,He placed a bet on the volatility of the Nikkei index. As long as the Nikkei index did not vary too much, Leeson would have won his bet.Leeson loses if the Nikkei index rises too high or falls too low.The fact was: Volatility on the Nikkei index was low at the time Leeson sold his position. As it turned out, the Nikkei index fell below the profitable range.,Leeson incurred further losses by buying futures on the Nikkei index in the hopes of a recovery that, to Barings regret, never occurred.,The Determinants of Option Values,American currency options prior to expiration priced according to the,Black-Scholes model,are:,Option value determinant Call,d/f,Put,d/f,1.Underlying exchange rate (S,d/f,) +,-,2.Exercise price (K,d/f,),-,+,3.Riskless rate in currency d (i,d,) +,-,4.Riskless rate in currency f(i,f,),-,+,5.Time to expiration (T),+ +,6. Exchange rate volatility,s,(s,d/f,) +,+,Option value = Time value + intrinsic value,Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/,1.69,1.70,1.71,1.72,1.73,1.68,1.67,1.66,0.0,1.0,2.0,3.0,4.0,5.0,Spot rate ($/),Option Premium,(US cents/),3.30,5.67,4.00,6.0,1.74,1.67,Total value,Intrinsic,value,Time value,- Valuation on first day of 90-day maturity -,0,Intrinsic Value, Time Value & Total Value for a Call,Option on British Pounds with a Strike Price of $1.70/,The Intrinsic Value of an Option,The intrinsic value of an option is the value of the option if,it is exercised today.,Call option value when exercised=Max(s,t,d/f,-k,d/f,),0,Put option value when exercised=Max(k,d/f,-s,t,d/f,),0,if a call or put option is,out-of-money, its intrinsic value is zero.,If an option is in-the-money, its intrinsic value is equal to the difference between the exercise price and the value of the underlying asset.,Ex,: For Sept 1.60 put option on British,Pound, spot rate is $1.5841, the,option premium is $0.0220, compute,intrinsic value and time value.,Intrinsic value,=X - S,=1.60 - 1.5841 = $0.0159,Time value,= P - intrinsic value,=0.0220-0.0159= $0.0061,The time value of an option,Time value,= Option value - intrinsic value,Intrinsic value,= value if exercised immediately,The,time value,of a currency option is a function of the following six determinants,Exchange rate,underlying the option,Exercise price,or striking price,Riskless rate,of interest,i,d,in currency d,Riskless rate,of interest,i,f,in currency f,Volatility,in the underlying exchange rate,Time to expiration,The two most critical variables are volatility in the underlying exchange rate and the time to expiration.,The time value of an option,Time value,= Option value - intrinsic value,As exchange rate volatility increases, the values of American call and put options increase.,As the time to expiration increases, the values of American call and put options increase.,That is, American option values are greater if volatility in the underlying asset increases or if the time to expiration is longer.,Because option holders continue to gain on one side of the exercise price but do not suffer continued losses on the other side, options become more valuable as the end-of-period exchange rate distribution becomes more dispersed,.,The time value of an option,As expiration, only that portion of the exchange rate distribution that expires in-the-money has value.,The out-of-the money call option has little value,because there is little likelihood of the spot rate climbing above the exercise price.,As the variability,of end-of-period exchange rates,increases,there is an increasing probability that the spot rate will close above the exercise price,.,As the more out-of-the money the option is , the time value is smaller.,价外期权越大，时间价值越小,由于价外期权行使的机会很低，立权人选择不进行套期保值。这种策略的风险是对应资产的价格可能狂升，使该期权到期时变成价内。如果这样，立权人被迫在市场上以比当期高得多的价格购买对应资产，从而遭受损失。,但是，期权订立时的价外越大，对应资产价格上升到价内的可能性越小，立权人风险就越低，因此其时间价值就越低。,Exchange rate volatility and out-of-the-money call option value,Call,T,d/f,S,T,d/f,S,T,d/f,Call,T,d/f,S,T,d/f,S,T,d/f,The time value of an option,An at-the-money call option gains if the spot rate closes farther above, the exercise price but does not lose if the spot rate closes farther belos the exercise price.,As the variability of end-of-period exchange rates increases, an area of the distribution falls farther in-the-money and the options is more valuable.,平价期权的时间价值最大,由于到期时平价期权被执行的概率为50%。那么期权立权人是否要购买对应资产进行套期保值呢？如果已经购买了，而期权到期时为价外，则遭受损失。如果不购买，而期权到期时为价内，再从市场上购买对应资产也要遭受损失。因此，,订立平价期权使立权人面临最大的不确定性。因此，此时时间价值最大。,Exchange rate volatility and at-the-money call option value,Call,T,d/f,S,T,d/f,S,T,d/f,Call,T,d/f,S,T,d/f,S,T,d/f,The time value of an option,For in-the-money call option, if an underlying exchange rate is below the exercise price at expiration, the option has zero value regardless of the how far the closing price falls below the exercise price.,On the other hand, as the spot rate increases,the call option continues to increase in value.,Thus, in-the-money call options benefit from higher volatility.,价内期权越多，时间价值越小,在这种情况下，期权很可能被行使，因此，,立权人需要买入对应资产进行套期,。这种策略有一种风险，对应资产的价格下降很快，期权到期时为价外，立权人就无法卖出对应资产（因为期权不被行使）。此时，在市场上卖出对应资产将遭受损失。,但是，,期权在订立时价内越多，这种风险发生的机会就越小，因此，当期权价内越多时，时间价值越小。,Exchange rate volatility and in-the-money call option value,Call,T,d/f,S,T,d/f,S,T,d/f,Call,T,d/f,S,T,d/f,S,T,d/f,Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/,上表表示：在到期日前，只要还有时间存在，期权就有时间价值。正是这一特征，使得美式期权的持有者很少有到期日前行使其期权，持有者可以将其手中持有的期权转卖而不会行使。,货币期权定价的敏感性分析,1.对即期汇率变化的敏感性（Delta ）:,Options delta,is also called the hedge ratio:,the change in option price with respect to a change in spot rate .,（期权价格对即期汇率变化的敏感性称为,Delta,值）,Ex,: ,=20%, S=$0.72, C,0,=$0.0395. If spot rate changes to $0.71, C,1,=$0.0335, find the Delta.,Delta = ,C/,S,= (0.0395-0.0335)/(0.72-0.71) =$0.6,该结果表明：如果给定,Delta,值,0.6,，那么，即期汇率变化,1,美分（,$0.01/,）,期权费变化将是,0.6,0.010.006$。如果初始期权费是$0.0395/,即期汇率减少1美分（从$1.72/减少到$1.71/ ），则新的期权费就应该为$0.0395-$0.006= $0.0335/,一般地，看涨期权的,Delta,值变化在,+1,和,0,之间。而看跌期权的,Delta,值变化在,1,和,0,之间,交易者根据,Delta,值对期权分类,The higher the delta, the greater the probability of the option expiring in-the-money。,当期权朝实值期权变化时，,Delta,值就上升趋向于,1.0,；,当期权朝虚值期权变化时，,Delta,值就下降趋向于,0,；,看涨期权费的分解,协定价格（$/,）,即期价格,看涨期权费 内在价值 时间价值,（美分/,）（,美分/,）+,（美分/, ）,Delta,值,1.70,1.70,1.70,1.75,1.70,1.65,6.37 5.00 1.37,3.30 0.00 3.30,1.37 0.00 1.37,0.71,0.50,0.28,2.,离到期日的时间（,theta,）,期权的价值随离到期日的时间长度而增加，预期期权费的变化与到期日时间变化的比值称为,Options,theta,：,the,change in option price with respect to a,Change in maturity,.,Ex:,=20%, S=$0.72, T=20 days,C,0,=$0.0288.,If T=18 days, C,1,= $0.0241, compute the theta.,theta=(0.0241-0.0288)/(18-20)= 0.00235,值得注意的是，,theta,值与时间的关系不是线性关系，而是时间的平方根。例如，,3,个月和,1,个月的实值期权的期权费关系是：,即,3,月期的期权费只是,1,月期期权费的,1.73,倍，,而不是,3,倍。离到期日越近，期权价值衰减越厉害！,货币期权定价的敏感性分析,时间价值的衰减对期权交易者来说非常重要。,交易者购买一个1月期或2月期到期的期权，其期权的价值迅速衰减。,同样地，他购买一个6月期的期权费比1月期的期权费贵2.45倍，而12月期的期权费仅比1月期的期权费贵3.46倍。,所以，,交易者通常发现较长到期日的期权有较好的价值！,Days remaining to maturity,Option Premium,(US cents/),A Call Option on British Pounds: Spot Rate = $1.70/,0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,90,80,70,60,50,40,30,20,10,0,In-the-money (ITM),call ($1.65 strike price),At-the-money (ATM),call ($1.70 strike price),Out-of-the-money (OTM),call ($1.75 strike price),0,Theta: Option Premium Time Value Deterioration,3.,对波动率的敏感性（,Lambda,）,期权的波动率定义为基础汇率每日百分比变化的标准差。,波动率对期权价值的重要性是因为,如果汇率的波动率增加，期权被行使的风险增加，期权费可能上升。,如，期权的年波动率为,12.6%,，则每日的波动率的百分比为：,Options Lambda,：期权费对波动率,1,个单位变化的敏感性用 表示,Ex:,=10%, S=$0.72, C,0,=$0.0266,If volatility ( ) rises to 20%, C,1,=$0.0395,Lambda=(0.0395 - 0.0266)/(0.2 - 0.1)= 0.129,货币期权定价的敏感性分析,Lambda: Option Premium Sensitivity to Volatility when the Spot Rate is $1.70/,4.,对利率变化差异的敏感性（,Options,Rho and Phi,）,从期权费的构成看出：期权费与利率呈正相关。由于国内利率与国外利率水平会影响到汇率，因此，我们定义：,国内利率水平的微小变化对期权费的影响称为,Rho,；,而国外利率水平的微小变化对期权费的影响称为,Phi,Rho =,C/,r,d,Rhi =,C/,r,f,Ex:,T=91,=20%, S=$0.72, r,d,=4.76%,C,0,=$0.0395.,If domestic interest rate increases to 5%,C,1,=$0.0392, compute the Rho.,Rho=(0.0392-0.0395)/(0.05-0.0476)= -0.125,货币期权定价的敏感性分析,Interest Differentials and Call Option Premiums when the Spot Rate is $1.70/,Summary of Option Premium Components,Summary of Option Premium Components,
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