Birkbeck College. MSc Financial Engineering. Pricing II Interest Rates I

Transcription

1 Birkbeck College MSc Financial Engineering Pricing II Interest Rates I Professor Dr. Rüdiger Kiesel Faculty of Economics Chair of Energy Trading & Finance Centre of Mathematics for Applications, University of Oslo Interest Rates Basic Concepts Short Rate21. ModelsFebruary Heath-Jarrow-Morton 2011 (HJM) model Contingent Claim1 / Pricing 89

2 1 Interest Rates Basic Concepts Economics of Interest Rates Basic Objects Market Rates 2 Short Rate Models Vasicek Model Cox-Ingersoll-Ross Model Multi-Factor Models 3 Heath-Jarrow-Morton (HJM) model 4 Contingent Claim Pricing Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim2 / Pricing 89

3 Agenda 1 Interest Rates Basic Concepts Economics of Interest Rates Basic Objects Market Rates 2 Short Rate Models 3 Heath-Jarrow-Morton (HJM) model 4 Contingent Claim Pricing Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 3 / 89 Economics of Interest Rates Basic Objects Market Rates

4 Basic Interest Rates Economic agents have to be rewarded for postponing consumption; in addition, there is a risk premium for the uncertainty of the size of future consumption. Investors, Firms, banks pay compensation for the willingness to postpone A common interest rate (equilibrium) emerges which allows to fulfill the aggregate liquidity demand. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 4 / 89 Economics of Interest Rates Basic Objects Market Rates

5 Basic Interest Rates Interest rates change with different maturities because market segmentation hypothesis different agents have different preferences for borrowing and lending in different segments of the yield curve. Agents can change their segment if the compensation for switching is high enough. expectation hypothesis today s interest rates (spot rates) are determined on the basis of the expected future rates plus a risk premium. liquidity preference hypothesis on average agents prefer to invest in short-term assets. Here Liquidity refers to closeness to maturity. Agents are compensated through a risk premium for investing in longer dated assets. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 5 / 89 Economics of Interest Rates Basic Objects Market Rates

6 fall von den jeweiligen Laufzeiten der Anleihen abhängen und vereinfacht gesagt beschre Yield Curves Chair for Energy Trading & Finance die Zinsstruktur genau diese Abhängigkeit zwischen Laufzeit und Zinssatz. Grafisch w die Zinsstruktur dabei meist in Form der sogenannten Zinsstrukturkurve veranschauli (Abb. 1.1). Abbildung 1.1: Aus dt. Bundeswertpapieren geschätzte Zinsstrukturkurve (Spotrates, jährliche Basis) vom Quelle: Bloomberg L.P. Aufgrund der Abhängigkeiten befinden wir uns bei der Modellierung von Zinssätzen in ei Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim6 / Pricing 89 Economics of Interest Rates Basic Objects Market Rates

7 Um einen Eindruck davon zu bekommen, wie sichprof. in Deutschland Dr. Rüdiger Kiesel die Zinsstruktur über die letzten Jahre verändert hat bzw. in welchen Bereichen sich die Zinssätze dabei bewegt haben, Yield Curves ist in Abb. 4.2 die historische Entwicklung der deutschen Zinsstrukturkurve dargestellt. Abbildung 4.2: Entwicklung der deutschen Zinsstrukturkurve Daten: Staatsanleihen mit 1,2,3,4,5,7 und 10 Jahren Laufzeit, monatlich, bis Laufzeit 1J 2J 3J 4J 5J 7J 10J Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim7 / Pricing 89 Mittelw. 0,0550 0,0574 0,0598 0,0616 0,0631 0,0653 0,0672 Economics of Interest Rates Basic Objects Market Rates

8 Fixed-rate Bond With a fixed-rate bond the seller promises the buyer to pay fixed coupons C over time, until the bond matures, and when it matures the seller will repay the principal amount borrowed. The price of a bond is determined by its cashflow and the discount factor (T = t + n maturity, N notional value) p c (t, T ) = = C (1 + r 1 ) 1 + C (1 + r 2 ) C + N (1 + r n ) n n C N (1 + r i ) i + (1 + r n ) n i=1 Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 8 / 89 Economics of Interest Rates Basic Objects Market Rates

9 Yield to Maturity The yield to maturity y can be calculated from the coupon bond prices p c (t, T ) = C (1 + y) 1 + C (1 + y) C + N (1 + y) n Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 9 / 89 Economics of Interest Rates Basic Objects Market Rates

10 Zero-Coupon Bonds p(t, T ) denotes the price of a risk-free zero-coupon bond at time t that pays one unit of currency at time T. We will use continuous compounding, i.e. a zero bond with interest rate r(t, T ) maturing at T will have the price p(t, T ) = e r(t,t )(T t). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 10 / 89 Economics of Interest Rates Basic Objects Market Rates

11 Forward Rates Given three dates t < T 1 < T 2 the basic question is: what is the risk-free rate of return, determined at the contract time t, over the interval [T 1, T 2 ] of an investment of 1 at time T 1? Time t T 1 T 2 Sell T 1 bond Pay out 1 Buy p(t,t 1) p(t,t 2 ) T 2 bonds Receive p(t,t 1) p(t,t 2 ) Net investment p(t,t 1) p(t,t 2 ) Table: Arbitrage table for forward rates Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 11 / 89 Economics of Interest Rates Basic Objects Market Rates

12 Forward Rates To exclude arbitrage opportunities, the equivalent constant rate of interest R over this period (we pay out 1 at time T 1 and receive e R(T 2 T 1 ) at T 2 ) has thus to be given by e R(T 2 T 1 ) = p(t, T 1) p(t, T 2 ). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 12 / 89 Economics of Interest Rates Basic Objects Market Rates

13 Various Interest Rates The forward rate at time t for time period [T 1, T 2 ] is defined as R(t, T 1, T 2 ) = log(p(t, T 1)) log(p(t, T 2 )) T 2 T 1 The spot rate for the time period [T 1, T 2 ] is defined as R(T 1, T 2 ) = R(T 1, T 1, T 2 ) The instantaneous forward rate is log(p(t, T )) f (t, T ) = T The instantaneous spot rate is r(t) = f (t, t) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 13 / 89 Economics of Interest Rates Basic Objects Market Rates

14 Rates The forward rate is the interest rate at which parties at time t agree to exchange K units of currency at time T 1 and give back Ke R(t,T 1,T 2 )(T 2 T 1 ) units at time T 2. This means, one can lock in an interest rate for a future time period today. The spot rate R(t, T 1 ) is the interest rate (continuous compounding) at which one can borrow money today and has to pay it back at T 1. The instantaneous forward and spot rate are the corresponding interest rates at which one can borrow money for an infinitesimal short period of time. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 14 / 89 Economics of Interest Rates Basic Objects Market Rates

15 Simple Relations The money account process is defined by { t } B(t) = exp r(s)ds. 0 The interpretation of the money market account is a strategy of instantaneously reinvesting at the current short rate. For t s T we have { } T p(t, T ) = p(t, s) exp f (t, u)du, s and in particular p(t, T ) = exp { T t f (t, s)ds }. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 15 / 89 Economics of Interest Rates Basic Objects Market Rates

16 Simple Spot Rate The simply-compounded spot interest rate prevailing at time t for the maturity T is denoted by L(t, T ) and is the constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from p(t, T ) units of currency at time t, when accruing occurs proportionally to the investment time. L(t, T ) = 1 p(t, T ) τ(t, T )p(t, T ) (1) Here τ(t, T ) is the daycount for the period [t, T ] (typically T t). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 16 / 89 Economics of Interest Rates Basic Objects Market Rates

17 Simple Spot Rate The bond price can be expressed as p(t, T ) = L(t, T )τ(t, T ). Other daycounts denoted by τ(t, T ) are possible. Notation is motivated by LIBOR rates (London InterBank Offered Rates). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 17 / 89 Economics of Interest Rates Basic Objects Market Rates

18 Agenda 1 Interest Rates Basic Concepts 2 Short Rate Models Vasicek Model Cox-Ingersoll-Ross Model Multi-Factor Models 3 Heath-Jarrow-Morton (HJM) model 4 Contingent Claim Pricing Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 18 / 89

19 Basic Structure We consider models for the short rate of type dr(t) = a(t, r(t))dt + b(t, r(t))dw (t), (2) with functions a, b sufficiently regular and W a real-valued Brownian motion. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 19 / 89

20 Term-Structure Equation Typically we assume that the short-rate dynamics satisfy (2) under the historical (statistical) probability measure P. Then for pricing purposes a pricing measure is used. It relates to the historical measure by means of a function λ(t, r(t)). The Q(λ)-dynamics of r are given by dr(t) = {a(t, r(t)) b(t, r(t))λ(t, r(t))}dt+b(t, r(t))d W (t). (3) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 20 / 89

21 Pricing Contingent Claims For a T -contingent claim X = Φ(r(T )) we know that arbitrage-free prices are obtained by [ ] E Q(λ) e T r(u)du t Φ(r(T )) F t = F (t, r(t)), The function F satisfies the partial differential equation F t + (a bλ)f r b2 F rr rf = 0 (4) and terminal condition F (T, r) = Φ(r). For a zero-coupon bond, we have the terminal condition F (T, r; T ) = 1. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 21 / 89

22 Risk Premium An application of Itô s formula to the process F (t, r(t); T ) yields df (t, r(t); T ) = F (t, r(t); T ) (m(t, T )dt + v(t, T )dw (t)), with the functions m(t, T ) = m(t, r(t); T ) and v(t, T ) = v(t, r(t); T ) given by m = F t + af r b2 F rr F and v = bf r F. The functions m, v 2 are mean and variance, respectively, of the instantaneous rate of return at time t on a bond with maturity date T. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 22 / 89

23 Risk Premium λ satisfies λ = F t + af r b2 F rr rf bf r, or in terms of m, v above, λ(t, r) = m(t, r; T ) r, T t. v(t, r; T ) The numerator m(t, r; T ) r is the risk premium for the T -bond, i.e. the rate of return over the risk-free rate, while in the denominator we have the volatility of the T -bond. So the quotient specifies the risk premium per unit of volatility. The ratio is called the market price of risk. It specifies the expected instantaneous rate of return on a bond per additional unit of risk. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 23 / 89

24 Martingale Modelling For pricing we assume that we model the short-rate dynamics directly under an equivalent pricing measure Q. Thus we assume that r has Q-dynamics dr(t) = a(t, r(t))dt + b(t, r(t))dw (t) (2) with W a (real-valued) Q-Wiener process. We obtain the price process Π X (t) of any T -contingent claim X by computing the Q-expectation, i.e. [ ] Π X (t) = E Q e T r(u)du t X F t. (5) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 24 / 89

25 Martingale Modelling Typically, the contingent claim is of the form X = Φ(r(T )) so the price process is given by Π X (t) = F (t, r(t)), where F is the solution of the partial differential equation F t + af r + b2 2 F rr rf = 0 (6) with terminal condition F (T, r) = Φ(r). In particular, T -bond prices are given by p(t, T ) = F (t, r(t); T ), with F solving (6) and terminal condition F (T, r; T ) = 1. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 25 / 89

26 Option Pricing Suppose we want to evaluate the price of a European call option with maturity S and strike K on an underlying T -bond. This means we have to price the S-contingent claim X = max{p(s, T ) K, 0}. We first have to find the price process p(t, T ) = F (t, r; T ) by solving (6) with terminal condition F (T, r; T ) = 1. Secondly, we use the risk-neutral valuation principle to obtain Π X (t) = G(t, r), with G solving G t +ag r + b2 2 G rr rg = 0 and G(S, r) = max{f (S, r; T ) K, 0}. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 26 / 89

27 Option Pricing So we are clearly in need of efficient methods of solving the above partial differential equations, or from a modelling point of view, we need short-rate models that facilate this computational task. Fortunately, there is a class of models, exhibiting an affine term structure (ATS), which allows for simplification. If bond prices are given as p(t, T ) = exp {A(t, T ) B(t, T )r}, 0 t T, with A(t, T ) and B(t, T ) are deterministic functions, we say that the model possesses an affine term structure. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 27 / 89

28 Option Pricing In such a model and a(t, r) = α(t) β(t)r b(t, r) = γ(t) + δ(t)r. Then A and B are given as solutions of ordinary differential equations, A t (t, T ) α(t)b(t, T ) + γ(t) 2 B2 (t, T ) = 0, A(T, T ) = 0, (1 + B t (t, T )) β(t)b(t, T ) δ(t) 2 B2 (t, T ) = 0, B(T, T ) = 0. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 28 / 89

29 ATS models Vasicek model: dr = (α βr)dt + γdw ; Cox-Ingersoll-Ross (CIR) model: dr = (α βr)dt + δ rdw ; Hull-White (extended Vasicek) model: dr = (α(t) β(t)r)dt + γ(t)dw ; Hull-White (extended CIR) model: dr = (α(t) β(t)r)dt + δ(t) rdw. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 29 / 89

30 Vasicek Model Dynamics Modelling under the risk-neutral measure dr(t) = κ[θ r(t)]dt + σdw (t), r(0) = r 0 (7) with r 0, κ, θ, σ positive constants. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 30 / 89

31 Vasicek Dynamics 0.11 Vasicek paths, x=0.05, k=1, θ=0.05, σ 1 =0.03, Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 31 / 89

32 Vasicek Dynamics 0.25 Vasicek paths, x=0.05, k=1, θ=0.05, σ=0.1, Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 32 / 89

33 Vasicek Model Distribution Chair for Energy Trading & Finance We can solve the SDE and find r(t) = r(s)e κ(t s) + θ(1 e κ(t s) ) + σ We can also calculate t s e κ(t u) dw (u) E(r(t) F s ) = r(s)e κ(t s) + θ(1 e κ(t s) ) t Var(r(t) F s ) = σ 2 e 2κ(t u) du = σ2 [1 e 2κ(t s)] s 2κ Rates in the Vasicek model are mean-revesting, however, negative rates are possible. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 33 / 89

34 Vasicek Model Bond Prices Computation of price of zero-coupon bond is explicitly possible by evaluating the risk-neutral expectation or solving the affine-term-structure equation. We obtain A(t, T ) = (θ σ2 σ2 )[B(t, T ) T + t] B(t, T )2 2κ2 4κ B(t, T ) = 1 κ [1 e κ(t t) ] Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 34 / 89

35 Term Structures: Vasicek Vasicek TS, r 0 =0.1, k=0.9, θ=0.2, σ= Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 35 / 89

36 Term Structures: Vasicek Vasicek TS, r 0 =0.1, k=0.1, θ=0.1, σ= Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 36 / 89

37 Term Structures: Vasicek Vasicek TS, r 0 =0.1, k=0.9, θ=0.2, σ= Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 37 / 89

38 Vasicek Long Rate Rates in the Vasicek Model are given by R(τ) = 1 τ {A(τ) B(τ)r t} where we set τ = T t and R(τ) = R(t, T ); A(τ) = A(t, T ); B(τ) = B(t, T ). Letting τ we find the limiting rate lim R(τ) = θ λσ τ κ σ2 2κ 2 (where λ denotes the market price of risk in case we start under an historic measure.) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 38 / 89

39 TS Shapes: Vasicek yield (%) Maturity (months) Starting rate Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 39 / 89

40 Correlation Consider the Vasicek model dr t = κ(θ r t )dt + σdw t with p(t, T ) = exp(a(t, T ) B(t, T )r t ) and so all rates are given by R(t, T ) = A(t, T ) (T t) + B(t, T ) (T t) r t := a(t, T ) + b(t, T )r t. Consider now the joint distribution of two rates, e.g. T 1 = t + 1 and T 2 = t Then Cov(a(t, T 1 )+b(t, T 1 )r t, a(t, T 2 )+b(t, T 2 )r t ) = b(t, T 1 )b(t, T 2 )Var(r t ). So, since Var(a(t, T i ) + b(t, T i )r t ) = b(t, T i ) 2 Var(r t ) we find that the correlation of R(t, T 1 ) and R(t, T 2 ) is 1! Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 40 / 89

41 Vasicek Model Forward Measure We can obtain the dynamics of r(t) under the T -Forward measure. Here dr(t) = [κθ B(t, T )σ 2 κr(t)]dt + σdw T (t) = κ(ˆθ r(t)) + σdw T (t) with W T a Q T -Brownian motion defined by dw T (t) = dw (t) + σb(t, T )dt. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 41 / 89

42 Vasicek Model Forward Measure Thus r(r) = r(s)e κ(t s) + M T (s, t) + σ Chair for Energy Trading & Finance t s e κ(t u) dw T (u) with ) ( M T (s, t) = (θ σ2 κ 2 1 e κ(t s)) [ + σ2 2κ 2 e κ(t t) e κ(t +t 2s)]. Therefore we have still a Gaussian distribution under Q T with E T {r(t) F s } = r(s)e κ(t s) + M T (s, t) Var T {r(t) F s } = σ2 [ 1 e 2κ(t s)]. 2κ Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 42 / 89

43 Vasicek Model Option Pricing Chair for Energy Trading & Finance The price of a European call options with strike X, maturity T written on a pure discount bond maturing at time S(> T ) is ZB(t, T, S, X) = p(t, S)Φ(h) Xp(t, T )Φ(h σ p ) (8) with 1 e2κ(t t) σ p = σ B(t, S) 2κ h = 1 ( ) p(t, S) log + σ p σ p p(t, T )X 2. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 43 / 89

44 The Cox-Ingersoll-Ross Model (CIR) Under the risk-neutral measure dr(t) = κ(θ r(t))dt + σ r(t)dw (t), r(0) = r 0 (9) with r 0, κ, θ, σ > 0. The condition 2κθ > σ 2 ensures that r remains positive. In this model the market price of risk should be assumed to be of form λ(t, r(t)) = λ r(t) then the structure is preserved under the change of measure. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 44 / 89

45 Dynamics: One-Factor 0.11 CIR paths, x=0.05, k=1, θ=0.05, σ=0.01, Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 45 / 89

46 Dynamics: One-Factor 0.35 CIR paths, x=0.05, k=1, θ=0.05, σ=0.04, Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 46 / 89

47 CIR Distribution It can be shown that r is distributed according to a non-central χ 2 -distribution. The conditional mean is given by ( E(r(t) F s ) = e κ(t s) r(s) + θ 1 e κ(t s)). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 47 / 89

48 CIR Bond Prices The price of a zero-coupon bond is (affine model) where log A(t, T ) = B(t, T ) = p(t, T ) = exp{a(t, T ) B(t, T )r t } h = κ 2 + 2σ 2 [ 2h exp{(κ + h)(t t)/2} ] 2κθ/σ 2 2h + (κ + h)(exp{(t t)h} 1) 2(exp{(T t)h} 1) 2h + (κ + h)(exp{(t t)h} 1) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 48 / 89

49 CIR Bond Prices Under the risk-neutral measure the bond price dynamic is dp(t, T ) = ( ) 1 log A(t, T ) B(t, T ) log p(t, T )dt p(t, T ) σp(t, T ) B(t, T ) log ( ) log A(t, T ) dw (t). p(t, T ) Observe that the bond price volatility is not deterministic, but depends on the current level of the bond price. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 49 / 89

50 Term Structures: One-Factor CIR TS, r 0 =0.05, k=1, θ=0.06, σ= Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 50 / 89

51 Term Structures: One-Factor CIR TS, r 0 =0.05, k=1, θ=0.05, σ= Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 51 / 89

52 Term Structures: One-Factor CIR TS, r 0 =0.05, k=1, θ=0.04, σ= Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 52 / 89

53 CIR Option Prices The price of a European call option with maturity T > t, strike X, written on an S-bond with S > T is given by ( ) ZBC(t, T, S, X) = p(t, S)χ 2 2F [ρ + ψ + B(T, S)] ; 4κθ σ 2, 2ρ 2 r(t) exp(h(t t)) ρ + ψ + B(T, S) ( Xp(t, T )χ 2 2F [ρ + ψ], 4κθ σ 2, 2ρ 2 r(t) exp{h(t t)} ρ + ψ ) with ρ = ρ(t t) := 2h σ 2 (exp[h(t t)] 1) ψ = κ + h σ 2 F = F (S T ) := log(a(t ; S)/X). B(T, S) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 53 / 89

54 CIR Forward Dynamics We can obtain the Q T -dynamics (dynamics under the T -forward measure) of r(t). T dr(t) = [κθ (κ + B(t, T )σ 2 )r(t)]dt + σ r(t)dw T (t) with dw T (t)dw (t) + σb(t, T ) r(t)dt. This is again of CIR-type and so the distribution of r(t) conditional on r(s) can be computed and is a non-central χ 2. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 54 / 89

55 CIR Libor Rates Consider the simply-compounded forward rate at time t with expiry T and maturity S defined as [ ] 1 p(t, T ) F (t; T, S) = τ(t, S) p(t, S) 1 where τ(t, S) is the year fraction between T and S. We compute the dynamics of p(t, T )/p(t, S) under the S-forward measure Q s. This must be a Q s -martingale, thus we only need to obtain the diffusion term. Using Itô s formula in 2-dimensions with f (x, y) = x y d [ p(t, T ) p(t, S) ] = = 1 p(t, S) p(t, T ) dp(t, T ) dp(t, S) + (...)dt p(t, S) 2 p(t, T ) p(t, S) ( B(t, T )σ r(t)dw s (t)) + p(t, T ) p(t, S) (B(t, S)σ r(t)dw s (t)) + (...) }{{} =0 p(t, T ) = p(t, S) (B(t, S) B(t, T ))σ r(t)dw s (t). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim55 Pricing / 89 dt

56 CIR Libor Rates Using and p(t, T ) = exp{(a(t, T ) A(t, S)) (B(t, T ) B(t, S))r(t)} p(t, S) r(t) = log [( p(t, T ) p(t, S) ) A(t, S) A(t, T ) ] /{B(t, S) B(t, T )} we find df (t; T, S) = σ(f (t; T, S) + 1 τ(t, S) ) A(t, S) (B(t, S) B(t, T )) log[(τ(t, S)F (t; T, S) + 1) A(t, T ) ]dw s (t) Observe that this is different from the dynamics assumed in the Black-type formula! Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 56 / 89

57 The 2-Factor CIR Model with δ 0 0, δ 1 > 0, δ 2 > 0 and with factors are given by r(t) = δ 0 + δ 1 Y 1 (t) + δ 2 Y 2 (t) dy 1 (t) = (µ 1 λ 11 Y 1 (t) λ 21 Y 2 (t))dt + Y 1 (t)dw 1 (t) dy 2 (t) = (µ 2 λ 21 Y 1 (t) λ 22 Y 2 (t))dt + Y 2 (t)dw 2 (t) In addition, µ 1 0, µ 2 0, λ 11 > 0, λ 22 > 0, λ 12 < 0, λ 21 < 0. Then we have Y 1 (t) 0 and Y 2 (t) 0. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 57 / 89

58 Dynamics: Two-Factor CIR2 paths, x 0 =0.04, y 0 =0.02, k 1 =1.2, k 2 =1, θ 1 =0.02, θ 2 =0.03, σ 1 =0.005, σ 2 = Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 58 / 89

59 Dynamics: Two-Factor CIR2 paths, x 0 =0.04, y 0 =0.02, k 1 =1.2, k 2 =1, θ 1 =0.02, θ 2 =0.03, σ 1 =0.005, σ 2 = Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 59 / 89

60 Dynamics: Two-Factor 0.35 CIR2 paths, x 0 =0.04, y 0 =0.02, k 1 =1.2, k 2 =1, θ 1 =0.02, θ 2 =0.03, σ 1 =0.04, σ 2 = Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 60 / 89

61 Quantile: Two-Factor Quantilfächer Mean Median 1% Quantil 10% Quantil 90%Quantil 99%Quantil Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 61 / 89

62 Term Structures: Two-Factor Chair for Energy Trading & Finance 1 CIR2 TS, x 0 =0.02, y 0 =0.02, k 1 =1.2, k 2 =1, θ 1 =0.02, θ 2 =0.03, σ Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 62 / 89

63 Term Structures: Two-Factor Chair for Energy Trading & Finance CIR2 TS, x 0 =0.01, y 0 =0.05, k 1 =1.2, k 2 =1, θ 1 =0.02, θ 2 =0.03, σ Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 63 / 89

64 Term Structures: Two-Factor Chair for Energy Trading & Finance 1 CIR2 TS, x 0 =0.04, y 0 =0.02, k 1 =1.2, k 2 =1, θ 1 =0.02, θ 2 =0.03, σ Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim Pricing 64 / 89

65 Agenda 1 Interest Rates Basic Concepts 2 Short Rate Models 3 Heath-Jarrow-Morton (HJM) model 4 Contingent Claim Pricing Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim65 Pricing / 89

66 Useful Relations Short-rate Dynamics: dr(t) = a(t)dt + b(t)dw (t), (10) Bond-price Dynamics: dp(t, T ) = p(t, T ) {m(t, T )dt + v(t, T )dw (t)}, (11) Forward-rate Dynamics: df (t, T ) = α(t, T )dt + σ(t, T )dw (t). (12) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim66 Pricing / 89

67 Useful Relations If p(t, T ) satisfies (11), then for the forward-rate dynamics we have df (t, T ) = α(t, T )dt + σ(t, T )dw (t), where α and σ are given by { α(t, T ) = vt (t, T )v(t, T ) m T (t, T ), σ(t, T ) = v T (t, T ). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim67 Pricing / 89

68 Useful Relations If f (t, T ) satisfies (12), then the short rate satisfies dr(t) = a(t)dt + b(t)dw (t), where a and b are given by { a(t) = ft (t, t) + α(t, t), b(t) = σ(t, t). (13) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim68 Pricing / 89

69 Useful Relations If f (t, T ) satisfies (12), then p(t, T ) satisfies dp(t, T ) = p(t, T ) {(r(t) + A(t, T ) + 1 ) 2 S(t, T ) 2 dt + S(t, T )dw (t) where T A(t, T ) = α(t, s)ds, t T S(t, T ) = σ(t, s)ds. (14) t Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim69 Pricing / 89

70 Heath-Jarrow-Morton (HJM) model The Heath-Jarrow-Morton model uses the entire forward rate curve as (infinite-dimensional) state variable. The dynamics of the instantaneous, continuously compounded forward rates f (t, T ) are exogenously given by df (t, T ) = α(t, T )dt + σ(t, T )dw (t). For any fixed maturity T, the initial condition of the stochastic differential equation is determined by the current value of the empirical (observed) forward rate for the future date T which prevails at time 0. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim70 Pricing / 89

71 Heath-Jarrow-Morton (HJM) model The exogenous specification of the family of forward rates {f (t, T ); T > 0} is equivalent to a specification of the entire family of bond prices {p(t, T ); T > 0}. Furthermore, the dynamics of the bond-price processes are dp(t, T ) = p(t, T ) {m(t, T )dt + S(t, T )dw (t)}, where m(t, T ) = r(t) + A(t, T ) S(t, T ) 2, with and T A(t, T ) = α(t, s)ds t T S(t, T ) = σ(t, s)ds. t Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim71 Pricing / 89

72 HJM Drift Condition We want to find an EMM equivalent measure (the risk-neutral martingale measure) such that Z(t, T ) = p(t, T ) B(t) is a martingale for every 0 T T. A risk-neutral EMM exists iff there exists a process λ(t), with 1 L(t) = e t λdw 1 t λ 2du. defines a Girsanov pair and 2 for all 0 T T and for all t T, we have α(t, T ) = σ(t, T ) T t σ(t, s)ds + σ(t, T )λ(t). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim72 Pricing / 89

73 Forward Risk-neutral Martingale Measures For many valuation problems in the bond market it is more suitable to use the bond price process p(t, T ) as numéraire. One needs an equivalent probability measure Q such that the auxiliary process Z (t, T ) = p(t, T ) p(t, T, t [0, T ], ) is a martingale under Q for T T. This measure is called such a measure forward risk-neutral martingale measure. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim73 Pricing / 89

74 Forward Risk-neutral Martingale Measures (FRN-EMM) Bond price dynamics under the original probability measure P are given as dp(t, T ) = p(t, T ) {m(t, T )dt + S(t, T )dw (t)}, with m(t, T ) from the HJM-drift condition. Application of Itô s formula to the quotient p(t, T )/p(t, T ) yields dz (t, T ) = Z (t, T ) { m(t, T )dt with + (S(t, T ) S(t, T ))dw (t)}, m(t, T ) = m(t, T ) m(t, T ) S(t, T )(S(t, T ) S(t, T )). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim74 Pricing / 89

75 FRN-EMM The drift coefficient of Z (t, T ) under any EMM Q is given as m(t, T ) (S(t, T ) S(t, T ))γ(t). For Z (t, T ) to be a Q -martingale this coefficient has to be zero, and replacing m with its definition we get (A(t, T ) A(t, T )) ( S(t, T ) 2 S(t, T ) 2) = (S(t, T ) + γ(t)) (S(t, T ) S(t, T )). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim75 Pricing / 89

76 FRN-EMM Written in terms of the coefficients of the forward-rate dynamics, this identity simplifies to T T = γ(t) α(t, s)ds T T σ(t, s)ds. T T σ(t, s)ds Taking the derivative with respect to T, we obtain T α(t, T ) + σ(t, T ) σ(t, s)ds = γ(t)σ(t, T ). T 2 Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim76 Pricing / 89

77 FRN-EMM There exists a forward risk-neutral martingale measure if and only if there exists an adapted process γ(t) such that for all 0 t T T α(t, T ) = σ(t, T ) (S(T, T ) + γ(t)) Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim77 Pricing / 89

78 Agenda 1 Interest Rates Basic Concepts 2 Short Rate Models 3 Heath-Jarrow-Morton (HJM) model 4 Contingent Claim Pricing Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim78 Pricing / 89

79 Gaussian HJM Framework Assume that the dynamics of the forward rate are given under a risk-neutral martingale measure Q by df (t, T ) = α(t, T )dt + σ(t, T )d W (t) with deterministic forward rate volatility. Then f (t, t) = r(t) t = f (0, t) + ( σ(u, t)s(u, t))du t + σ(u, t)d W (u), 0 which implies that the short-rate as well as the forward rates f (t, T ) have Gaussian probability laws. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim79 Pricing / 89 0

80 Options on Bonds Consider a European call C on a T -bond with maturity T T and strike K. So we consider the T -contingent claim Its price at time t = 0 is X = (p(t, T ) K) +. C(0) = p(0, T )Q (A) Kp(0, T )Q T (A), with A = {ω : p(t, T ) > K} and Q T resp. Q the T - resp. T -forward risk-neutral measure. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim80 Pricing / 89

81 Options on Bonds has Q-dynamics d Z = Z Z(t, T ) = p(t, T ) p(t, T ) { S(S S )dt (S S )d W } (t), so a deterministic variance coefficient. Now Q (p(t, T ) K) = Q ( p(t, T ) p(t, T ) K ) = Q ( Z(T, T ) K). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim81 Pricing / 89

82 Options on Bonds Since Z(t, T ) is a Q T -martingale with Q T -dynamics d Z(t, T ) = Z(t, T )(S(t, T ) S(t, T ))dw T (t), we find that under Q T Z(T, T ) = p(0, T { ) p(0, T ) exp { exp 1 2 T 0 T 0 (S S )dwt T } (S S ) 2 dt The stochastic integral in the exponential is Gaussian with zero mean and variance Σ 2 (T ) = T 0 (S(t, T ) S(t, T )) 2 dt. } Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim82 Pricing / 89

83 Options on Bonds So Q T (p(t, T ) K) with = Q T ( Z(T, T ) K) = N(d 2 ) ( ) d 2 = log p(0,t ) Kp(0,T ) 1 2 Σ2 (T ). Σ 2 (T ) Repeat the argument to get The price of the call option is given by C(0) = p(0, T )N(d 2 ) Kp(0, T )N(d 1 ), with parameters given as above. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim83 Pricing / 89

84 Swaps Consider the case of a forward swap settled in arrears characterized by: a fixed time t, the contract time, dates T 0 < T 1,... < T n, equally distanced T i+1 T i = δ, R, a prespecified fixed rate of interest, K, a nominal amount. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim84 Pricing / 89

85 Swaps A swap contract S with K and R fixed for the period T 0,... T n is a sequence of payments, where the amount of money paid out at T i+1, i = 0,..., n 1 is defined by X i+1 = Kδ(L(T i, T i ) R). The floating rate over [T i, T i+1 ] observed at T i is a simple rate defined as 1 p(t i, T i+1 ) = 1 + δl(t i, T i ). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim85 Pricing / 89

86 Swaps Chair for Energy Trading & Finance Using the risk-neutral pricing formula we obtain (we may use K = 1), n [ Ti ] Π(t, S) = E Q e r(s)ds t δ(l(ti 1, T i 1 ) R) Ft i=1 [ [ n Ti ] r(s)ds T = E Q E Q e i 1 F T i 1 i=1 ( Ti 1 ) ] e r(s)ds 1 t p(t i 1, T i ) (1 + δr) F t = n ( ) p(t, T i 1 ) (1 + δr)p(t, T i ) i=1 = p(t, T 0 ) n c i p(t, T i ), i=1 with c i = δr, i = 1,..., n 1 and c n = 1 + δr. So a swap is a Interest linear Rates combination Basic Concepts Short of zero-coupon Rate Models Heath-Jarrow-Morton bonds, and (HJM) we model obtain Contingent itsclaim price 86 Pricing / 89

87 Caps An interest cap is a contract where the seller of the contract promises to pay a certain amount of cash to the holder of the contract if the interest rate exceeds a certain predetermined level (the cap rate) at some future date. A cap can be broken down in a series of caplets. A caplet is a contract written at t, in force between [T 0, T 1 ], δ = T 1 T 0, the nominal amount is K, the cap rate is denoted by R. The relevant interest rate (LIBOR, for instance) is observed in T 0 and defined by p(t 0, T 1 ) = δl(t 0, T 0 ). Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim87 Pricing / 89

88 Caplets A caplet C is a T 1 -contingent claim with payoff X = Kδ(L(T 0, T 0 ) R) +. Writing L = L(T 0, T 0 ), p = p(t 0, T 1 ), R = 1 + δr, we have L = (1 p)/(δp), (assuming K = 1) and ( ) 1 p + X = δ(l R) + = δ δp R ( ) 1 + ( ) 1 + = p (1 + δr) = p R. Interest Rates Basic Concepts Short Rate Models Heath-Jarrow-Morton (HJM) model Contingent Claim88 Pricing / 89

89 Caplets The risk-neutral pricing formula leads to [ T1 ] Π C (t) = E Q e r(s)ds ( ) + t 1p R Ft [ [ T1 r(s)ds = E Q E Q e T 0 = E Q [ = E Q [ = R E Q [ T0 p(t 0, T 1 ) e t T0 e t ] F T 0 r(s)ds (1 pr ) + Chair for Energy Trading & Finance ( T0 ) ] + e r(s)ds 1 t p R F t r(s)ds ( 1 p R ) + F t ] ] Ft ( T0 ) ] + e r(s)ds 1 t R p F t. So a caplet is equivalent to R put options on a T 1 -bond with Interest Rates Basic Concepts Short Rate Models maturity T 0 and strike 1/R Heath-Jarrow-Morton (HJM) model Contingent Claim89 Pricing / 89.