Interest Rate Options
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Interest Rate Options Saurav Sen April 2001 Contents 1. Caps and Floors 2 1.1. Deﬁntions . 2 1.2. Plain Vanilla Caps . 2 1.2.1. Caplets . 3 1.2.2. Caps . 4 1.2.3. Bootstrapping the Forward Volatility Curve . 4 1.2.4. Caplet as a Put Option on a Zero-Coupon Bond . 5 1.2.5. Hedging Caps . 6 1.3. Floors . 7 1.3.1. Pricing and Hedging . 7 1.3.2. Put-Call Parity . 7 1.3.3. At-the-money (ATM) Caps and Floors . 7 1.4. Digital Caps . 8 1.4.1. Pricing . 8 1.4.2. Hedging . 8 1.5. Other Exotic Caps and Floors . 9 1.5.1. Knock-In Caps . 9 1.5.2. LIBOR Reset Caps . 9 1.5.3. Auto Caps . 9 1.5.4. Chooser Caps . 9 1.5.5. CMS Caps and Floors . 9 2. Swap Options 10 2.1. Swaps: A Brief Review of Essentials . 10 2.2. Swaptions . 11 2.2.1. Deﬁnitions . 11 2.2.2. Payoﬀ Structure . 11 2.2.3. Pricing . 12 2.2.4. Put-Call Parity and Moneyness for Swaptions . 13 2.2.5. Hedging . 13 2.3. Constant Maturity Swaps . 13 2.3.1. Deﬁnition . 13 2.3.2. Pricing . 14 1 2.3.3. Approximate CMS Convexity Correction . 14 2.3.4. Pricing (continued) . 15 2.3.5. CMS Summary . 15 2.4. Other Swap Options . 16 2.4.1. LIBOR in Arrears Swaps . 16 2.4.2. Bermudan Swaptions . 16 2.4.3. Hybrid Structures . 17 Appendix: The Black Model 17 A.1. Model Speciﬁcation . 17 A.2. Pricing Vanilla Calls . 17 A.3. Pricing Vanilla Puts . 19 A.4. Digital Call Pricing . 20 A.5. LIBOR-in-Arrears Calculations . 20 1. Caps and Floors 1.1. Deﬁntions A cap is a call option on the future realisation of a given underlying LIBOR rate. More speciﬁcally, it is a collection (or strip) of caplets, each of which is a call option on the LIBOR level at a speciﬁed date in the future. Similarly, a ﬂoor is a strip of ﬂoorlets, each of which is a put option on the LIBOR level at a given future date. Caps and ﬂoors are widely traded OTC instruments. As explained below, they provide protection against widely ﬂuctuating interest rates – a cap, for instance, is insurance against rising interest rates. Caps and ﬂoors also reﬂect market views on the future volatility of LIBOR rates. 1.2. Plain Vanilla Caps In this section, we discuss plain vanilla caps in further detail. For concreteness, suppose the underlying interest rate is the τ− maturity LIBOR. Let L (t, T, τ) denote1 the forward LIBOR at time t for the accrual period [T,T + τ] . The spot LIBOR at time T is then, by deﬁnition, L (T, T, τ) . This rate ﬁxes at time T, and a dollar invested at this rate pays 1+τL (T, T, τ) at time T +τ. The maturity τ is expressed in terms of fractions of a year – for example, τ = 0.25 for the 3-month LIBOR. 1Notation: In L (t, T, τ) , the ﬁrst argument is current time, the second argument is the start date for the accrual period, and the third argument is the length of the accrual period. 2 1.2.1. Caplets As mentioned earlier, a cap is a strip of caplets. The price of a cap is the sum of its constituent caplet prices, and so we focus on these ﬁrst. A caplet is a call option on L. Speciﬁcally, a caplet with maturity date T and strike rate K has the following payoﬀ: at time T + τ, the holder of the caplet receives2 + ζT = τ (L (T, T, τ) − K) Note that the caplet expires at time T, but the payoﬀ is received at the end of the accrual period, i.e. at time T + τ. The payoﬀ is day-count adjusted. The liabilites of the holder of this caplet are always bounded above by the strike rate K, and clearly if interest rates increase, the value of the caplet increases, so that the holder beneﬁts from rising interest rates. By the usual arguments, the price of this caplet is given by the discounted risk-adjusted expected payoﬀ. If {P (t, T ): T ≥ t} represents the observed term structure of zero-coupon bond prices3 at time t, then the price of the caplet is given by ˜ + ζt = τP (t, T + τ) Et (L (T, T, τ) − K) In this equation4, the only random term is the future spot LIBOR, L (T, T, τ) . The price of the caplet therefore depends on the distributional assumptions made on L (T, T, τ) . One of the standard models for this is the Black model, described in the appendix. According to this model, for each maturity T, the risk-adjusted relative changes in the forward LIBOR L (t, T, τ) are normally distributed with speciﬁed constant volatility σT , i.e. dL (t, T, τ) = σ dW˜ (t) L (t, T, τ) T As shown in the appendix, this implies a lognormal distribution for L (T, T, τ) , and under this modelling assumption the price of the T – maturity caplet is given by T T ζt = τP (t, T + τ) L (t, T, τ) N d1 − KN d2 where 1 2 T log (L (t, T, τ) /K) ± 2 σT (T − t) d1,2 = √ σT T − t and Z z 1 − 1 u2 N (z) = √ e 2 du 2π −∞ x if x > 0 2Notation: x+ = max {x, 0} = 0 if x ≤ 0 3Notation: In P (t, T ) , t represents current time and T is maturity date. A dollar at time T is worth P (t, T ) dollars at time t. 4 ˜ Notation: Et [.] is the expectation operator, conditioned on all information at time t, hence the subscript, and computed using the risk-adjusted probabilities P˜, hence the superscript. 3 1.2.2. Caps To construct a standard cap on the τ – maturity LIBOR with strike K and maturity T, we proceed as follows. Suppose the current time is t. Starting with T, we proceed backwards in steps of length τ, and let n be the number of complete periods of length τ between t and T. Thus, we get a set of times T0 = t + δ; δ < τ T1 = T0 + τ T2 = T1 + τ = T0 + 2τ . Tn = T = T0 + nτ We then construct a portfolio of n caplets, struck at K, with maturities {T0,T1, ..., Tn−1} . These are called the ﬁxing dates or caplet maturity dates. The dates {T1,T2, ..., Tn} are called payment dates. The cap is then just equal to this strip of caplets. The price of a cap is equal to the price of its constituent caplets. If ζi (t) denotes the price at time t of a caplet with maturity date Ti (and payment date Ti + τ), then the price of the cap is n−1 n−1 X X Ti Ti ζ (t, T ) = ζi (t) = τP (t, Ti+1) L (t, Ti, τ) N d1 − KN d2 i=0 i=0 The only quantity that cannot be directly observed in this pricing formula is the set of forward rate volatilities, σTi . Thus ζ (t, T ) = ζ t, T ; σT0 , σT1 , ..., σTn−1 and a given set of forward rate volatilities produces a unique price. If we can ﬁnd a single number σ such that ζ (t, T ) = ζ t, T ; σT0 , σT1 , ..., σTn−1 = ζ (t, T ; σ, σ, ..., σ) then this σ is called the implied or Black volatility for the T – maturity cap. The observed prices of caps of various maturities are inverted numerically to obtain a term structure of Black volatilities, and these implied volatilites are quoted on the market. 1.2.3. Bootstrapping the Forward Volatility Curve The forward volatility curve describes information about individual caplet volatilities con- tained in the term structure of implied cap volatilities. It therefore unravels the information contained in the caps, and is useful in pricing other vanilla or exotic options on LIBOR rates. The technique used to extract caplet volatilities is called bootstrapping. The idea is essen- tially that a 10-year cap contains all the caplets in a 5-year cap. We work recursively, using 4 the caplet volatilities in short maturity caps to infer the volatility for the next maturity caplet. Suppose we have a cap volatility term structure as follows. Maturity (yrs) Implied Cap Volatility 1 κ1 2 κ2 . n κn The volatility of the 1-year caplet is just σ1 = κ1. To get the volatility of the 2-year caplet, we use the fact that the 2-year cap consists of a 1-year cap and a 2-year caplet added to it. 2-yr Caplet Price = 2-yr Cap Price – 1-yr Cap Price Since both the quantities on the right hand side are known, we can use this equation to infer an implied volatility for the 2-year caplet. Proceeding in exactly the same manner, we can infer the individual caplet volatilities for all maturities. Interpolation between quoted maturities is either piecewise linear or smoothed. 1.2.4. Caplet as a Put Option on a Zero-Coupon Bond A caplet is a call option on an interest rate, and since bond prices are inversely related to interest rates, it is natural to be able to view a caplet as a put option on a zero coupon bond. Speciﬁcally, the payoﬀ of a caplet is τ (L (T, T, τ) − K)+ This payoﬀ is received at time T + τ.