Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

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1 Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon bond, or discoun bond: asse which pays, wih cerainy, 1 a ime T. Price a ime is P (, T ). Pull-o-par: lim T P (, T )=P(T,T)=1. Wih fixed rae r: P (, T ) = exp( r[t ]). Yield-o-mauriy: average coninuously compounded reurn on zero-coupon bond, over remaining lifeime: log P (T,T) log P (, T ) R(, T ) = T log P (, T ) =. T Wih fixed rae, R(, T )=r. Yield curve, erm srucure: plo of R(, T ) agains (T ). Wih fixed rae, yield curve is fla line. Insananeous rae, shor rae: log P (, T ) r = lim R(, T )= T T. T = 1

Pull-o-par: lim T P (, T )=P(T,T)=1. Wih fixed rae r: P (, T ) = exp( r[t ]).

2 Forward conrac: agreemen a ime <T, o buy an S- bond a ime T<Sa price K. Replicaed by porfolio of +1 S-bond and K T-bonds. Value a ime should be zero, so K = P (, S)/P (, T ). Reurn (yield) over period [T,S] is herefore log P (S, S) log K S T = log P (, S) log P (, T ). S T Noe ha his reurn is known a ime! The limi, as S approaches T, of his reurn is he insananeous forward rae, log P (, S) log P (, T ) f(, T ) = lim S T S T log P (, T ) =. T Noe ha f(, ) =r. Also, f(, T )=R(, T )+(T ) R(, T ). T The erm srucure may also be described by f(, T ) as a funcion of (T ). The definiions of R(, T ) and f(, T ) in erms of P (, ) may be invered o obain P (, T ) = exp ( (T )R(, T )) ( T ) = exp f(, s)ds. In general, P (, T ) can no be obained from r alone! 2

The limi, as S approaches T, of his reurn is he insananeous forward rae, log P (, S) log P (, T ) f(, T ) = lim S T S T log P (, T ) =. T Noe ha f(, ) =r.

3 Term srucure models We only consider single-facor models (one Brownian moion). Ingrediens: A cash bond B paying he shor rae (r ): db = r B d, ( ) so B = exp 0 r sds. An Iô process for he zero-coupon bonds: dp (, T ) = P (, T )[µ P (, T )d +Σ(, T )dw ] = P (, T ) [r d +Σ(, T )d W ], wih W a P-Brownian moion and W a Q-Brownian moion. Under Q, P (, T )/B is a maringale. and α(, T ) = µ P (, T ) σ(, T )Σ(, T ), T µ P (, T ) = r T α(, s)ds Σ(, T )2. The marke price of risk does no depend on T : wih γ = µ P (, T ) r Σ(, T ) W = W + 0 γ s ds, = 1 1 T Σ(, T ) α(, s)ds. 2 Σ(, T ) Noe ha under risk-neuraliy, everyhing depends on he erm srucure of volailiies σ(, T ). The SDE for P (, T ) implies, via Iô s lemma, SDE s for R(, T ), f(, T ) and r. Heah-Jarrow-Moron sar wih Iô processes for f(, T ): df (, T ) = α(, T )d + σ(, T )dw = σ(, T )Σ(, T )d + σ(, T )d W, where σ(, T )= Σ(, T ), Σ(, T )= T T σ(, s)ds, 3

Under Q, P (, T )/B is a maringale. and α(, T ) = µ P (, T ) σ(, T )Σ(, T ), T µ P (, T ) = r T α(, s)ds + 1 2 Σ(, T )2.

4 Shor-rae models Usually he single facor is associaed wih he shor rae r. Shor-rae models are usually formulaed in risk-neural form, i.e., as a SDE dr = ρ(r,)d + ν(r,)d W, where W is a Q-Brownian moion. (Baxer and Rennie denoe his by W in Secion 5.4). ( ) Le B = exp 0 r sds be he cash bond price. Under he risk-neural measure, P (, T )/B is a maringale, so ha [ P (, T ) = B E Q B 1 T P (T,T) ] F [ ] = E Q B BT 1 F ( T ) ] = E Q [exp r s ds F. Therefore, a single-facor (risk-neural) model for r implies P (, T ), and hence he erm srucure, as a funcion of r. We disinguish: Endogenous erm srucure models: here he curren erm srucure follows from he SDE for r, which may be differen from he acual erm srucure a ime. Well-known examples are he Vasicek model: dr =(θ αr )d + σd W, and he Cox-Ingersoll-Ross model dr =(θ αr )d + σ r d W. These are also called equilibrium models. Exogenous erm srucure models: here he drif ρ(r,) is adjused such ha he curren erm srucure is fied exacly. The simples example is he Ho-Lee model: dr = θ d + σd W, where θ is such ha f(0,t)=r σ2 T 2 + T 0 θ sds maches he iniial erm srucure. These are also called no-arbirage models. 4

Under he risk-neural measure, P (, T )/B is a maringale, so ha [ P (, T ) = B E Q B 1 T P (T,T) ] F [ ] = E Q B BT 1 F ( T ) ] = E Q [exp r s ds F.

5 Ineres rae derivaives We assume again he general single-facor model dp (, T )=r P (, T )d +Σ(, T )P (, T )d W, where W is a Q-Brownian moion. We consider: Forward conracs Coupon-bearing bonds Floaing rae bonds Bond opions Swaps Swapions Caps, floors, collars Sock opions wih sochasic ineres raes Some of hese urn ou o have a price ha depend only on P (, T ), independen of model assumpions (i.e., independen of Σ(, T )). Forward conracs The forward price of a zero-coupon T 2 -bond, o be delivered a T 1, is simply F (, T 1,T 2 )=P(, T 2 )/P (, T 1 ). Coupon-bearing bonds Suppose ha a bond has a principal of 1 dollar, and pays coupons a imes T i = iδ, i =1,...,n,a rae k; a ime T n he principal amoun is repaid. This means ha he payoff is kδ a ime T i,i<n, and 1+kδ a ime T n. Since he noarbirage value a ime, of a cerain paymen of x a ime T i is { Ti } ) E Q (exp r s ds x F = P (, T i )x, he bond value a ime beween T j and T j+1 becomes n V = kδp(, T i )+P(, T n ). i=j+1 Since we expec ha V 0 =1(he par value), we find ha he coupon rae should be k = 1 P (, T n) n i=1 δp(, T i). 5

only on P (, T ), independen of model assumpions (i.e., independen of Σ(, T )).

6 Floaing-rae bonds Suppose ha he coupon rae i no fixed, bu equal o he LIBOR rae a he previous paymen ime T i 1. The δ-period LIBOR rae L() is defined by P (, + δ)(1 + δl())=1, or L() = 1 ( ) 1 δ P (, + δ) 1. Hence he payoff a ime T i is 1/P (T i 1,T i ) 1 for i<n, and he payoff a ime T n is 1/P (T n 1,T n ). Then he no-arbirage value V 0 of his bond should be 1. The replicaing sraegy is: A ime 0, buy1/p (0,T 1 ) zero coupon bonds mauring a T 1 (cosing 1); A ime T 1 = δ, sell he T 1 -bonds (yielding 1/P (0,T 1 )), and buy 1/P (T 1,T 2 ) of T 2 -bonds (cosing 1). Hence he payoff is 1/P (0,T 1 ) 1; A ime T i, sell he curren posiion of 1/P (T i 1,T i ) of T i -bonds, and buy 1/P (T i,t i+1 ) of T i+1 -bonds. This gives exacly he righ payoff, and coss 1. Bond opions A European call opion on a zero-coupon T -bond, sruck a k wih exercise ime S, is worh C = B E Q ( B 1 S [P (S, T ) k]+ F ). When he shor rae r is a Gaussian process, hen boh B and P (, T ) will have a log-normal disribuion. This can be used o obain a Black-Scholes-ype formula C = P (, S) {F (, S, T )Φ(d 1 ) kφ(d 2 )}, where F (, S, T ) is he forward price, and log(f (, S, T )/k) d 1,2 = σ ± 1 S 2 σ S, where σ 2 is he condiional variance (condiional on F )of log(p (S, T )/P (, T ))/ S, i.e., S σ 2 = 1 Σ(u, T ) 2 du. S For example, in he Ho-Lee model, σ = σ(t S). Opions on coupon-bearing bonds will only have a closedform price in single-facor models where r is a diffusion process. See Baxer & Rennie, p.170, or Hull, pp

Then he no-arbirage value V 0 of his bond should be 1.

7 Swaps A fixed-versus variable ineres rae swap (on a principal of 1) has payoff, a imes T i, equal o δ[l(t i 1 ) k], where k is he fixed swap rae. This means ha he swap can be replicaed by a porfolio of a shor posiion in a fixed coupon bond, and a long posiion of a floaing coupon bond. Since he swap should have inial value of zero, we have n 1 kδp(, T i ) P (, T n )=0, or i=1 k = 1 P (, T n) n i=1 δp(, T i), which is also he coupon rae ha ses he bond value equal o par. Swapions A swapion is an opion o ener a swap. Is value is he same as he value of a call opion on a fixed-coupon bearing bond, sruck a 1. Caps, floors, collars For a cap, he payoff a T i is δ[l(t i 1 ) k] +. The payoff of one caple can be replicaed as follows: Buy (1 + kδ) pu opions on a T i -bond, sruck a 1/(1+kδ) wih exercise ime T i 1. A ime T i 1, his yields (1 + kδ)[1/(1 + kδ) P (T i 1,T i )] + =[1 (1 + kδ)p (T i 1,T i )] +. Then pu his money in he bank, o yield he LIBOR rae L(T i 1 ), so ha he payoff a ime T i becomes [1 + δl(t i 1 ) (1 + kδ)] + = δ[l(t i 1 ) k] +. Thus he value of a cap can be derived from he value of he pu opion. Similarly, he payoff of a floor is δ[k L(T i 1 )] + a ime T i, he value of which can be derived from a call opion. A swap is essenially he sum of a cap and a floor, a he same rae k; his resuls in a pu-call pariy. A collar is also a combinaion of a cap and a floor, bu a differen raes (cap rae bigger han floor rae). 7

Since he swap should have inial value of zero, we have n 1 kδp(, T i ) P (, T n )=0, or i=1 k = 1 P (, T n) n i=1 δp(, T i), which is also he coupon rae ha ses he bond value equal o par.

8 Sock opions wih sochasic ineres raes Suppose we wish o price a European call opion on a sock wih price S, and ha he ineres rae is sochasic. In paricular, le ds = r S d + σs d W 1 dp (, T ) = r P (, T )d +Σ(, T )P (, T )d W 2, and furhermore define he cash bond as usual. Here ( W 1, W 2 ) is a bivariae Q-Brownian moion, wih correlaion ρ. Ifr is a Gaussian process, such ha B and P (, T ) are log-normal, hen he price of a European call opion becomes C = P (, T ) {F (, T )Φ(d 1 ) kφ(d 2 )}, where F (, T ) = P (, T ) 1 S is he forward sock price, and log(f (, T )/k) d 1,2 = ˆσ ± 1 T 2 ˆσ T, where ˆσ 2 = 1 T [ σ 2 +Σ(s, T ) 2 2ρσΣ(s, T ) ] ds. T This resul was derived by Meron (1973). 8

Here ( W 1, W 2 ) is a bivariae Q-Brownian moion, wih correlaion ρ.

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random