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1 OPTIONS 1. FX Options Terminology The Four Basic Positions Standard Options Exotic Options Asian Option (Average Rate Option, ARO) Compound Option Barrier Options (Trigger Options) Digital Options Factors affecting the Option Price Profit and Loss Profiles Call Put Strategies Straddle Strangle Butterfly Spread Option Pricing Models Call/ Put-Parity Diagrams of Synthetic Options Positions Diagrams of Synthetic Underlying Positions FINANCE TRAINER International Options / Page 1 of 38

2 OPTIONS 2. Risk Factors Delta and Delta Hedging Gamma Theta Vega (Kappa) Epsilon (Rho) Skew ( Smile Curve ) and Risk Reversal The Skew of Implied Volatility ( Smile Curve ) Risk Reversal FINANCE TRAINER International Options / Page 2 of 38

3 1. FX Options Ever since the mid-eighties currency options have become an additional liquid instrument in the FX market. An FX option gives the right - but not the obligation - to the option buyer to buy (or to sell) a defined currency amount at an agreed rate (strike price) at expiry date. This means that the holder has the right to exercise the option if this gives him an advantage compared to the actual market rates. A Call option gives the right to buy, a Put option gives the right to sell a currency. The seller of the option receives a premium for giving this right to the buyer. This premium has to be paid on the day the option is traded. The seller of an option has the obligation to buy (or to sell) a defined currency amount at an agreed rate at expiry date. For this obligation he receives a premium. Premium Seller of the option Buyer of the option Right FINANCE TRAINER International Options / Page 3 of 38

4 Call EUR/USD 10 mio Strike: Expiry date: 25 th September Premium: 3 USD Ct. Explanation: Call: deal type EUR/USD: underlying: the instrument, the buyer of the option has the right to buy 10 mio: volume of the base currency (the buyer of the call has the right to buy 10 mio EUR) Strike 1.25: at this price the buyer of the call has the right - but not the obligation - to buy the underlying Expiry date: the last day on which the option seller accepts the exercise of the option Premium: price the buyer has to pay to the seller of the option (3 USD Ct. per EUR) 1.1 Terminology Dates in the option contract The trading date is the day when the option is dealt. Premium payment date is the day when the option premium has to be paid, usually value 2 days. Exercise date is the day when the option buyer exercises the option Expiration date is the last day on which the option seller accepts the exercise of the option. FINANCE TRAINER International Options / Page 4 of 38

5 Settlement date for American options is 2 bank days after exercise date and for European options 2 bank days after the expiry date. In-the-money (ITM)/ at-the-money (ATM)/ out-of-the money (OTM) An option at-the-money (ATM) has a strike price around the actual market rate. If the strike price is compared to the spot rate, the option is called at-the-money spot. By comparing the strike with the outright rate, the option is called at-the-money forward. An option is in-the-money (ITM) if the strike is better than the market rate. For a Call this means that the strike is below the market rate. The Put is in-the-money if the strike is higher than the market rate. An option is out-of-the-money (OTM) if the market rate is better than the strike rate. For a Call out-of-the-money this means a lower market rate than the strike. A Put is out-of-themoney if the market rate is higher than the strike. OTC/ exchange traded options FX Options are almost exclusively traded OTC ( over the counter ). In opposition to exchange traded options where only standardized periods and strikes can be traded there are no restrictions in the OTC market. In practice OTC options have stood up to exchange traded options because the original deals have to be regarded individually. 1.2 The Four Basic Positions DEFINITION PUT CALL LONG BUY/HOLD RIGHT to SELL RIGHT to BUY SHORT SELL/WRITE OBLIGATION to BUY OBLIGATION to SELL FINANCE TRAINER International Options / Page 5 of 38

6 By buying a Call option you acquire the right to buy an agreed amount of a currency at the expiry date. If the market rate on the expiry date is lower than the agreed price, the option will not be exercised and the currency can be bought at the current market rate. During the term of the option you can also profit from low market rates by buying the currency at any time at 'low' rates. Application: Option for the importer, who has to buy the foreign currency (Note: USD Call EUR Put!), hedging instrument against rising FX rates; hedging a short position; trading a long position with limited risk. By selling a Call option the seller has the obligation to sell the currency at the strike price at expiry, if the buyer decides to exercise. For undertaking this risk the seller receives a premium. Application: By selling a Call against a long position, the option premium reduces the average buying price. This strategy however gives no hedge against declining rates and cuts the profit potential in case of higher rates. By buying a Put option you buy the right to sell the currency at the agreed price at expiry. In case of a higher rate at expiry the buyer of the option does not exercise and may sell at the higher market rate. Possible higher rates during the term can be locked in by selling (outright) the currency. Application: Option for the exporter who hast to sell the foreign currency (Note: USD Put EUR Call!), hedging instrument against falling FX rates; hedge against a long spot position; trading short position. The seller of a Put option takes on the obligation to buy the currency at the agreed price at expiry; if the buyer of the option decides to exercise. For taking this risk the seller receives a premium. Application: A company being short a currency can bring down the average price by selling Puts, e.g. a travel agency which needs foreign currency day-to-day. FINANCE TRAINER International Options / Page 6 of 38

7 General rule: the Call in the base currency is at the same time always the Put in the quote currency (and vice versa), e.g. a EUR Call is at the same time a USD Put (the right to buy EUR equals the right to sell USD). 1.3 Standard Options A European style option is an option that can only be exercised on the expiry date. An American style option is an option that can be exercised at any trading day during the life of the option (usually with exchange traded options). A Bermudan style option can only be exercised at certain dates during the option period. It is a mixture of European and American style option. 1.4 Exotic Options In recent years additional to European and American style options a number of different variations of options have been developed. These mutations are called exotic options. Below the most important exotic options are described Asian Option (Average Rate Option, ARO) These are options, which refer to the average rate of the underlying exchange rate that exist during the life of the option. This average will be used to determine the intrinsic value of the option by comparison with the predetermined fixed strike. If the option is a call option and the average rate exceeds the strike, the buyer will receive a cash flow (i.e. the difference between the average rate and the strike). For a put option, the average must be below the strike Compound Option A Compound option is an option on an option: the buyer has the right to buy a plain vanilla call or put option with a predetermined strike at a predetermined date and at a predetermined price (i.e. the options premium). FINANCE TRAINER International Options / Page 7 of 38

8 1.4.3 Barrier Options (Trigger Options) These options are standard options with an additional barrier (trigger) level. The options right ceases to exist (respectively starts to exist) once the spot rate reaches the barrier level. As barrier options might expire prematurely (respectively they might never start to exist), these options are always cheaper than plain vanilla options. Basically there are two types Knock-out options: A knock-out option is a standard option, which expires worthless if a formerly specified exchange rate (barrier, trigger) is dealt in the spot market before expiration. In the knock-out option the spot rate moves towards out-of-the-money in order to reach the outstrike, i.e. the option is OTM when the barrier is hit. spot: down & out call: call strike outstrike the option expires if spot falls below up & out put: put strike outstrike the option expires if spot rises over Barrier options where the spot rate moves towards in-the-money in order to reach the outstrike are called reverse knock-out option or kick-out option. up & out call: call strike outstrike the option expires if spot rises above down & out put: put strike outstrike the option expires if spot falls below Knock-in options: A knock-in option is a standard option, which only starts to exist if a formerly specified exchange rate (barrier, trigger) is dealt in the spot market before expiration. In the knock-in option the spot rate moves towards out-of-the-money in order to reach the instrike, i.e. the option is OTM when the barrier is hit. FINANCE TRAINER International Options / Page 8 of 38

9 spot: down & in call: call strike instrike the option appears if spot falls below up & in put: put strike instrike the option appears if spot rises over Barrier options where the spot rate moves towards in-the-money in order to reach the instrike are called reverse knock-in option or kick-in option. up & in call: call strike instrike the option appears if spot rises above down & in put: put strike instrike the option apears if spot falls below Variants of barrier options: Double knock-out options (double knock-in options) have two barrier levels. They expire worthless (start to exist) if one of them is reached in the spot market. Usually the barrier level is valid during the whole life of barrier options. However, there are some variants where the barrier is just valid at the expiry date or during a predetermined period Digital Options A digital option is a transaction where a specified amount will be paid if the spot rate is above the strike at expiry for calls (or below the strike for puts). Usually the intervention path of spot between the trade date and expiry is irrelevant and the determining factor whether or not the spot is above or below the strike at the time of expiry. However, there are some variants where the strike is valid during the whole life of the option. One Touch / No Touch Options One touch is a transaction where a specified amount will be paid only if spot rate is dealt at the touchstrike or an exceeding exchange rate before expiration. One touch is also called lock-in or touch digital. No touch options will pay a specified amount if spot rate is not dealt at the touchstrike. They are also called lock-out FINANCE TRAINER International Options / Page 9 of 38

10 Double No Touch (Range Binary) A transaction where a specific amount will be paid only if spot is not dealt at, or at levels exceeding the predefined two exchange rates before expiration. 1.5 Factors affecting the Option Price Strike price Intrinsic value Outright rates Term Option premium Time value Volatility If an option has an intrinsic value this means that the strike price is better than the outright rate. All Calls with lower strikes than the outright rate and all Puts with higher strikes than the outright rate have an intrinsic value (ITM). The option premium however is always higher than the intrinsic value. This difference is the time value. It is highest for longer option periods higher expected fluctuations of the underlying The volatility is the measure for the variability or the price range of the exchange rates or underlying prices. There can be differentiated between historical and implied volatility. The historical volatility is calculated out of historical data and is mainly used for risk management calculations. For calculating actual option prices the implied volatility is needed. This volatility is a measure for the market participants expectations concerning the future price range of the underlying. In the professional FX options market, traders only trade the volatility as it is the only number which is up to the trader s opinion. FINANCE TRAINER International Options / Page 10 of 38

11 GBP/USD Spot Outright Premium GBP Put Strike Cent Intrinsic value GBP Put 2 Cent Time value GBP Put 1 Cent Premium Quotations Option premiums are quoted either in per cent of the base currency or in BP of the quote currency. GBP/USD Put Strike premium 2.0 % If you buy the Put with contract volume of GBP 5.0 m you pay a premium of GBP 100,000 (= 5 m x 2.0 %) GBP/USD Put Strike premium 3 Ct. If you buy the Put with contract volume of GBP 5.0 m you pay a premium of USD 150,000 (= 5 m x 0.03 USD) Reuters page with FX options volatilities: FINANCE TRAINER International Options / Page 11 of 38

12 Interpretation of Volatility The fair option price is calculated generally speaking by a statistic model. The main factor is the volatility as a measure for the fluctuation resp. uncertainty. Statistically volatility equals the annualized standard deviation (sigma). Also for risk management calculations the standard deviation is a main factor. How can the standard deviation be interpreted? A standard deviation is the range where 2/3 (exactly 68.26%) of all values (e.g. EUR/USD rates) can be found. In other words one could say that with a probability of 68.26% the value (e.g. EUR/USD rate) will not change more than one standard deviation. The term annualized means that the deviation refers to the period of one year, the so-called holding period is 1 year. frequen cy 68.26% rate 1 standard deviation In the above example the annualized standard deviation is 0.12 resp. 10%, i.e % of all values can be found within the interval from +/ resp. 10% for the period of one year. The volatility (meaning the annualized standard deviation) can be transformed into another holding period by multiplying it by the square root of the holding period (resp. dividing it when the holding period is reduced). FINANCE TRAINER International Options / Page 12 of 38

13 On your Reuters screen you find the EUR/USD vol with 10%. As the implied vol is an annualized standard deviation, this means that the market expects that EUR/USD will not change more than 10% with a probability of 68.26% within one year. What is the standard deviation from one day to the next day (i.e. holding period 1 day)? In order to transform the holding period from 1 year (250 trading days) to 1 day the volatility has to be devided by the square root of the holding period: Vola1year 10% Vol 1D = Vola 1 D = = 0.63% Interpretation: The market expects that the EUR/USD fluctuation from one trading day to the other will not be more than 0.63% with a probability of 68.26%. What is the standard deviation for a holding period of 10 trading days? In order to scale the 1 day-volatility up to a 10 days-volatility it has to be multiplied by the square root out of 10: Vola 10 D = Vola1D 10 Vola 10 T = 0.63% 10 = 2.00% FINANCE TRAINER International Options / Page 13 of 38

14 1.6 Profit and Loss Profiles Call Long Call with strike A and break-even B (strike + premium) + 0 A Spot at expiry of the option - B Loss if spot < Strike + Premium Profit if spot > Strike + Premium The break-even is reached, if the spot rate is at the strike plus premium on expiration day. The maximum loss is the premium. Between point A and point B, exercising the option is profitable but not enough to cover the cost of the premium, therefore you make a loss. Buy EUR Call USD Put: Strike premium: 1 Ct. P&L in USD with the following EUR/USD Spot values at expiry: Spot at expiry Long Call Premium Total (B/E) The higher spot is at expiry, the bigger is the inner value. Premium is the same at every rate (0.01 USD), it is paid immediately when traded. The maximum loss is 0.01 USD (premium), the potential profit is theoretically unlimited. FINANCE TRAINER International Options / Page 14 of 38

15 Short Call with strike A and break-even B (strike + premium) + B 0 A Spot at expiry of the option - Loss if spot > Strike + Premium Profit if spot < Strike + Premium At expiry the seller of the option makes a profit as long as the spot price stays below the strike price plus premium. The potential loss is theoretically unlimited. The maximum gain is the premium. Between point A and B the spot rate is above the strike price, but the seller gains more from the premium as he loses from the option. Sell EUR Call USD Put: Strike , premium: 1 Ct P&L in USD with the following EUR/USD Spot prices at expiration day: Spot at expiry Long Call Premium Total (B/E) The higher spot is at expiry, the bigger is the loss. Premium is the same at every rate (0.01 USD), it is paid immediately when traded. The maximum profit is 0.01 USD (premium), the potential loss is theoretically unlimited. FINANCE TRAINER International Options / Page 15 of 38

16 1.6.2 Put Long Put with Strike A and break-even B (strike premium) + B 0 Spot at expiry of the option - A Loss if spot > Strike Premium Profit if spot < Strike Premium The break-even is reached if the spot price at expiry is at the strike price minus premium. The maximum loss is the premium. Between Point A and B the spot price is below the strike price but the buyer of the option gains less from exercising the option than he has paid for it. Anyway the buyer reduces his premium costs. Buy EUR Put USD Call: Strike , premium: 1 Ct. P&L in USD with the following EUR/USD spot prices at expiry: Spot at expiry Long Put premium total The lower spot is at expiry, the bigger is the inner value. Premium is the same at every rate (0.01 USD), it is paid immediately when traded. The maximum loss is 0.01 USD (premium), the potential profit is theoretically unlimited (up to the value of the strike). FINANCE TRAINER International Options / Page 16 of 38

17 Remark: In the ACI exams the potential profits (losses) of long (short) positions are termed to be substantial! Short Put with Strike A and break-even B (strike premium) + B 0 Spot at expiry of the option A Loss if spot < Strike Premium Profit if spot > Strike Premium For the seller the option is profitable until the spot price at expiration is below the strike price. In this case the gain from the premium is less than the loss from the option exercise buy the buyer of the option. The loss is theoretically limited by a spot price with zero. But because a short Put option is from the risk orientated sight a long position in the underlying (= risk of falling prices) and long FX-positions have unlimited risk, the risk of short Put positions is also unlimited. The maximum gain is the premium. Between Point A and B the spot price is below the strike but the seller gains more from the premium as he loses from the option. Sell EUR Put USD Call: Strike , premium: 1 Ct. P&L in USD with the following EUR/USD spot prices at expiry: Kassakurse am Verfalltag Short Put premium total FINANCE TRAINER International Options / Page 17 of 38

18 The lower spot is at expiry, the bigger is the loss. Premium is the same at every rate (0.01 USD), it is paid immediately when traded. The maximum profit is 0.01 USD (premium), the potential loss is theoretically unlimited (up to the value of the strike). The tables show that the results for buyers and sellers are inverted. 1.7 Strategies Straddle A Straddle is the purchase resp. the sale of a Call and a Put with the same strike. Long Straddle Short Straddle Purchase / sale purchase Call, purchase Put sale Call, sale Put Strikes same, mostly ATM same, mostly ATM Maturity same same Why is a Straddle used? A Long Straddle gains with exchange-rate fluctuations (= volatility) independent of the direction. Because the position is purchased ATM a Long Straddle is an aggressive position. If the underlying does not fluctuate much the relatively high premium is lost. A Short Straddle is used if low volatilities are expected. If the expected volatility is met, the premium is earned, if there are stronger fluctuations than expected the loss is proportional higher the higher the fluctuations are. Long Straddle Long Call and long Put, same term, same strike, normally ATM. FINANCE TRAINER International Options / Page 18 of 38

19 Short Straddle Short Call and short Put, same term, same strike, normally ATM Strangle A Strangle is the purchase resp. the sale of a Call and a Put with different strike prices. Long Strangle Short Strangle Purchase / Sale buy Call, buy Put sell Call, sell Put Strikes different, OTM different, OTM Maturities same same Why are Strangles dealt? The Long Strangle gains if there are strong exchange rate fluctuations independent of the direction. In comparison to the Straddle the maximum loss is smaller, because the options are purchased OTM. Because of the OTM purchase of the option the leverage-effect of the Strangle is higher compared to the Straddle. The Short Strangle gains if there are no strong exchange rate fluctuations. The effects are contrary to the Long Strangle. Long Strangle Long Call and long Put, same term, different strikes, usually OTM. FINANCE TRAINER International Options / Page 19 of 38

20 Short Strangle Short Call and short Put, same term, different strikes, usually OTM Butterfly A butterfly consists of a straddle and an opposite strangle. Long Butterfly Short Butterfly Long / Short short straddle + long strangle long straddle + short strangle Strikes see straddle, strangle see straddle, strangle Maturities same same Why are Butterflies dealt? A Short Butterfly position gains if there are strong exchange rate fluctuations independent of the direction. In comparison with a long strangle or straddle position, the maximum gain of the position is limited as the butterfly position includes a short strangle to reduce the positions total premium The Long Butterfly position yields a profit if there are little to none exchange rate fluctuations. The loss potential is limited, as opposed to short strangle or straddle positions. Long Butterfly Short Straddle and Long Strangle, same maturity FINANCE TRAINER International Options / Page 20 of 38

21 Short Butterfly Long Straddle and Short Strangle, same maturity Spread A Spread is the purchase (sale) of an option and the sale (purchase) of another option at the same time. Contrary to the Straddle or Strangle this strategy is a buy and sell strategy strategy with only one option type Vertical spread Horizontal spread *) Diagonal spread *) Purchase / sale buy / sell Call buy / sell Call buy / sell Call or or or buy / sell Put buy / sell Put buy / sell Put Strikes different, one ITM, the same different, one ITM, the other OTM other OTM Maturities same different different *) not often used in practice The most usual Spread-Strategy is the vertical Spread. There is differentiated between Bull and Bear Spreads. FINANCE TRAINER International Options / Page 21 of 38

22 Bull Spread Bull Spreads are usually formed with Call options, with one Call purchased in or ATM and one Call sold OTM. If a Bull spread is traded with Put options, one Put is written ITM and the other Put is purchased OTM. Long Call ITM basis A, short Call OTM basis B, same term A B Bear Spread A bear Spread is usually formed with a purchased Put in or ATM and a Put, which is sold OTM. As well Bear Spreads can be constructed with Call options (sell ITM Call plus buy OTM Call). Long Put ITM basis A, short Put OTM basis B, same term A B FINANCE TRAINER International Options / Page 22 of 38

23 1.8 Option Pricing Models The option market started to boom with the development of option pricing models. These models determine the price of an option as a function of variables like market data, volatility, strike price, term and interest rates. The main principle in option pricing is the calculation of the fair option price, i.e. the price where no arbitrage is possible. The best-known valuation model in option markets is the Black-Scholes model, published in 1973 by F. Black and M. Scholes and originally used for pricing share options. Generally the following assumptions are made: The stock prices are subject to a log-normal distribution. During the option period no dividends are allowed. The annualised, riskless interest rate is constant during the option period. Markets are efficient, the hedge portfolio can be traded continuously. Options are European-style. Volatility remains constant during the option period. The main difference between FX and share options is that the foreign currency interest rates have to be integrated in the Black-Scholes model as continuous dividend payments. For FX options the option price is influenced by the interest rates in both currencies. The differences in the assumptions and in the valuation formula of the Black & Scholes model were published in an essay by Mark B. Garman and Steven W. Kohlhagen (December 1982). The Garman-Kohlhagen valuation model still is - even if slight modifications were done - the most common FX option valuation model. In 1979 Cox, Ross and Rubinstein developed also a model for evaluating interest rate options. In contrary to the Garman-Kohlhagen and the Black-Scholes model the Cox, Ross and Rubinstein model assumes a discrete random variable, i.e. it does not assume a normal distribution like the other two models but a binomial distribution. With the binomial distribution two different points of time are regarded, which are the points of beginning and end of a period resp. a time interval. Regarding the fixed, actual rate it is FINANCE TRAINER International Options / Page 23 of 38

24 required that the rate at the end of the period can have exactly two different values, either a maximum or minimum value. Excursus: The option price formula The price of a Call is the following: {[ ONd ( 1) ] [ SNd ( 2) ]} 1 CALL = T 1+ ib B B With: d 1 O ln + S = V * T 2 ( 0,5 * V * T) 1 2 d 0,5 2 = d1 V T S = Strike O = Outright V = Volatility T = Term of the option (in % of a year) I B = Interest rate p.a. in decimals, base currency N(..) = Cumulative normal distribution B = Day base (360 or 365) FINANCE TRAINER International Options / Page 24 of 38

25 1.9 Call/ Put-Parity The possibility to rebuild every Call or Put position with a combination out of underlying and put resp. call is called call/put-parity Diagrams of Synthetic Options Positions Long Call Long Put Short Call Short Put Long Put + Buy Outright Long Call + Sell Outright Short Put + Sell Outright Short Call + Buy Outright Option Outright Synthetic options position Long Put + Buy Outright = Long Call Long Call + Sell Outright = Long Put Short Put + Sell Outright = Short Call Short Call + Buy Outright = Short Put FINANCE TRAINER International Options / Page 25 of 38

26 1.9.2 Diagrams of Synthetic Underlying Positions Also underlying positions can be rebuilt with options. Long Underlying Short Underlying + + Long Call + Short Put Short Call + Long Put - - Call Put Synthetic underlying position Short Put + Long Call = Long Underlying Short Call + Long Put = Short Underlying The call/ put-parity can be used for option price calculations. If the parity is not kept arbitrage will be possible. Generally you have the following rules: Call = Put + 1+ r ( O S) Q T B Q FINANCE TRAINER International Options / Page 26 of 38

27 Call = Put + 1+ r ( O S) Q T B Q The put price can be derived from arbitrage considerations. With a call and selling the underlying you can build a risk profile that is the same as for a put. Note: it has to be a European style option. You are looking for a price for a USD Put CHF Call premium USD Call spot outright interest rate quote currency 2.50 % period 90 days base 360 ( ) = p ut price = = CHF If the market price of the Put with Strike is, e.g , a risk-free profit can be realised by selling the Put, selling the outright and buying the Call. Spot rate at expiry Premium Sell put Sell outright at Buy call Total premium Total premium incl. interests Total: FINANCE TRAINER International Options / Page 27 of 38

28 As the theoretic Put price is 11 BP cheaper than the market price, these 11 BP can be arbitraged. Note: the options premium has to be paid in the beginning of the option period, the results from option and outright turn up in the end of the option period. Therefore the (in this case) paid premium has to be compounded for the option period. As the result of 11 BP turns up in the end of the option period, it has to be discounted for today. FINANCE TRAINER International Options / Page 28 of 38

29 2. Risk Factors The profit and loss results described so far always assume that the option is held till expiry. On the following pages we describe the factors influencing the options position during the term of the option. 2.1 Delta and Delta Hedging If the underlying price increases, the Call price increases too. The Delta of a Call shows by how much the Call price increases if the underlying increases. Change in option price DELTA = Change in underlying The Delta shows the change in the option price once there is a small change in the underlying. Mathematically the Delta is the first derivation of the option price formula by the underlying. As the Call changes in the same direction as the underlying, the Delta of a Call is positive. By similar reasoning, the Delta of a Put is negative. The Delta of an option has values in the range of -1 and +1 (resp. 100% and +100 %). OTM options: absolute delta from 0 to 0,5 ITM options: absolute delta from 0,5 to 1 ATM options: absolute delta is approx. 0,5 A Call with a Delta of 1, implies that the option price increases by 1 unit. A Delta of 0, means that there is no change in the option price if the underlying price changes. FINANCE TRAINER International Options / Page 29 of 38

30 For the different option types the following Deltas can be observed: Delta sign Call Put Long options ( + ) ( ) Short options ( ) ( + ) A Delta of +0.5 (50%) for a GBP/USD Call means that if GBP/USD increases by 1 Cent, the Call price increases by 0.5 Cent. Delta Hedging Delta plays also an important role in hedging. Delta Hedging is the hedging of an option position by a position in the underlying. The gain/loss of the option is offset by the loss/gain in the underlying. The amount of the underlying is calculated by multiplying Delta with the number of option contracts. In order to Delta-hedge a long Call or short Put you sell the underlying, for a short Call and long Put you buy the underlying. You have bought a USD Call/ CHF Put USD 1,000,000, Strike The Delta is 0.5. What is your Delta-hedge? 1. Options position: long USD call 2. With a long Call you are long the underlying, amount = volume x delta, i.e. 1,000,000 x 0.5 = USD 500,000. So you are long spot USD 500, Therefore you have to sell USD 500,000 in the spot market. Call Put long/buy short/sell long/buy short/sell Underlying: long short short long Delta Hedge: sell buy buy sell FINANCE TRAINER International Options / Page 30 of 38

31 2.2 Gamma As the Delta shows the change in the option price for a small change in the underlying, it can only serve as a snapshot calculation. The next question is, how Delta changes if the price of the underlying changes. This factor is called Gamma. Gamma = Change in delta Change in underlying Since Gamma measures the change of Delta it is strongest where Delta is most volatile. This is at-the-money and with short time to maturity. Is Delta near 0 or (+/-) 1, a change in the underlying does not influence considerably the Delta position since the option stays still deep out-of-the-money or in-the-money. The Gamma shows the expected change in Delta for a small change in the price of the underlying. Delta can be compared to the speed and Gamma to the acceleration. Gamma can also be taken as a measure of stability of the Delta. Gamma sign Call Put Long options ( + ) ( + ) Short options ( ) ( - ) Gamma has the strongest effect for ATM options and Options with a short period FINANCE TRAINER International Options / Page 31 of 38

32 2.3 Theta The longer the term, the more expensive the option. Therefore the price of an option has to decrease with the lapse of time (all other factors being stable). THETA = Change in option price Change in underlying Theta is strongest for at-the-money and short-term options. Theta is for short time to maturities very strong (= time decay very high). If the life of the option is reduced by one day it will have little influence on the option price if the remaining time is one year. If though the remaining term is very short, the time value of the option deteriorates very quickly. In case of one day to maturity the whole time value is gone at the following day. For deep in-the-money options the premium consists mainly of the intrinsic value, for deep out-of-the-money value a symbolic premium is paid. In both cases the time value of the option is very low and changes in the time to maturity have negligible influence on the option price. The change in the option price with the passage of time is Called Theta. A positive Theta means, that the value of the option position is getting better as time goes on. A negative Theta means that with time passage, the position value decreases. Theta sign Call Put Long options (premium paid) ( - ) ( - ) Short options (premium received) ( + ) ( + ) Theta has the strongest effect for ATM options and Options with a short period FINANCE TRAINER International Options / Page 32 of 38

33 2.4 Vega (Kappa) One of the most important influencing factors on the option price is the volatility. The question in risk measurement is, how much a change in volatility implies a change in the option price. This change is Called Vega (or Kappa). Change in option price Kappa = Change in volatility A positive Vega (long option) tells us that the option position is improving in value if volatility increases. A negative Vega (short option) means that we are losing in our option valuation as the volatility decreases. Higher volatilities lead to higher Call and Put premiums. For the seller of options this means that he is losing money and that the buyer of options is gaining money if volatility increases. For deep in-the-money options the premium consists mainly of the intrinsic value. Since the intrinsic value does not change if the volatility changes, a change of the volatility does not influence the option price strongly in this case. Also deep out the money options are hardly influenced by volatility changes since the low premium will not change if the insecurity in the markets rise. Vega sign Call Put Long options ( + ) ( + ) Short options ( - ) ( - ) Vega has the strongest effect for ATM options and Options with a long period Option strategies are often described by their volatility view. According to this, volatility is 'bought' or 'sold'. Buying volatility means that you profit from an increase in volatility; to sell volatility means that you profit from a decline in volatility. If volatility is the only undetermined measure in the option pricing formula, most option strategies may be reduced to views about the volatility. FINANCE TRAINER International Options / Page 33 of 38

34 2.5 Epsilon (Rho) As shown in the section on option pricing, interest rates (of both currencies) have an influence on the option price. The epsilon of FX options shows the influence of a change in interest differential on the option premium. EPSILON (RHO) = Change in option price Change in underlying Price change Call Put Base interest rate ( - ) ( + ) Base interest rate ( + ) ( - ) Variable interest rate ( + ) ( - ) Variable interest rate ( - ) ( + ) Epsilon (Rho) has the strongest effect for ITM options and Options with long periods Epsilon (Rho) is not unambiguous, as sometimes it is calculated with interest rate of base currency, sometimes with interest rate of quoted currency and sometimes on interest difference. FINANCE TRAINER International Options / Page 34 of 38

35 3. Skew ( Smile Curve ) and Risk Reversal 3.1 The Skew of Implied Volatility ( Smile Curve ) Skew means that the market quotes different implied volatilities for different strike prices. The reason for this skew is that options with certain strikes might be favoured in the market. Market participants will always buy options which earn the maximum profit when markets move as expected. If the market expects a rate rise, OTM calls (i.e. options with higher strikes) will be favoured, if the market expects the rate to fall, OTM puts (i.e. options with lower strikes) will be favoured. Therefore when believing in rising rates and thus due to increasing demand the implied volatilities for higher strikes will be higher than for lower strikes. As in the interbank options market only OTM options ( low deltas ) are traded, in this case the low delta calls (regarding the base currency) are favoured compared to the low delta puts ( calls over puts ). If the market expects rates to fall, the demand for lower strikes will increase ( puts over calls ). If the market participants expect stronger fluctuations in prices in both directions, both higher strikes (low delta calls) and lower strikes (low delta puts) will be favoured with no particular preference for one or the other direction. If hardly any fluctuations are expected, market participants are not willing to pay premiums for far-away strikes, low delta calls and puts are not interesting. FINANCE TRAINER International Options / Page 35 of 38

36 Illustration of typical volatility curves for different market expectations: Vol - rising rates Vol - falling rates Strike Strike - strong fluctuations - hardly any fluctuations Vol Vol Strike Strike In practice you can often also find combinations of these volatility curves. The reason for the favouring and thus higher implied volatilities for certain strikes lies in the impact of the gamma-curve on the position. When trading on certain price moves one should always go with an ascending gammacurve, i.e. gamma should increase when reaching expected spot levels, not decrease. Gamma expresses by how much Delta changes when the underlying rate changes. Gamma is highest for ATM options. For example a long call has a positive delta and profits from a rise of the underlying rate. Additionally the positive gamma of the long position leads to an increasing delta when rates rise (acceleration of profits). An OTM call moves in ATM-direction when rates rise. Thus gamma increases and the acceleration gets even stronger, i.e. you have an ascending gamma-curve. FINANCE TRAINER International Options / Page 36 of 38

37 Therefore: When expecting rising rates: BUY options with higher strikes (OTM calls) 12 Gamma Spot Strike Expectation Spot When expecting falling rates: BUY options with lower strikes (OTM puts) Gamma Strike Spot Expectation Spot FINANCE TRAINER International Options / Page 37 of 38

38 3.2 Risk Reversal These different expectations can be traded in the interbank options market. The quotation of a risk reversal expresses whether calls ( upside ) or puts ( downside ) are favoured. The quotation shows whether the market participants expect an upward move, a downward move or no particular move in one direction (viewed from the actual level). A risk reversal is a spread strategy, i.e. one option is bought and another one is sold. It is always a (OTM) call and a (OTM) put, i.e. either one buys the call and sells the put or sells the call and buys the put. Usually both options have the same delta, e.g. 25%. The quotation is in volatility. But not the whole volatility for each option is traded but only the difference (spread) between both volatilities. A quotation for a EUR/USD risk reversal might be: 3-monats R/R 25 delta calls over First one can easily see that the 25 delta calls are favoured and thus have to be calculated with a higher implied volatility. The spread is a bid and offer quotation (based on the favoured option), i.e. if one sells the call and buys the put, the volatility of the call lies 0.5% above the volatility of the put. If one buys the call and sells the put, the volatility of the call lies 0.8% above the volatility of the put. For our example volatilities would be (starting from a volatility level of 11.0%): Sell risk reversal at +0.5 Buy risk reversal at +0.8 sell call 25 delta at vol 11.5% buy call 25 delta at vol 11.8% buy put 25 delta at vol 11.0% sell put 25 delta at vol 11.0% FINANCE TRAINER International Options / Page 38 of 38

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