25857 Interest Rate Modelling

Transcription

1 25857 Interest Rate Modelling UTS Business School University of Technology Sydney Chapter 21. The Paradigm Interest Rate Option Problem May 15, /22

2 Chapter 21. The Paradigm Interest Rate Option Problem 1 Interest Rate Caps, Floors and Collars Interest Rate Caps Interest Rate Floors Interest Rate Collars 2 Payoff Structure of Interest Rate Caps and Floors 3 Relationship to Bond Options 4 The Inherent Difficulty of the Interest Rate Option Problem 2/22

3 Interest Rate Modelling Interest Rate Caps, Floors and Collars Interest Rate Caps Interest Rate Caps An interest rate cap is an agreement written on some reference rate R (e.g. 6-month LIBOR) that sets the borrowing rate at the market rate R if R < R Cap and limits the rate to R Cap if the market rate R > R Cap. The date at which the comparison between R and R Cap is made is known as the reset date. Payoff R Cap R Figure 1: Payoff on a long position in an interest rate cap 3/22

4 Interest Rate Caps, Floors and Collars Interest Rate Caps Interest Rate Caps R R Cap time Figure 2: Illustrating the actual and capped interest rates 4/22

5 Interest Rate Caps, Floors and Collars Interest Rate Caps Interest Rate Caps An interest rate cap is a call option on the interest rate; is an insurance against the interest rate on an underlying floating rate asset rising above a certain level. If the interest rate rises above the cap rate, the buyer of the cap effectively receives a payoff which is the difference between the current market rate and the cap rate. There can be a series of rate resets over the life of the cap. Hence, the cap is a portfolio of call options on R. Each component of the cap is known as a caplet. For the caplet over the period between t i and t i+1, the cap rate R Cap is compared with the reference rate at t i (i.e., R i ). However, the payoff for this period is settled at t i+1 (i.e., payment in arrears). 5/22

6 Interest Rate Modelling Interest Rate Caps, Floors and Collars Interest Rate Floors Interest Rate Floors An interest rate floor is the reverse of an interest rate cap. A long position would be of interest to a lender who wants to guarantee that a lending rate will not fall below a certain pre-specified rate. If R > R Floor the lender receives the market rate R. If R < R Floor then the lender receives the floor rate R Floor. Payoff R Floor R Figure 3: Payoff on a long position in an interest rate floor 6/22

7 Interest Rate Caps, Floors and Collars Interest Rate Floors Interest Rate Floors R R Floor Time Figure 4: Illustrating the actual and floor rates 7/22

8 Interest Rate Caps, Floors and Collars Interest Rate Collars Interest Rate Collars Payoff R Floor R Cap R Figure 5: Long a cap and short a floor Long an interest rate cap + short an interest rate floor. The price for a collar < just a cap or a floor; Offset the cost of a cap with the premium from the floor. Effectively, selling off some of the cap s downside protection. 8/22

9 Payoff Structure of Interest Rate Caps and Floors Payoff Structure of Interest Rate Caps and Floors The present value at time t i of a caplet payoff received at time t i+1 for a $1 principal amount is: PV(caplet payoff i ) = τ max[r i R Cap,0] (1) 1+R i τ = τ max[r i R Cap,0]P(t i,t i+1 ) (2) where τ = t i+1 t i, and the price at t i of a pure P(t i,t i+1 ) = discount bond maturing at time t i+1. 9/22

10 Payoff Structure of Interest Rate Caps and Floors Payoff Structure of Interest Rate Caps and Floors The underlying reference interest rate R is quoted so that (1+R i τ) 1 = P(t i,t i+1 ), (3) which allows us to use the prices of pure discount bonds as discount factors in equation (2). Conversely for an interest rate floor, the present value at time t i of the floorlet payoff for period (t i,t i+1 ) is : PV(floorlet payoff i ) = τ max[r Floor R i,0] 1+R i τ = τ max[r Floor R i,0]p(t i,t i+1 ). Figure (6) illustrates the payoff situation for an n-period interest rate cap. 10/22

11 Payoff Structure of Interest Rate Caps and Floors Payoff Structure of Interest Rate Caps and Floors R R Cap k th caplet value of cap at T = 0 τ 2τ kτ (k +1)τ nτ Observe R k Payment = τlmax(r k R Cap,0) (n 1) caplets = the cap; Principal amount = L, Cap rate = R Cap Interest reset dates = τ,2τ,...,kτ,(k +1)τ,...,nτ. (e.g. τ = 3 months) Figure 6: The payoff structure of an interest rate cap 11/22

12 Relationship to Bond Options Relationship to Bond Options This section demonstrates that the problem of pricing interest rate caplets and floorlets reduces to the problem of pricing put and call options on bonds. Hence interest rate caps and floors can be reduced to a portfolio of options on bonds. The significance of this observation lies in the fact that bonds are traded instruments and so we may use them to form the hedging portfolios that are the basis of derivative security pricing methodology. This interpretation is necessary since we cannot form hedging portfolios directly with the underlying reference interest rates as these are not traded instruments. 12/22

13 Relationship to Bond Options Relationship to Bond Options From eqn. (1), the caplet payoff can be written as : PV(caplet payoff i ) [ ] τ = 1+R i τ max Ri τ R Cap τ,0 τ [ ] τ = 1+R i τ max 1+Ri τ (1+R Cap τ),0 τ [ = 1max 1 1+R ] Capτ 1+R i τ,0 [ ] 1 = (1+R Cap τ)max 1+R Cap τ 1 1+R i τ,0. (4) 13/22

14 Relationship to Bond Options Relationship to Bond Options Let X c = 1/(1+R Cap τ), eqn. (4) becomes: [ ] 1 PV(caplet payoff i ) = (1+R Cap τ)max X c 1+R i τ,0 = (1+R Cap τ)max[x c P(t i,t i+1 ),0]. Hence, the caplet payoff is equivalent (within the proportionality factor 1+R Cap τ) to the payoff of a bond put option maturing at time t i on an underlying bond maturing at time t i+1 with exercise price being X c. 14/22

15 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem This section briefly highlights why it is that the interest rate option problem is a so much more difficult problem than the option pricing problem in a world of deterministic interest rates. There are essentially two main reasons. The first is that the number of underlying assets is infinite. The second is that it is not obvious which underlying asset to use. 15/22

16 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem Consider the first issue, namely the infinite dimensional nature of the problem. The basic object whose dynamics we seek to model is the yield curve out to a range of maturities. Figure (7) graphs the yield curve in Australia during out to a maturity of 15 years. Interest rate derivatives derive their value from the evolution of this surface. In principle this surface is an infinite dimensional object, though in later chapters we shall see that its dynamic evolution can be captured reasonably well by a finite number of fixed maturity forward rates. 16/22

17 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem Figure 7: The term structure of interest rates in Australia /22

18 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem On the second issue we note that one may use at least three different quantities to describe the par rates in figure (7), r(t) = inst. interest rate agreed at t for borrowing starting at t, P(t,T) = price at time t of pure discount bond maturing at time T, f(t,t) = inst. interest rate agreed at t for borrowing starting at T. 0 t T Figure 8: The time line for interest rate processes 18/22

19 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem The relationship between these quantities is illustrated in figure (9), the details of which will become clearer in subsequent chapters. We use a? in the expectation operator linking r(t) and P(t,T) to indicate that at this point the probability measure with respect to which this expectation is calculated is not known. Other quantities such as yields and the discretely compounded rates that are quoted in markets can also be determined in terms of r(t), f(t,t) or P(t,T). 19/22

20 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem r ( t) = f ( t, t) r(t) T? r( sds ) t PtT (, ) = E t e.1 f(t,t) P(t,T) = t P( t, T ) e T f ( t, s) ds Figure 9: Illustrating the relationship between spot rate, forward rate and bond price 20/22

21 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem The paradigm problem of pricing an interest cap (or floor) reduces to the problem of pricing an option on a bond. A further difficulty is that this process is in fact a two pass process. Suppose T B is the date of bond maturity and T C is the bond option maturity date. We first need to solve for the bond price over the interval T B to T C to determine the possible option payoffs at T C. We then need to solve the bond option pricing problem over the interval T C to 0. 21/22

22 The Inherent Difficulty of the Interest Rate Option Problem The Inherent Difficulty of the Interest Rate Option Problem X T C }{{}}{{} Solve option Solve bond pricing p.d.e pricing p.d.e subject to: Boundary condition at T C T B time Figure 10: Illustrating the two pass nature of bond option pricing 22/22