Interest rates stochastic models Ioane Muni Toke Ecole Centrale - PowerPoint PPT Presentation

Interest rates stochastic models Ioane Muni Toke Ecole Centrale Paris Option Math´ ematiques Appliqu´ ees Majeure Math´ ematiques Financi` eres December 2010 - January 2011 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 1 / 61

Course outline Lecture 1 Basic concepts and short rate models Lecture 2 From short rate models to the HJM framework Lecture 3 Libor Market Models Lecture 4 Practical aspects of market models - I (E.Durand, Soci´ et´ e G´ en´ erale) Lecture 5 Practical aspects of market models - II (E.Durand, Soci´ et´ e G´ en´ erale) Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 2 / 61

Useful bibliography This short course uses material from : Brigo D. and Mercurio F. (2006). Interest rates models - Theory and Practice , 2nd edition, Springer. Martellini L. and Priaulet P. (2000). Produits de taux d’int´ et , erˆ Economica. Shreve S. (2004). Stochastic Calculus for Finance II: Continuous-Time Models , Springer. Original research papers (references below). Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 3 / 61

Part I Basic concepts Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 4 / 61

Spot interest rates r ( t ) : Instantaneous (interbank) rate, or short rate. P ( t , T ) : Price at time t of a T -maturity zero-coupon bond R ( t , T ) : Continuously-compounded spot interest rate R ( t , T ) = − ln P ( t , T ) i.e. P ( t , T ) = e − R ( t , T )( T − t ) (1) T − t L ( t , T ) : Simply-compounded spot interest rate 1 − P ( t , T ) 1 L ( t , T ) = P ( t , T )( T − t ) i.e. P ( t , T ) = (2) 1 + L ( t , T )( T − t ) Y ( t , T ) : Annually-compounded spot interest rate 1 1 Y ( t , T ) = P ( t , T ) 1 / ( T − t ) − 1 i.e. P ( t , T ) = (1 + Y ( t , T )) ( T − t ) (3) Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 5 / 61

Term structure of interest rates Reproduced from “Danish Government Borrowing and Debt 1998”, Danmarks National Bank, 1999. Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 6 / 61

Forward interest rates Forward-rate agreement : exchange of a fixed-rate payment and a floating-rate payment L ( t , T , S ) : Simply-compounded forward interest rate 1 � P ( t , T ) � P ( t , S ) − 1 L ( t , T , S ) = S − T (4) 1 + ( S − T ) L ( t , T , S ) = P ( t , T ) i.e. P ( t , S ) f ( t , T ) : Instantaneous forward interest rate � T � � f ( t , T ) = − ∂ ln P ( t , T ) i.e. P ( t , T ) = exp − f ( t , u ) du (5) ∂ T t Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 7 / 61

Swap rates Exchange of fixed-rate cash flows and floating-rate cash flows Exchanges at dates T α +1 , . . . , T β , with τ i = t i − T i − 1 Value at time t of a receiver swap : β Π RS ( t , α, β, N , K ) = − N ( P ( t , T α ) − P ( t , T β ) + N � τ i KP ( t , T i ) i = α +1 (6) Swap rate S α,β ( t ) = P ( t , T α ) − P ( t , T β ) (7) � β i = α +1 τ i P ( t , T i ) Link with simple forward rates 1 − � β 1 j = α +1 1+ τ j L ( t , T j − 1 , T j ) S α,β ( t ) = (8) � β � i 1 i = α +1 τ i j = α +1 1+ τ j L ( t , T j − 1 , T j ) Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 8 / 61

Caps, floors and swaptions Cap : Payer swap in which only positive cash flows are exchanged Floor : Receiver swap in which only positive cash flows are exchanged Caplet (floorlet) : One-date cap (floor), i.e. contract with payoff at time T i N τ i [ L ( T i − 1 , T i ) − K ] + . (9) Swaption : A European payer swaption is an option giving the right to enter a payer swap ( α, β ) at maturity T , i.e. contract with payoff at time T if T = T α � + β � � N τ i P ( T α , T i ) [ L ( T α , T i − 1 , T i ) − K ] . (10) i = α +1 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 9 / 61

Part II Short-rate models Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 10 / 61

Table of contents The Vasicek model 1 The CIR model 2 The Hull-White (extended Vasicek) model 3 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 11 / 61

The Vasicek model Table of contents The Vasicek model 1 The CIR model 2 The Hull-White (extended Vasicek) model 3 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 12 / 61

The Vasicek model Model definition Original paper Vasicek, O. (1977). “An equilibrium characterization of the term structure”, Journal of Financial Economics , 5 (2), 177–188. Dynamics of the short rate Short rate r ( t ) follows an Ornstein-Uhlenbeck process dr t = κ [ θ − r t ] dt + σ dW t (11) with κ, θ, σ positive constants. Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 13 / 61

The Vasicek model Dynamics of the short rate Proposition In the Vasicek model, the SDE defining the short rate dynamics can be integrated to obtain � t r ( t ) = r ( s ) e − κ ( t − s ) + θ (1 − e − κ ( t − s ) ) + σ e − κ ( t − u ) dW u (12) s Short rate r ( t ) is normally distributed conditionally on F s . Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 14 / 61

The Vasicek model Price of zero-coupon bonds Proposition In the Vasicek model, the price of a zero-coupon bond is given by P ( t , T ) = A ( t , T ) e − B ( t , T ) r ( t ) (13) with  ( θ − σ 2 2 κ 2 )( B ( t , T ) − ( T − t )) − σ 2 � � 4 κ B ( t , T ) 2 A ( t , T ) = exp    1 − e − κ ( T − t )  B ( t , T ) =   κ (14) Explicit pricing formula. Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 15 / 61

The Vasicek model Continuously-compounded spot rate Proposition In the Vasicek model, the continuously-compounded spot rate is written R ( t , T ) = R ∞ +( r t − R ∞ )1 − e − κ ( T − t ) σ 2 4 κ 3 ( T − t )(1 − e − κ ( T − t ) ) 2 (15) + κ ( T − t ) with R ∞ = θ − σ 2 (16) 2 κ 2 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 16 / 61

The CIR model Table of contents The Vasicek model 1 The CIR model 2 The Hull-White (extended Vasicek) model 3 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 17 / 61

The CIR model Model definition Original paper Cox J.C., Ingersoll J.E. and Ross S.A. (1985). ”A Theory of the Term Structure of Interest Rates”. Econometrica 53 , 385–407. Dynamics of the short rate Short rate is given by the following SDE dr t = κ [ θ − r t ] dt + σ √ r t dW t (17) with κ, θ, σ positive constants satisfying σ 2 < 2 κθ . Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 18 / 61

The CIR model Dynamics of the short rate Proposition In the CIR model, the short rate r ( t ) follows a noncentral χ 2 distribution Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 19 / 61

The CIR model Pricing of zero-coupon bonds Proposition In the CIR model, the price of a zero-coupon bond is P ( t , T ) = A ( t , T ) e − B ( t , T ) r ( t ) (18) with � 2 κθ  � κ + γ ( T − t ) σ 2 2 γ e 2   A ( t , T ) =  2 γ + ( κ + γ )( e γ ( T − t ) − 1)     2( e γ ( T − t ) − 1) (19) B ( t , T ) =  2 γ + ( κ + γ )( e γ ( T − t ) − 1)      � κ 2 + 2 σ 2 γ =  Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 20 / 61

The Hull-White (extended Vasicek) model Table of contents The Vasicek model 1 The CIR model 2 The Hull-White (extended Vasicek) model 3 Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 21 / 61

The Hull-White (extended Vasicek) model Model definition Original paper J Hull and A White (1990). ”Pricing interest-rate-derivative securities”. The Review of Financial Studies 3 , 573–592. Dynamics of the short rate Short rate is given by the following SDE dr t = [ b ( t ) − ar t ] dt + σ dW t (20) with a and σ positive constants. Non-time-homogeneous extension of the Vasicek model. Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 22 / 61

The Hull-White (extended Vasicek) model Model definition Proposition This model can exactly fit the term-structure observed on the market by setting b ( t ) = ∂ f M ∂ T (0 , t ) + af M (0 , t ) + σ 2 2 a (1 − e − 2 at ) (21) Dynamics of the short rate The short rate SDE can then be integrated to obtain : � t r ( t ) = r ( s ) e − a ( t − s ) + α ( t ) − α ( s ) e − a ( t − s ) + σ e − a ( t − u ) dW u (22) s with α ( t ) = f M (0 , t ) + σ 2 2 a 2 (1 − e − at ) 2 . Ioane Muni Toke (ECP - OMA Finance) Interest rates stochastic models Dec. 2010 - Jan. 2011 23 / 61