Exotic interest-rate options Marco Marchioro www.marchioro.org November 10th, 2012 Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 1 Lecture Summary • Exotic caps, floors, and swaptions • Swaps with exotic floating-rate legs • No-arbitrage methods (commodity example) • Numeraires and pricing formulas • stochastic differential equations (SDEs) • Partial differential equations (PDEs) • Feynman-Kac formula • Numerical methods: analytical approximations Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 2 Common exotic interest-rate options There are all sorts of exotic interest-rate options. The least exotic, i.e. commonly traded, types are • Caps and floors with a digital payoff • Caps and floors with barriers • Bermuda and American Swaptions Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 3 Caps and floors with a digital payoff A vanilla Cap has caplets with payoff Payoff CapletV = NT (6m; 9m) max [(3m-Libor − K); 0] (1) A digital Cap has caplets with payoff 8 > NT (6m; 9m) R for 3m-Libor ≥ K Payoff < CapletD = (2) :> 0 for 3m-Libor < K Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 4 Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 5 Caps and floors with barriers A barrier Cap has caplets with payoff Payoff CapletV = NT (6m; 9m) max [(3m-Libor − K); 0] (3) • Knock-in: paid if 3m-Libor touches a barrier rknock-in • Knock-out: paid if 3m-Libor does not reach rknock-out Discrete barriers are checked at certain given barrier dates Continuous barriers are checked on each daily Libor fixing Digital payoffs are also available. A rebate is paid upon knock out. Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 6 Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 7 Bermuda and American Swaptions Very similar to standard Swaptions (known as European swaptions) can be exercised at other dates • Bermuda swaptions: exercised dates are discrete (typically with the same tenor as one of the legs) • American Swaption can be exercised at any time Both are found in two different types of payoffs: co-terminal swap and constant-maturity swap Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 8 Questions? Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 9 More exotic interest-rate options • Caps and floors on Constant-Maturity-Swaps rates (CMS) • Look-back options • TARN (Target Accumulator Redemption Note) swap legs • Spread options Usually these options are gift-wrapped within a swaps paying a fixed or a Libor rate Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 10 Exotic swap legs • Constant-maturity swaps (CMS) • Spread options • Ratchet swap • Range accruals • Look-back options • TARN (Target Accumulator Redemption Note) swap legs Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 11 Constant-maturity swaps (CMS) A swap with a standard fixed-rate leg and a floating-rate leg that pays at times Ti, with i = 1; : : : ; n cash flows given by Ci = N τi−1 CMSM (Ti−1) (4) where • τi is the year fraction between dates Ti and Ti+1 • CMSM (Ti) is the fair rate, observed at time Ti, of a vanilla swap (fixed leg against Libor leg) with maturity date Ti + M Constant-maturity swaps usually have Caps or Floors on the swap rate Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 12 Swaps with spread options A swap with a standard fixed-rate leg and a floating-rate leg that pays a cash flow given by the difference of two swap rates. For example the floating leg fixing at time Ti−1, and paying at time Ti, payments are given by Ci = N τi−1 max rfloor; CMS10Y (Ti−1) − CMS2Y (Ti−1) • We are betting on the swap curve to steepen • Usually a multi-factor model is needed to price this instrument Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 13 Swap with ratchet options Also known as cliquet option. A swap with a standard fixed-rate leg and a floating-rate leg that pays at times T2;:::;Tn Caplet-like coupons, Ci = N τi−1 max 0;Li−1 − Ki−1 (5) where Li is the Libor rate observed at time Ti, and Ki is the strike satisfying Ki−1 = max Ki−2;Li−2 for i = 3;::: (6) Note: coupon rates are not decreasing. Usually K1 = F wd12 Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 14 Swaps with range accruals A swap with a floating-rate leg paying coupons Ci = N τi−1 hri (7) where the average rate hri is given by 1 hri = X f[L(t )] (8) K k k2[Ti−1;Ti) K business dates between Ti−1 and T1, f being the range function of Libor rate 8 r for L ≤ L ≤ Lmax <> in min f(L) = (9) > : rout otherwise Lmin minimum Libor rate and Lmax maximum Libor rate Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 15 Look-back options A swap with a floating-rate leg paying coupons on the maximum Libor fixing over a certain past period Ci = N τi−1 rmax(Ti−1;Ti) (10) where rmax = max [L(t )] (11) k2[Ti−2;Ti−1) k • We always receive the best Libor fixing in a range • Sometimes look-back periods extend further back than one coupon Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 16 TARN (Target Accumulator Redemption Note) swap legs A swap with a floating-rate leg paying coupons up to a maximum accumulated rate rmax Ci = N τi−1 ci (12) where c1 = min [rmax;L0] c2 = min [rmax − c1;L1] c3 = min [rmax − c1 − c2;L2] ::: Coupon rates are subtracted to the maximum rate until a zero rate is reached Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 17 Questions? Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 18 Crash course on commodity contracts (Example of no-arbitrage methods) • Introduction to commodities • The forward-spot and the forward-forward relationship • Convenience yields • Example of commodity derivatives: The forward contract and the futures spread Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 19 Exposure to commodity prices If you are managing an hedge fund and want some exposure on the price of live cattle what can you do? A. Become a cowboy overnight and buy some live cattle B. Enter in a derivative contract that gives you an exposure to the live-cattle price (e.g. a futures contract) Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 20 Commodity futures contracts • Commodity futures are among the oldest financial instrument traded on any trading floor • Futures on the cotton price, for example, have traded on the market for longer than a century • Refer to specialized literature for more details • In this talk we describe how to generate simulated spot prices and convenience-yield curves (to be used in the computation of risk figures) Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 21 Copper, Corn, and WTI Oil We consider three samples commodities • Copper (a.k.a Dr. Copper): non-ferrous industrial metal quoted, in $ per ton, on the London Metal Exchange (LME) • Corn: an agriculture commodity quoted, in $-cents per bushel (27,216 kg), on the Chicago Board of Trade (CBoT) • West-Texas-Intermediate Oil: an energy commodity quoted, in $ per barrel (158.987 liters), on the New York Mercantile Exchange (NYMEX) Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 22 Commodity futures contracts A contract that allows the delivery of a commodity at a certain future date (however settled daily according to the close price). E.g. on 2011-11-01 we observed the following market quotes Commodity Maturity Date Price copper 2012G 2012-02-24 7993.00 copper 2012H 2012-03-27 7995.00 copper 2012J 2012-04-25 7995.50 Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 23 Commodity forward contracts A commodity forward contract is struck between two parties that agree to buy/sell a given commodity at a future date at a pre- determined price (the strike price). Settlement could be physical or cash. In case of cash settlement the payoff P at the maturity date T (for the long side) is given by the difference between the value of the underlying commodity spot price S(T ) and a strike price K. For W lots we have, h i P = W · S(T ) − K (13) Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 24 Arbitrage-free strategies An arbitrage-free strategy is ... A series of physical or financial transactions that starting with a portfolio with a zero value end up with a risk-less portfolio. The no-arbitrage assumption states that the final portfolio value is zero with 100% probability (otherwise we could buy the cheaper part and sell the dearer one making a risk-free profit with some probability). Advanced Derivatives, Interest Rate Models 2010{2012 c Marco Marchioro Exotic interest-rate options 25 Spot-forward relationship (1/2) For a storable commodity.