Lecture 5: Review of interest rate models Xiaoguang Wang STAT 598W January 30th, 2014 (STAT 598W) Lecture 5 1 / 46 Outline 1 Bonds and Interest Rates 2 Short Rate Models 3 Forward Rate Models 4 LIBOR and Swaps (STAT 598W) Lecture 5 2 / 46 Bonds and interest rates Definition A zero coupon bond with maturity date T , also called a T -bond, is a contract which guarantees the holder 1 dollar to be paid on the date T . The price at time t of a bond with maturity date T is denoted by p(t; T ). We assume the following: There exists a (frictionless) market for T -bonds for every T > 0. The relation p(t; t) = 1 holds for all t. For each fixed t, the bond price p(t; T ) is differentiable w.r.t time of maturity T . (STAT 598W) Lecture 5 3 / 46 Interest Rates At time t, we can make a contract guaranteeing a riskless rate of interest over the future interval [S; T ]. Such an interest rate is called a forward rate. Definition The simple forward rate (or LIBOR rate) L, is the solution to the equation p(t; S) 1 + (T − S)L = p(t; T ) whereas the continuously compounded forward rate R is the solution to the equation p(t; S) eR(T −S) = p(t; T ) (STAT 598W) Lecture 5 4 / 46 Interest Rates Definition The simple forward rate for [S; T ] contracted at t, henceforth referred to as the LIBOR forward rate, is defined as p(t; T ) − p(t; S) L(t; S; T ) = − (T − S)p(t; T ) The simple spot rate for [S; T ], or the LIBOR spot rate, is defined as p(S; T ) − 1 L(S; T ) = − (T − S)p(S; T ) (STAT 598W) Lecture 5 5 / 46 Interest Rates The continuously compounded forward rate for [S; T ] contracted at t is defined as log p(t; T ) − log p(t; S) R(t; S; T ) = − T − S The continuously compounded spot rate, R(S; T ) is defined as log p(S; T ) R(S; T ) = − T − S The instantaneous forward rate with maturity T , contracted at t, is defined by @ log p(t; T ) f (t; T ) = − @T The instantaneous short rate at time t is defined by r(t) = f (t; t) (STAT 598W) Lecture 5 6 / 46 Some useful facts The money account is defined by R t r(s)ds Bt = exp 0 For t ≤ s ≤ T we have Z T p(t; T ) = p(t; s) × exp − f (t; u)du s and in particular Z T p(t; T ) = exp − f (t; s)ds t (STAT 598W) Lecture 5 7 / 46 Relations between short rates, forward rates and zero coupon bonds Assume we have dr(t) = a(t)dt + b(t)dW (t) dp(t; T ) = p(t; T )m(t; T )dt + p(t; T )v(t; T )dW (t) df (t; T ) = α(t; T )dt + σ(t; T )dW (t) Then we must have ( α(t; T ) = vT (t; T )v(t; T ) − mT (t; T ) σ(t; T ) = −vT (t; T ) and ( a(t) = fT (t; t) + α(t; t) b(t) = σ(t; t) (STAT 598W) Lecture 5 8 / 46 Continue And we also should have 1 dp(t; T ) = p(t; T ) r(t) + A(t; T ) + kS(t; T )k2 dt 2 +p(t; T )S(t; T )dW (t) where ( R T A(t; T ) = − t α(t; s)ds R T S(t; T ) = − t σ(t; s)ds (STAT 598W) Lecture 5 9 / 46 Fixed Coupon Bonds Fix a number of dates, i.e. points in time, T0; ··· ; Tn. Here T0 is interpreted as the emission date of the bond, whereas T1; ··· ; Tn are coupon dates. At time Ti ; i = 1; ··· ; n, the owner of the bond receives the deterministic coupon ci . At time Tn the owner receives the face value K. Then the price of the fixed coupon bond at time t < T1 is given by n X p(t) = K × p(t; Tn) + ci × p(t; Ti ) i=1 And the return of the ith coupon is defined as ri : ci = ri (Ti − Ti−1)K (STAT 598W) Lecture 5 10 / 46 Floating rate bonds If we replace the coupon rate ri with the spot LIBOR rate L(Ti−1; Ti ): ci = (Ti − Ti−1)L(Ti−1; Ti )K If we set Ti − Ti−1 = δ and K = 1, then the value of the ith coupon at time Ti should be 1 − p(Ti−1; Ti ) 1 ci = δ = − 1 δp(Ti−1; Ti ) p(Ti−1; Ti ) which further discounted to time t < T0 should be p(t; Ti−1) − p(t; Ti ) Summing up all the terms we finally obtain the price of the floating coupon bond at time t n X p(t) = p(t; Tn) + [p(t; Ti−1) − p(t; Ti )] = p(t; T0) i=1 This also means that the entire floating rate bond can be replicated through a self-financing portfolio. (Exercise for you) (STAT 598W) Lecture 5 11 / 46 Interest Rate Swap An interest rate swap is a basically a scheme where you exchange a payment stream at a fixed rate of interest, known as the swap rate, for a payment stream at a floating rate (typically a LIBOR rate). Denote the principal by K, and the swap rate by R. By assumption we have a number of equally spaced dates T0; ··· ; Tn, and payment occurs at the dates T1; ··· ; Tn (not at T0). If you swap a fixed rate for a floating rate (in this case the LIBOR spot rate), then at time Ti , you will receive the amount KδL(Ti−1; Ti ) and pay the amount KδR where δ = Ti − Ti−1. (STAT 598W) Lecture 5 12 / 46 Pricing of IRS Theorem The price, for t < T0, of the swap above is given by n X Π(t) = Kp(t; T0) − K di p(t; Ti ) i=1 , where di = Rδ; i = 1; ··· ; n − 1; dn = 1 + Rδ If, by convection, we assume that the contract is written at t = 0, and the contract value is zero at the time made, then p(0; T0) − p(0; Tn) R = Pn δ i=1 p(0; Ti ) (STAT 598W) Lecture 5 13 / 46 Yield In most cases, the yield of an interest rate product is the "internal rate of interest" for this product. For example, the continuously compounded zero coupon yield y(t; T ) should solve p(t; T ) = e−y(T −t) × 1 which is given by log p(t; T ) y(t; T ) = − T − t For a fixed t, the function T ! y(t; T ) is called the (zero coupon) yield curve. (STAT 598W) Lecture 5 14 / 46 Yield to maturity for coupon bonds The yield to maturity, y(t; T ), of a fixed coupon bond at time t, with market price p, and payments ci , at time Ti for i = 1; ··· ; n is defined as the value of y which solves the equation n X −y(Ti −t) p(t) = ci e i=1 For the fixed coupon bond above, with price p at t = 0, and yield to maturity y, the duration, D is defined as n P T c e−yTi D = i=1 i i p which can be interpreted as the "weighted average of the coupon dates". With the notations above we have dp = −D × p dy (STAT 598W) Lecture 5 15 / 46 Yield Curve Definition The zero-coupon curve (sometimes also referred to as "yield curve") at time t is the graph of the function ( L(t; T ); t ≤ T ≤ t + 1 (years) T 7! Y (t; T ); T > t + 1 (years) where the L(t; T ) is the spot LIBOR rate and Y (t; T ) is the annually compounded spot interest rate. Such a zero-coupon curve is also called the term structure of interest rates at time t. Under different economic environments, the shape of zero-coupon curve can be very different, such as the "normal curve", ”flat curve", "inverted curve", "steep curve" and so on. (STAT 598W) Lecture 5 16 / 46 Zero-bond curve Definition The zero-bond curve at time t is the graph of the function T 7! P(t; T ); T > t which, because of the positivity interest rates, is a T -decreasing function starting from P(t; t) = 1. Such a curve is also referred to as term structure of discount factors. (STAT 598W) Lecture 5 17 / 46 Outline 1 Bonds and Interest Rates 2 Short Rate Models 3 Forward Rate Models 4 LIBOR and Swaps (STAT 598W) Lecture 5 18 / 46 Model set up In this section we turn to the problem of how to model an arbitrage free family of zero coupon bond price process fp(·; T ); T ≥ 0g. To model that, we first assume that short rate under the objective probability measure P, satisfies the SDE dr(t) = µ(t; r(t))dt + σ(t; r(t))dW¯ (t) And the only exogenously given asset is the money account, with price process B defined by the dynamics dB(t) = r(t)B(t)dt We further assume that there exists a market for zero coupon T -bond for every value of T . (STAT 598W) Lecture 5 19 / 46 Incomplete Market Question: Are bond prices uniquely determined by the P-dynamics of the short rate r? Answer: No! The market is incomplete. The short rate, as an "underlying", is not tradable.