Lecture 17Exotic Options, Stochastic Models & Numerical MethodsOptions, Futures, Derivatives / April 3, 2008 1Compound OptionsCompound options are options on options. There are four main kinds• call on a call– At time T1the holder of the compound option is entitled to pay the first strike price K1andreceive a call option.– The call option then gives the holder the right to buy the underlying asset for the secondstrike price K2on the second date T2.– The compound option will be exercised on the first exercise date only if the value of theoption on that date is greater than the first strike price.Let M(a, b; ρ) be the cumulative bivariate normal distribution function that the first variablewill be less than a and the second variable will be less than b when the coefficient of correlationbetween the two is ρ.Let S∗be the asset price at the time T1for which the option price at time T1equals K1. Ifthe actual asset price is above S∗at time T1, the first option will be exercised; if it is notabove S∗, the option expires worthless .Options, Futures, Derivatives / April 3, 2008 2The price isS0e−qT2M(a1, b1;qT1/T2) − K2e−rT2M(a2, b2;qT1/T2) − e−rT1K1N(a2)wherea1=lnS0S∗+ (r − q +σ22)T1σ√T1, a2= a1− σpT1b1=lnS0K2+ (r − q +σ22)T2σ√T2, b2= b1− σpT2• put on a callThe price isK2M(a1, b1;qT1/T2) − S0e−qT2e−rT2M(a2, b2;qT1/T2) + e−rT1K1N(a2)• call on a put• put on a putWhy?Options, Futures, Derivatives / April 3, 2008 3Consider first the lognormally distributed stock. If we want to find the price of a call option on Sthen recall that the Black-Scholes formula can be found by solving the PDE or by evaluatingE [max{V − K2, 0}] =Z∞K2(S − K2) g(S)dSHere again S is lognormally distributed and g(S) is the probability density function of S (howlikely an end stock value will be struck).We can now use Rubinstein’s argument for the price of a call option on a call option. Recall thatthe call on call has maturity T1< T2where T2is the time of maturity of the base call option.The price of the underlying call option isc(ST1, T1) = ST1e−qT2−T1N(dT1) − K2e−r(T2−T1)N(dT1− σpT2− T1)wheredT1=lnST1K2+ (r − q +σ22)(T2− T1)σ√T2− T1Options, Futures, Derivatives / April 3, 2008 4Using risk-neutral world principle, the current value of C, the call on the call, we findC = e−rT1Eˆmax{c(ST1, T1) − K1, 0}˜In integral form we findC = e−rT1Z∞−∞max{c(S0eu, T1) − K1, 0}f(u)duwhere u = lnST1S0, f(u) =1σ√2πT1e−12v2andv =u − µtσ√T1µ = r − q +σ22Since ST1is lognormal, then u is normal and f(u) is the normal density function.Setx =lnS0X+ (r − q +σ22)(T1)σ√T1y =lnS0K1+ (r − q +σ22)(T2)σ√T2Options, Futures, Derivatives / April 3, 2008 5where X solvesXe−q(T2−T1)N(dT1) − K1e−r(T2−T1)N(zT1− σpT2− T1) = 0Then we evaluate the integral by breaking it up into three pieces:S0e−rT1Z∞logXS0euN(dT1)f(u)du = S0e−qT2N2(x, y; ρ)Ke−rT1Z∞logXS0N(dT1− σpT2− T1)f(u)du = Ke−rT2N2(x − σpT1, y − σpT2), ρ)Ke−rT1Z∞logXS0f(u)du = Ke−rT1N(x − σpT1)Options, Futures, Derivatives / April 3, 2008 6Barrier OptionsBarrier options are options where the payoff depends on whether the underlying asset’s pricereaches a certain level during a certain period of time• A number of different types of barrier options regularly trade in the OTC market. They areattractive to some market participants because they are less expensive than the correspondingregular options.• Barrier options can be classified as one of two types:• A knock-out option ceases to exist when the underlying asset price reaches a certain barrier• A knock-in option comes into existence only when the underlying asset price reaches a barrierA down-and-out call is one type of knock-out option. It is a regular call option that ceases to existif the asset price reaches a certain barrier level H. The barrier level is below the initial asset price.The corresponding knock-in option is a down-and-in call which comes into existence once theasset reaches a certain barrier value.Options, Futures, Derivatives / April 3, 2008 7Recall thatc = S0e−qTN(d1) − Ke−rTN(d2)p = Ke−rTN(−d2) − S0e−qTN(−d1)whered1=lnS0K+ (r − q +σ22)Tσ√Td2= d1− σ√TWe can give a few pricing formulas. The price of a down-and-in call iscdi= S0e−qT„HS0«2λN(y) − Ke−rT„HS0«2λ−2N(y − σ√T )whereλ =r − q +σ22σ2y =lnH2S0Kσ√T+ λσ√TOptions, Futures, Derivatives / April 3, 2008 8Since the value of a regular call equals the value of a down-and-in call plus the value of adown-and-out call, the value of a down-and-out cll is given bycdo= c − cdiThere are seven other similar barrier options.Note that barrier options can have quite different properties from regular v anilla options. Forexample, the vega can sometimes be negative. Consider an up-and-out call option when the assetprice is close to the barrier lev el.As a volatility increases the probability that the barrier will be hit increases. As a result, a volatilityincrease can cause the price of the barrier option to decrease in these circums tances.Options, Futures, Derivatives / April 3, 2008 9Binary OptionsBinary options are options with discontinuous payoffs.• A cash-or-nothing call pays nothing if the asset price ends up below the strike price at time Tand pays a fixed amount Q if it ends up above the strike price.• In a risk-neutral world, the probability of the asset price being above the strike price beingabove the strike price at the maturity of an option isN(d2)The value of the cash-or-nothing call is therefore,Qe−rTN(d2)• A cash-or-nothing put pays off Q if the asset price is below the strike price and nothing if it isabove the strike price. The value of a cash-or-nothing put isQe−rTN(−d2)Options, Futures, Derivatives / April 3, 2008 10• An asset-or-nothing call pays nothing if the underlying asset ends up below the strike priceand pays an amount equal to the asset price if it ends up above the strike price. Therefore, theasset-or-nothing call is worthS0e−rTN(d2)• An asset-or-nothing call pays nothing if the underlying asset ends up above the strike priceand pays an amount equal to the asset price if it ends up below the s trike price. Therefore, theasset-or-nothing call is worthS0e−rTN(−d2)• A regular European call option is equivalent to a long position in an asset-or-nothing call and ashort position in a cash-or-nothing