FX Options 1 Outline Part I Chapter 1: basic FX options, standard terminology, mechanics Chapter 2: Black-Scholes pricing model; some option pricing relationships 2 Outline Part II Chapter 3: Binomial trees; use for pricing more complex options Chapter 4: quiz 3
Chapter 1 Basic FX options Standard terminology Mechanics of options 4 FX Option Basic Terms Contract that gives one party right to purchase from ( call ), or sell to ( put ), the other party a defined quantity of one currency against a defined quantity of another currency on or before a specified date. If right to exercise arises only on the last day, option is European, while if the option can be exercised any time, option is American. Date on which right to exercise lapses is called expiration date Time until expiration is maturity or tenor Price per unit at which two currencies are exchanged is strike Amount of each currency exchanged is notional amount for that currency Premium is price paid by option buyer to option seller for acquiring the option. 5 FX Option Sample Term Sheet Sterling call /US Dollar Put Notional Amount 10,000,000 Strike 1.50 Trade Date January 1, 2009 Expiration Date July 1, 2009 Style Premium European $250,000 6
Option Exercise Owner 10MM $15MM Seller 7 FX Option Additional Insights So a call on Sterling against US dollars is also a put on US dollars against Sterling. You can choose to name this instrument either way, but initially we recommend you give it a dual name i.e. Sterling call/dollar put until you are sure not to get confused about this. Obviously owner would not exercise option at expiration unless /$ spot > 1.50; otherwise he is better off purchasing in spot market. But owner of option, even a European one, may not wait until expiration to see if he has a gain; he can sell the option in secondary market if market conditions are favorable. 8 Long Call (excl. premium) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 500,000-1.00 1.20 1.40 1.60 1.80 2.00 9
Long Call (incl. premium) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 500,000 - (500,000) 1.00 1.20 1.40 1.60 1.80 2.00 10 Long Put (excl. premium) 4,000,000 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 500,000-1.00 1.20 1.40 1.60 1.80 2.00 11 4,000,000 Long Put (incl. premium) 3,500,000 3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 500,000 - (500,000) 1.00 1.20 1.40 1.60 1.80 2.00 12
Short Call (excl. premium) 1,000,000.00 500,000.00 - -500,000.00 1.00 1.20 1.40 1.60 1.80 2.00-1,000,000.00-1,500,000.00-2,000,000.00-2,500,000.00-3,000,000.00 13 Short Call (incl. premium) 1,000,000 500,000 - (500,000) 1.00 1.20 1.40 1.60 1.80 2.00 (1,000,000) (1,500,000) (2,000,000) (2,500,000) (3,000,000) 14 Short Put (excl. premium) - 1.00 (500,000) 1.20 1.40 1.60 1.80 2.00 (1,000,000) (1,500,000) (2,000,000) (2,500,000) (3,000,000) (3,500,000) 15
Short Put (incl. premium) 500,000 - (500,000) 1.00 1.20 1.40 1.60 1.80 2.00 (1,000,000) (1,500,000) (2,000,000) (2,500,000) (3,000,000) (3,500,000) 16 Maximum Gain Call Put Long Unlimited Strike x Notional* Short Premium x Notional Premium x Notional */ But note that if we express the gain in Sterling, our result would get infinitely large as Sterling became increasingly worthless. 17 Chapter 2 Option pricing formula Important pricing characteristics of basic FX options 18
Garman-Kohlhagen Model for FX Options Spot 1.5000 Call Price 0.0413 Fwd 1.5000 Put Price 0.0413 Strike 1.5000 R T 5.00% R B 5.00% Vol 10% Tenor 0.50 C = S exp [-r B t] N(d 1 ) - X exp [-r T t] N(d 2 ) P = X exp [-r T t] N(-d 2 ) - S exp [-r B t] N(-d 1 ) 19 Continuous Compounding If I know effective rate, I can get continuously compounded rate r simply via the equation r = LN(1+R), where R is the effective rate So for example if an instrument pays a nominal interest rate of 5% that is semiannually compounded, I can derive the continuously compounded rate via the following two steps; Convert 5% (s.a.) into effective rate by writing in Excel = EFFECT(5%,2), which yields 5.06%; and Convert this into the continuously compounded rate by writing in Excel =LN(1+5.06%), which yields 4.94% The FV of $1 continuously compounded for t years at rate r is EXP(rxt) The PV of $1 discounted for t years at a c.c. rate r is EXP(-rxt) 20 Volatility (1) Cannot be observed precisely, nor is it constant; so must input estimate of what it is expected to be Reflects how much currency bounces around, versus moving smoothly Following are generally correct; Vol is higher for free-floating than for controlled currencies; Higher for EM currencies than for G7; and Usually for G7 currencies fluctuates between 5% and 15% (annualized), and averages close to 10% 21
Volatility (2) Must always input volatility on annualized basis If I know the vol for a period T which is not one year, I can convert it into an annualized volatility via this equation: Ann. Vol VolT 1/ T where T is expressed in years For example if monthly vol is 4%, annualized vol is AnnVol. 4% 1/(1/12) 4% 12 13.86% 22 Forward Formula for forward is also based on continuously compounded interest rates You would get same result if you used the discrete rate and the more familiar formula Forward = Spot x ((1+R T x t)/ (1+R B x t). 23 Option Pricing Parameters Spot Strike 1.50 1.50 Rate in (base currency) Rate in $ (term currency) Vol Tenor (in years) 5% 5% 10% 0.5 24
Option Moneyness In-the-money At-the-money Out-of-the-money Calls F>X F=X F<X Puts F<X F=X F>X Our definitions make no reference to spot, which we think is the better approach especially for European options which cannot be exercised prior to expiration Think of forward as market s best estimate of where spot will most likely be at expiration For American options, alternative definition is more defensible since can exercise at any time. Best advice is simply to always specify what you mean, by saying in-the-money-spot or in-themoney-forward, etc 25 Approximate Pricing for European ATMF Options Premium 0.4 Spot Vol 1 ( RT T ) T Formula does not distinguish between puts and calls which are therefore worth the same when both are ATMF, as confirmed in Cells G6 and G7 of the worksheet This is illustration of put-call parity, which, in its narrow version, says a European call and put whose strike equals the forward will have equal premium. 26 Sensitivities for ATMF Options Spot Movement Effect on Call Value Effect on Put Value Spot Vol Vol Tenor (Usually) (Usually) Tenor (Usually) (Usually) 27
Sensitivities Relationships for Options that are ITMF or OTMF If call is ITM, put is OTM (if they have the same strike), and vice versa So the ITM one is almost always more expensive than the other Like ATM options, calls gain and puts lose value when spot rises, and vice-versa Like ATM options, calls and puts gain value when vol rises, and vice-versa when it falls, but the relationship is no longer linear Like ATM options, calls generally gain value and puts generally lose value when the tenor is extended, and vice versa when the tenor is reduced. But approximate rule mentioned previously, that value increases by square root of time, no longer holds at all 28 DITM Options and Effect of Tenor Increase 3 month call with strike 1.25 is worth $0.2469, while 6-month call is worth $0.2440 Call we started with was so DITM that it has become in essence a forward A forward purchase contract at $1.25 when the current forward rate is $1.50 would have value equal to PV of this difference of $0.25 As we lengthen it to 6 months, we would be discounting same $0.25 but for longer period Of course this assumes 3-month rate and 6-month rate are equal 29 Put-Call Parity Spot 1.5500 Call Price 0.0708 Fwd 1.5500 Put Price 0.0220 Strike 1.5000 Call - Put 0.0488 R T 5.00% PV of Fwd at 1.50 0.0488 R B 5.00% Vol 10% Tenor 0.50 C = S exp [-r B t] N(d 1 ) - X exp [-r T t] N(d 2 ) P = X exp [-r T t] N(-d 2 ) - S exp [-r B t] N(-d 1 ) 30
Put-Call Parity Call K Put Where: CallK is the premium for a call with strike K, PutK is the premium for a put with strike K, F is the forward rate K is the strike R is the term currency interest rate, and T is the Tenor of the option K ( F K) EXP( R T ) ( F K) e RT 31 Put-Call Parity: Explanation If you are long a call and short a put at strike K, you are certain to take delivery at expiration: if S>K, you will exercise call, while if S <K counterparty will exercise her put So you are long a forward contract at price K, the value of which we know is the present value of (F-K), which is the RHS of our equation above It follows that Put price can always be derived from Call price, and vice-versa: Call Put K K Put K Call K ( F K) EXP( RT ) and ( F K) EXP( RT ) 32 FX Options (Part II) 33
Outline Chapter 3: Binomial trees Chapter 4: quiz Part II 34 Chapter 3 Pricing with binomial trees Price wider spectrum of options 35 Binomial Model: Intuitive Explanation (1) Over a short time interval, spot will move to one of two levels, symmetrically distributed around forward for that interval U EXP[( RT RB 2) t e [( RT RB 2 ) t 2 T 2 T D EXP[( RT RB 2) t e [( RT RB 2 ) t 2 T ] ] 2 T ] ] If you set vol to zero, you get U D EXP( RT RB) t e ( RT RB ) t This is amount by which you need to multiply spot to get forward 36
Binomial Model: Intuitive Explanation (2) Reminder: U EXP[( RT RB 2 2) t T ] D EXP[( RT RB 2 2) t T ] e [( RT RB 2 ) t 2 T ] e [( RT RB 2 ) t 2 T ] Vol Term at the end is added in formula for UP and subtracted for DOWN Model is saying that spot is expected on average to trend to its forward by expiration date, and than sudden bullish new will have positive effect (relative to forward), while sudden bearish news will have negative effect (relative to forward) Note T adjustment to vol, to convert annualized vol to vol for our time interval σ2/2 term is explained in downloadable PDF To get next 2 possible values at any time, just multiply current value by U and D 37 Binomial Tree (6 Nodes) Spot 1.5000 Forward 1.5000 Maturity (yrs) 0.5 Up Move 1.0289 Term Rate 5.00% Down Move 0.9711 Base Rate 5.00% Time Node 0.0833 Volatility 10.00% Strike 1.5000 FX Price Tree 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 1.2583 38 Binomial Model: Intuitive Explanation (3) Also, since S x U x D = S x D x U, spot level under many different paths reaches same level, so tree is said to be recombining, which diminishes enormously number of nodes and possible outcomes [Recombining tree with N intervals has N+1 outcomes, while non-recombining tree has 2^N outcomes] We can associate a probability with each outcome based on simple multiplication and use of combinatorial : probability of arriving at any one level is 6 6Ck 0. 5 where k ranges from 0 to 6, the zero corresponding to the highest final outcome and the 6 to the lowest final outcome 39
Spot at expiration: probability distribution Spot 1.5000 Maturity (yrs) 0.5 Term Rate 5.00% Base Rate 5.00% Volatility 15.00% Forward 1.5000 Up Move 1.0433 Down Move 0.9567 Time Node 0.0833 Strike 1.5000 35.00% 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% 1.00 1.25 1.50 1.75 2.00 40 FX Price Tree Prob Coeff 1.5000 1.5216 1.5436 1.5659 1.5885 1.6114 1.6347 1.56% - 1.4784 1.4997 1.5213 1.5433 1.5656 1.5882 9.38% 1.00 1.4570 1.4780 1.4994 1.5210 1.5430 23.44% 2.00 1.4360 1.4567 1.4777 1.4991 31.25% 3.00 1.4153 1.4357 1.4564 23.44% 4.00 1.3948 1.4150 9.38% 5.00 1.3747 1.56% 6.00 41 Call Final Payoff FX Price Tree 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 1.2583 European Call 0.2792 0.1794 0.0852 0.0000 0.0000 0.0000 0.0000 42
Backward Induction FX Price Tree 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 1.2583 European Call 0.2284 0.2792 0.1317 0.1794 0.0422 0.0852 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 43 European Call Pricing FX Price Tree: 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 1.2583 European Call: 0.0401 0.0621 0.0927 0.1325 0.1793 0.2284 0.2792 0.0185 0.0320 0.0537 0.0867 0.1317 0.1794 0.0052 0.0105 0.0211 0.0424 0.0852 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 44 European Put Pricing FX Price Tree: 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 0.0401 0.0197 0.0063 0.0005 0.0000 0.0000 0.0000 0.0609 0.0332 0.0122 0.0009 0.0000 0.0000 0.0892 0.0545 0.0236 0.0019 0.0000 Call Price: 0.0401 0.1246 0.0858 0.0455 0.0037 Put Price: 0.0401 0.1644 0.1268 0.0877 Call - Put 0.0000 0.2035 0.1669 PV of Fwd a 0.0000 0.2417 45
Pricing American Options with Binomial Trees Binomial approach to Pricing American options has three steps: proceed as before to develop your trees for the evolution of the spot price and the value of a European option at each node of the tree; then compare the value at each node to the payoff the option holder would have if he chose to exercise his option at that point instead of waiting until maturity; If the latter amount exceeds the value you calculated for the European option then you substitute the latter amount in that cell 46 American Call Pricing FX Price Tree: 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 1.2583 American Call: 0.0403 0.0624 0.0934 0.1337 0.1808 0.2293 0.2792 0.0186 0.0320 0.0538 0.0870 0.1323 0.1794 0.0052 0.0105 0.0211 0.0424 0.0852 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 47 American Put Pricing FX Price Tree: 1.5000 1.5433 1.5878 1.6337 1.6808 1.7293 1.7792 1.4567 1.4988 1.5420 1.5865 1.6323 1.6794 1.4147 1.4555 1.4975 1.5407 1.5852 1.3738 1.4135 1.4543 1.4963 1.3342 1.3727 1.4123 1.2957 1.3331 1.2583 American Put: 0.0405 0.0198 0.0063 0.0005 0.0000 0.0000 0.0000 0.0615 0.0334 0.0123 0.0009 0.0000 0.0000 0.0901 0.0549 0.0237 0.0019 0.0000 0.1262 0.0865 0.0457 0.0037 0.1658 0.1273 0.0877 0.2043 0.1669 0.2417 48
Put-Call Parity for American Options? Long call and short put is a synthetic forward if options are European, since and only one option is certain to be exercised at expiration Same cannot be said for American options: if I am long a call and short a put, I might still exercise both options, but at different times Thus this second combination is not a synthetic forward 49 Comparing Prices Spot 1.5000 Forward: 1.4777 Maturity (yrs): 0.5 Up Move: 1.0133 Term Rate: 5.00% Down Move: 0.9856 Base Rate: 8.00% Time Node: 0.0192 Volatility: 10.00% Strike: 1.5000 European Call 0.0313 European Call (GK) 0.0310 European Put 0.0531 European Put (GK) 0.0528 American Call 0.0336 American Put 0.0531 50 When is Right to Exercise Early Worth Extra? Intuitive answer is that you would exercise American option early if yield on currency delivered to you exceeds yield of currency you must deliver So it makes sense to early exercise Sterling call if Sterling yields 8% while US dollar yields 5% so early exercise feature makes American option more expensive than European But it makes no sense to early exercise Sterling call if Sterling yields 5% while US dollar yields 8% so early exercise feature makes American call more expensive than European call Generally the two types of options will also be worth the same if the two yields are comparable 51
Tenor Extension: European v. American Calls Tenor 0.5 yrs 1yr 2yrs 3yrs 5yrs 10yrs 20yrs European Call 0.0313 0.0373 0.0416 0.0415 0.0354 0.0225 0.0070 American Call 0.0336 0.0427 0.0523 0.0577 0.0624 0.0659 0.0653 European call initially gains in value until it peaks around 2-3 years and begins declining in value after that, as expected American call always more valuable than European call of same tenor, and longer the tenor the more so, since the longer the opportunity to exercise into the high yielding sterling currency American call generally becomes more expensive the longer you make it, which is intuitively obvious since longer call has all the same rights as shorter one and some additional ones 52 Tenor Extension for American Call: Proof that longer is always more expensive Take two otherwise identical American options, Options 1 and 2, except that their tenors T1 and T2 are different, with T1 > T2 Assume P1 < P2, where P1 and P2 are the premia Arbitrageur sells Option 2, buys Option 1 and pockets upfront P2 - P1 If Option 2 is exercised against him, he exercises Option 1 so is fully hedged, and still walks away with his upfront profit If Option 2 expires without being exercised, he is still long Option 1 and still has his upfront profit This is a risk-free money-machine, so our initial assumption was impossible 53 Then why is 20-yr American cheaper than 10-yr American? Our argument on previous slide is airtight Yet 20-year appears a little cheaper than 10-yr, and 100-yr far cheaper still Problem is that our nodes have become ridiculously long almost 4 years so our model has crashed and is spitting out garbage Increasing nodes significantly would cure this problem 54
Chapter 4 Quiz 55 Question 1 When I sell an American put option, which of the following three factors vol rising, spot rising, time passing would be beneficial for me? a) Vol rising and time passing b) Vol rising and spot rising c) Spot rising and time passing d) All 3 factors 56 Pricing Relationships for ATMF Options Spot Spot Movement Effect on Call Value Effect on Put Value Vol Vol Tenor Tenor (Usually) (Usually) (Usually) (Usually) 57
Question 2 What is the most accurate way of stating the put-call parity principle? a) A call and put that have the same strike must have the same premium b) A call and put whose strike equals the spot must have the same premium c) A European call and put whose strike equals the forward must have the same premium d) The premium of a European call minus the premium of a European put that have the same strike must equal the forward 58 Question 3 In comparing American and European FX options with otherwise identical terms, which is the most accurate statement: a) The American option is always more expensive b) They are usually worth the same c) They are worth the same unless the currency of the underlying has a lower yield than the other currency d) They are worth the same unless the currency of the underlying has a higher yield than the other currency 59 Question 4 Which of the following is most accurate about an out-of-the-money European option s price as we extend its maturity? a) The price increases as the option gets longer b) The price initially increases but starts to decrease when the maturity becomes very long c) The price increases as the option gets longer until it eventually reaches a maximum d) The price increases as the option gets longer but at a gradually decelerating pace 60
Qn 4 Solution Since option is OTM, increasing its tenor initially increases its value as it gives more hope of going ITM by expiration date However, value eventually starts declining when tenor becomes long enough, due to impact of discounting payoff over such a long period Sooner or later discounting impact dominates any benefit of lengthening tenor 61 Question 5 Which of the following is false for a put seller? a) A put seller can earn no more than the premium b) A put seller suffers a loss if the spot goes down c) A put seller can lose no more than the initial spot d) The loss to a seller of a put on Sterling against US dollars, if expressed in US dollars, cannot exceed the strike 62 Qn 5 Solution (a) and (b) are clearly correct (d) is correct given the precise wording: a seller of a put on Sterling, at a strike of USD 1.75, say, can lose no more than USD 1.75 if the put is exercised, and this happens only if Sterling has devalued so much as to have become virtually worthless. At this point he receives one worthless sterling and delivers USD 1.75, which becomes his maximum loss. Note that the loss of USD 1.75 becomes a huge amount of Sterling, so his loss is not subject to any limit if expressed in Sterling. (c) is the false statement, since the initial spot level has no bearing on the seller s potential loss; it is the strike that is relevant, not the spot. 63
Question 6 Estimate without using a calculator or computer the premium of a 4-year European ATMF GBP call/usd put on a notional of GBP 100MM if Spot is 1.75, vol is 10% and rates in both currencies are equal and close to zero a) $7,000,000 b) $14,000,000 c) $18,000,000 d) $25,000,000 64 Qn 6 Solution Our shortcut formula for the premium of an ATMF option is 0.4 Spot Vol Premium 1 ( RT T ) T Since RT is close to zero, we can ignore the denominator We also know that since the forward is 1.75 and the rates are roughly equal, spot must also be close to 1.75 Thus applying the above formula gives Premium = 100MM x 0.4 x 1.75 x 10% x 4 = $14MM So the correct answer is (b) 65