Debt Instruments Set 9 Backus/Novemb er 30, 1998 Interest-Rate Options 0. Overview Fixed Income Options Option Fundamentals Caps and Flo ors Options on Bonds Options on Futures Swaptions Debt Instruments 9-2 1. Fixed Income Options Options imb edded in b onds: { Callable b onds { Putable b onds { Convertible b onds Options on futures { Bond futures { Euro currency futures OTC options { Caps, o ors, and collars { Swaptions Debt Instruments 9-3 2. Option Basics Big picture { Options are everywhere Sto ck options for CEOs and others Corp orate equity: call option on a rm Mortgages: the option to re nance { Options are like insurance Premiums cover the down side, keep the up side Customers like this combination Insurer b ears risk or shares it diversi cation or reinsurance { Managing cost of insurance Out-of-the-money options are cheap er insurance with a big deductable Collars: sell some of the up side Aggregate: basket option cheap er than basket of op- tions { Managing option b o oks Customer demands may result in exp osed p osition Particular exp osure to volatility: puts and calls b oth rise with volatility Hedging through replication is another route Debt Instruments 9-4 2. Option Basics continued Option terminology { Basic terms Options are the right to buy a cal l or sell a put at a xed price strike price The thing b eing b ought or sold is the underlying This righttypically has a xed expiration date A short p osition is said to have written an option { Kinds of options European options can b e exercised only at expira- tion American options can b e exercised any time Bermuda options can b e exercised at sp eci c dates eg, b onds callable only on coup on dates Debt Instruments 9-5 2. Option Basics continued Features of options { Leverage cheap source of exp osure { Nonlinear payo s Payo s vary with underlying in- and out-of-the money Translates into variable duration convexity rears its ugly head Creates risk management hazards { Volatility has p ositive e ect on b oth puts and calls Another risk management hazard! { They're state-contingent claims no way around it, but nothing new either Debt Instruments 9-6 3. Approaches to Valuation Why use a pricing mo del? No choice | the instruments de- mand that wevalue uncertain cash ows state-contingent claims. What pricing mo del? Go o d question. Interest rate trees { Been there... and it hasn't changed { We'll return to them shortly The Black-Scholes b enchmark Black's formula { Underlying: an arbitrary b ond with say maturity m { Parameters: n-p erio d Europ ean call with strike price k { Formula: 1=2 Call Price = pN x d kN x n n with p = current price of underlying f = forward price of underlying d = n-p erio d discount factor n N = normal cdf 2 log f=k+n =2 x = 1=2 n Debt Instruments 9-7 3. Approaches to Valuation continued Remarks on Black-Scholes for xed income { Formula based on log-normal price of underlying normal continuously comp ounded sp ot rates p ossibility of negative sp ot rates { Volatility varies systematically with maturities of op- tion and underlying \term structure of volatility" Sample swaption volatility matrix : Option Swap Maturity Maturity 1yr 2yr 5yr 10 yr 1m 15.50 16.00 16.75 15.25 3m 17.50 18.50 18.25 18.25 1yr 21.50 21.25 19.25 16.50 5yr 20.00 19.00 17.50 15.50 Source: Tradition, Inc, global swap broker, Jan 2, 1996. Remark: \hump" is typical { Despite problems, a common b enchmark dealers often quote volatility instead of price Debt Instruments 9-8 3. Approaches to Valuation continued Prop erties of Black-Scholes option prices most of these generalize to other settings { The Delta: Call Price Delta = = N x; p whichvaries b etween zero and one nonlinear. { Volatility is the only unobservable we use call prices to \imply" it { If volatility rises, so do es the call price puts, to o Debt Instruments 9-9 4. Caps, Flo ors, and Collars Terminology: { A cap pays the di erence b etween a reference rate and the cap rate, if p ositive. Series of call options on an interest rate { A oor pays the di erence b etween the o or rate and a reference rate, if p ositive. Series of put options on an interest rate { A col lar is a long p osition in a cap plus a short p osition in a o or. Contract terms: { Cap and/or o or rate { Reference rate typically LIBOR { Frequency of payment { Notional principal amount on whichinterest is paid Approaches: { Apply Black's formula { Interest rate tree we did this earlier { An uncountable numb er of other mo dels Debt Instruments 9-10 4. Caps, Flo ors, and Collars continued Example 1: two-year semiannual 7 cap on 6-m LIBOR Payments shifted back one p erio d: + r 7 =2 Notional Principal 1+r=2 three such semi-annual payments, excluding the rst Short rate tree same as b efore: 8.913 P 7.869 P P P P 6.913 6.587 P P P P P 5.869 6.036 P P P P P P P 4.913 4.587 P P P P 3.869 P P P 2.913 Price path for cap for 100 notional: 0.916 P 0.858 P P P P 0.416 0.000 P P P P P 0.000 0.202 P P P P P P P 0.000 0.000 P P P P 0.000 P P P 0.000 Debt Instruments 9-11 4. Caps, Flo ors, and Collars continued Example 1 continued: the e ects of volatility Short rate tree = 2, same implied sp ot rates: 12.001 P 9.912 P P P P 8.001 7.602 P P P P P 5.912 6.036 P P P P P P P 4.001 3.602 P P P P 1.912 P P P 0.001 Price path for cap for 100 notional: 2.359 P 2.740 P P P P 0.481 1.712 P P P P P 0.234 0.892 P P P P P P P 0.000 0.115 P P P P 0.000 P P P 0.000 Remarks: { Volatility increase cap prices they're b ets on extreme events, and higher makes them more likely { Similar in this resp ect to Black-Scholes Debt Instruments 9-12 4. Caps, Flo ors, and Collars continued Example 2: 2-Year FRN with collar 7 cap, 4 o or E ectiveinterest rates b oxes indicate cap/ o or binds: 7.000 P 7.000 P P P P 6.587 6.913 P P P P P 6.036 5.869 P P P P P P P 4.587 4.913 P P P P 4.000 P P P 4.000 Price path for note: 99.08 P 99.14 P P P P 100.00 99.58 P P P P P 100.00 99.88 P P P P P P P 100.00 100.16 P P P P 100.33 P P P 100.54 Remarks: { Di erences from 100 indicate impact of collar { Giving up low rates partially o sets cost of cap 0:12 = 100 99:88 < 0:20 { Issuers would generally adjust cap and o or to get a price of 100 Debt Instruments 9-13 5. Options on Bonds Earliest and most common interest-rate option? Examples of callable corp orate b onds { Apple Computer Corp oration's 6-1/2s, issued February 10, 1994, due February 15, 2004. Callable at \make whole." { Ford Motor Company's 6.11 p ercent b onds, issued Septem- b er 22, 1993, due January 1, 2001. Callable at \make whole under sp ecial circumstances." { Intel Overseas Corp oration's 8-1/8s, issued April 1, 1987 really, due March 15, 1997, callable at par. Par in this situation means par plus accrued interest: the rm pays the relevantinterest as well as the face value. The b onds were called March 15, 1994, at 100. { Texas Instruments' 9s of July 99, issued July 18, 1989, due July 15, 1999. Callable on or after July 15, 1996, at par. Debt Instruments 9-14 5. Options on Bonds continued Example: call option on 2-year 5 b ond the usual rate tree Price path of b ond: 102.50 P 100.63 P P P P 102.50 99.77 P P P P P 101.58 100.00 P P P P P P P P 102.50 101.65 100.94 P P P P P P P P 102.54 102.81 P P P P P P P 102.50 103.58 P P P P 103.53 P P P 102.50 18-month Europ ean option callable at 102.5 | \par" Price path is: 0.000 P 0.000 P P P P 0.000 0.000 P P P P P 0.010 0.000 P P P P P P P P 0.134 0.021 0.000 P P P P P P P P 0.227 0.042 P P P P P P P 0.525 0.000 P P P P 1.028 P P P 0.000 No des in boxes indicate cash ows from exercise, other no des indicate value in earlier p erio ds.