The market for exotic options Development of exotic products increased flexibility for risk transfer and hedging highly structured expression of expectation of asset price movements facilitation of trading in new risk dimension such as the correlation between key financial variables Modest volumes of trading and a relative lack of liquidity. These are associated with the difficulty in pricing, hedging / replicating (due to complex risk profiles). 1
Path dependent options asset price t 0 T time The payoff of the option contract depends on the realization of the asset price within the whole life or part of the life of the option. 2
Most common types of path dependent options i Option is knocked out or activated when the asset price breaches some threshold value Barrier Options. i Average value of the asset prices over a certain period is used as the strike Asian Options. i The strike price is determined by the realized maximum value of the asset price over a certain period Lookback Options. 3
Barrier options up-barrier asset price barrier level knock-out time Extinguished or activated upon achievement of relevant asset price level. 4
Features * barrier periods may cover only part of the option s life * discretely monitored * can be in both European and American exercise format * barrier variable other than the underlying asset price * two-sided barriers (up-down) and sequential breaching * rebate may be paid upon knock out Advantage To achieve savings in premium; no need to pay for states believed to be unlikely to occur. 5
idelta: i gamma: it is typically positive (for a call) but it becomes negative as it approaches the barrier demonstrate very high gamma when the asset price is close to barrier i i vega: theta: usually higher than the non-barrier counterpart pattern of time decay is not smooth, with sharp discontinuity when close to barrier 6
Hedging difficulties circuit breaker effect upon knock out Market manipulation near barrier to trigger knock-out. Soros (1995) knock-out options relate to ordinary options the way crack relates to cocaine. 7
More complex versions of barrier options The option could have two barrier levels (double barriers), one above the and below the current level of the index. The knockout condition then be (i) touching either one, or (ii) sequential breaching. The barrier level could be based on another market (external barrier), say, the knock out of FTSE-100 option could be subject to the S&P 500 trading below a given level. The barrier condition could exist for only part of the life time of the option (partial barriers). Variable rather than a fixed barrier. 8
9 Down-and-out call option The call option is nullified when the asset price hits a down barrier B during the life of the option. The price formula for the continuously monitored down-and-out barrier call option is given by = τ τ τ σ, ), ( ), ( 2 1 2 2 S B c B S S c S c E E r where c E (S, τ) is the price of the vanilla counterpart. in. and down out and down vanilla c c c + = The second term then gives the price of the corresponding down-and-in call option. τ σ, 2 1 2 2 S B c B S E r
Static hedging of barrier securities Examine how barrier securities can be broken into vanilla European options. This represents the static hedging of path-dependent barrier securities with vanilla options. To replicate a knock-out call, the writer may buy an ordinary call at the same strike and sell an ordinary put with the strike lowered to the point where the net investment equals to the knock-out call premium. 10
Difficulties with dynamic hedging of barrier securities 1. The underlying asset as the dynamic hedging instrument is insensitive to changes in volatility. Option s vega for barrier securities is usually high. Vega risk is unhedgeable except with other option-like securities. 2. Barrier options often have regions of high gamma, which greatly increase the hedging error associated with dynamic hedging. 11
Parisian Option Knock-out of the option is activated only when the asset price stays in the knock-out region consecutively or cumulatively for some pre-specified length of time. Proportional Step Option e ρτ c Payoff is given by e ρτ c max(s T X, 0), where is the amortization factor, ρ is the killing rate and τ c is the total occupation time staying in the knock-out region. 12
Digital options (binary) A pre-determined fixed payout if the option is at- or in-the-money (also called all-or-nothing, bet or lottery options). Primarily European in style. Suited to markets where support and resistance levels are found, say, in the currency and bond markets. If an investor believes that a currency will not fall below a certain level, he can write a digital option to earn premium. Writer faced with greater hedging challenges due to large gamma. 13
Bermudan options The holder can exercise only at pre-set dates, somewhat a hybrid of European and American options. Hence, the premium is in between those of European and American options. A corporate client can use this type of option to manage his periodic balance sheet more proactively without paying extras for unwanted protection tailored to manage a client s risk exposure that may arise at infrequent time intervals. 14
Asian options Asian options are averaging options whose terminal payoff depends on some form of average. 1 n Arithmetic averaging = S i n i= 1 n S i i 1 Geometric averaging = = Used by investors who are interested to hedge against the average price of a commodity over a period, rather than the end-of-theperiod price e.g. Japanese exporters to the US, who are receiving stream of US dollar receipts over certain period, may use the Asian currency option to hedge the currency exposure. To minimize the impact of abnormal price fluctuation near expiration (avoid the price manipulation near expiration, in particular for thinlytraded commodities). 1 n 15
Asian Averaging Options max S AVE X,0 Average rate call: ( ) Average strike call: max( S T SAVE,0) Uses S 1 M AVE = S( ti ) or SAVE = 1 2 M ) M i= 1 1 [ S( t ) S( t ) KS( t ] M i Exposure as a future series of asset prices e.g. cost of production is sensitive to the prices of raw material. i To prevent abnormal price manipulation on expiration date, arising perhaps from a lack of depth in the market. 16
Fixed strike Asian call: max( A X,0) Cash settlement agreement. The option premium is expected to be lower than that of the vanilla Options since the volatility of the average asset value should be lower than that of the terminal asset value; The delta and gamma tend to zero as time is approaching expiration. Floating strike Asian call: max( S T A,0) Set the strike to the average of prices over a period so as to avoid the exposure of market. The delta and gamma tend to that of the vanilla option with identical expiration data and strike equal to the average. 17
Shout options The payoff upon shouting is another derivative with contractual specifications different from the original derivative. The embedded shout feature in a call option allows its holder to lock in the profit via shouting while retaining the right to benefit from any future upside move in the payoff. The terminal payoff of a shout call option is the form: C = max(s T K, L K, 0), where K is the strike price, S T is the terminal stock price and L is some ladder value installed at shouting. The ladder value L is set to be the prevailing stock price S t at the shouting instant t. 18
Shout feature The terminal payoff is guaranteed to be at least S t K. Obviously, the holder should shout only when S t > K. The number of shouting rights throughout the life of the contract may be more than one. Some other restrictions may apply, say, the shouting instants are limited to some predetermined times. 19
Reset feature This is the right given to the derivative holder to reset certain contract specifications in the original derivative. Strike reset strike reset to a lower strike for a call or to a higher strike for a put. Maturity reset extension of the maturity of a bond. Constraints on reset A limit to the magnitude of the strike adjustment. Triggered by underlying price reaching certain level. Reset allowed only on specific dates or limited period. 20
Example - Reset strike put option The strike price is reset to the prevailing stock price upon shouting. The shouting payoff is given by max(s t S T, 0) = max(s T K, S t K, 0) (S T K). The shout call option can be replicated by the reset strike put and a forward contract (put-call parity relation). 21
Example Extendible bonds Gives the holder the option of extending the term of the instrument, on or before a fixed date at a pre-determined coupon rate. The 5.5 percent Government of Canada extendible bond was issued on October 1, 1959. It was exchangeable on or before June 1, 1962 into 5.5 percent bonds maturing October 1, 1975. The three year initial bond was extendible into a 16 year bond at the holder s option. 22
Example - S&P 500 index bear warrants with a three-month reset Launched in the Chicago Board Options Exchange and the New York Stock Exchange (late 1996). These warrants are index puts, where the strike price is automatically reset to the prevailing index value if the index value is higher than the original strike price on the reset date three months after the original issuance. 23
Altering the terms of executive stock options expiration expiration expiration total lengthened unchanged reduced Strike price increase 2 2 Strike price lowered above market price 59 89 0 148 Strike price lowered to market price 294 309 13 616 Strike price reduced below market price 11 11 Total (%) 45% 51% 2% 100% 24
Lookback options Reset the strike to the realized lowest or highest market price during the lookback period. Payoff of the following forms: max [0, T ] [0, T ] [0, T ] [0, T ] ( S X,0), max( S S,0), ( S S X,0), etc. max max Partial lookbacks: selects a subset of the period from commencement to expiry as the lookback period. The premium increases with the length of the lookback period. Strike bonus rollover hedging strategy For the floating strike put, whenever a new maximum asset price is realized, replace the old put with a new put that has strike equal to the new maximum. T max min 25
Uses of lookback options Offshore debt or equity investments where the investor wishes to achieve the best currency over the relevant time period and wishes to uncouple the timing of the investment from the currency rate setting. Perspectives of holder Most advantageous if the realized volatility of the underlying asset price is higher that the implied volatility. There will be a sharp move in the underlying asset price but is unsure when and for how long the price will move. 26
Installment Options Characteristics i Option premium is paid periodically (monthly or quarterly) over the option s life. i Holder has the right to stop making payment, terminating the option on the date of first missed payment (this occurs if the option is not worth the present value of the remaining payments) Uses i Appeal to an investor who is willing to pay a little extra for the opportunity to terminate payment and reduce losses (this is most significant when an over-the-counter option premium can be difficult to recover). 27
Contingent premium options ithe option buyer pays the premium only if the option is in-the-money at expiration. profit/loss Worst scenario: option expires slightly in-the-money ivalue of contingent premium option is given by the value of the comparable standard option adjusted for the probability that the seller will receive the premium and for the discount appropriate for the delay in premium payment. X standard call break-even point contingent premium call asset price 28
Uses of contingent premium options Enables the buyer to acquire insurance against large one way price movement at no initial cost. Decomposed into a plain vanilla option minus a digital option so as to produce an initial zero cost structure. 29
Chooser options The presence of two maturity dates. The holder decides whether the option is to be a put or a call option at the first maturity date (choice date). The call and the put may or may not have identical strikes and expiry dates. Similar to a straddle (simultaneous purchase of put and call options with the same strike), except the holder has to choose at the choice date. The need to produce reduced premium options for investors has been the driving force behind chooser options. 30
Uses of chooser options Conditions of extreme volatility in asset price. Hedging where the direction of the exposure is unknown. Example During 1991 Gulf conflict, both oil producers and purchasers utilized chooser options to hedge against the extremely volatile conditions prevailing in the market for crude oil. 31
Swing options in energy derivatives In the electricity and natural gas markets, many contracts incorporate flexibility-of-delivery options. Swings permit the option holder to repeatedly exercise the right to receive greater or smaller amounts of energy, subject to some constraints. Local effect: The exercise of a right modifies the delivery amount only on the date of exercise. Global effect: The exercise of a right modifies the delivery volume from the exercise date onward. 32
Forward start options It is a standard call or put that begins life after the elapse of a specified time. The buyer receives an option with strike set ATM at the time of the option creation rather than at the transaction date. Forward start options carry most of the vega risk from the transaction date until the forward creation date. Forward start options can be useful where a buyer feels the current traded volatility level is favorable, but does not have a need for such an option until a specific time in the future. 33
Options on options Gives the holder the right but not the obligation to buy or sell another option at a pre-specified price. Uses To hedge in contingency situation such as contract tendering, where currency or interest rate hedges are required only if the entity is successful in its bid. a call on a call option a put on a call option a call on a put option a put on a put option 34
Multifactor options Multifactor options faces price risk from two or more sources. Types maximum / minimum options - payout depends on the best or worst performance of two or more assets exchange options - right to exchange one asset for another spread options - struck on the difference between prices of two assets 35
Callable American call The callable feature embedded in a financial derivative gives the right to the issuer to buy back the derivative from the holder at a predetermined recall price K. Upon recall the holder may choose to exercise the American call or sell it back to the issuer. Notice period requirement: The decision of the holder to exercise or to receive the cash is made at the end of the notice period. Question: What should be the optimal calling policy adopted by the issuer so that the value of the American call is minimized among all possible recall policies? 36
Callable Options Consider a 3-year call option with a fixed strike. After the first year and at every 6-month interval thereafter, the issuer has the right to call back the option. Upon calling, the holder is forced to exercise at the intrinsic value, or if the option is out-of-the-money, then the call option is terminated without any payment. Questions:- 1. Explain why the price of this callable option lies within the prices of the 1-year and 3-year non-callable counterparts. 2. What is the impact of dividend yield on the optimal calling policy of this callable option? 37
Note with embedded options Customer pays notional of 100 today. We pay a coupon of x% (p.a.) in 3 months. If spot price is above 100 at the end of the 3-month period, then the deal is terminated and we pay back 100 to him on that date. If the spot price is below 100, then a further coupon of 2% (p.a.) is paid in 6 months. The final redemption amount that the customer would obtain is given by Customer gets notional S/100 if S < 90 or S > 110, otherwise he would get back the notional. 38
The problem is to work out x%. The interesting thing is the barrier condition at the end of 3 months. The final payout for the customer can be decomposed into a combination of call option, put option and binary options. 39