Term Structure Models Workshop at AFIR-ERM Colloquium, Panama, 2017

Transcription

1 Term Structure Models Workshop at AFIR-ERM Colloquium, Panama, 2017 Michael Sherris CEPAR and School of Risk and Actuarial Studies UNSW Business School UNSW Sydney UNSW August 2017 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

2 Workshop Aims Overview of commonly used term structure models, both continuous time and discrete time and methods use to fit models. Provide participants with an understanding of models and financial and actuarial applications rather than full mathematical details. Key topics covered (some more detailed than others) will be: Yield curves, Market instruments, Static Term Structure Concepts of no-arbitrage and risk neutral versus real world probabilities, Main continuous time single factor short rate models, Affine models, Dynamic Nelson-Siegel model, Arbitrage-free Nelson Siegel model, HJM (Heath, Jarrow, Morton) framework, Market models - LMM (Libor Market Models), BGM (Brace, Gatarek, Musiela), SABR (stochastic alpha, beta, rho), Discrete time models and binomial implementation including forward induction. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

3 References: Cairns, A. J. (2004), Interest Rate Models: An Introduction, Princeton University Press. Diebold, F. X. and Rudebusch, G. D., (2013), Yield Curve Modeling and Forecasting: the Dynamic Nelson-Seigel Approach, Princeton University Press. Filipovi c, Damir (2009), Term-Structure Models A Graduate Course, Springer. van Deventer, D. R. and Imai, K. (1997), Financial Risk Analytics: A Term Structure Approach for Banking, Insurance and Investment Management, Irwin Professional Publishing. Svoboda, S. (2004), Interest Rate Modelling, Palgrave Macmillan. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

4 My Background Late 70 s and early 80 s: Industry experience in banking, life insurance and pensions - bond markets (RBA open market operations), corporate finance (tax based financing), swaps, bill futures arbitrage. Mid 80 s through late 90 s: Professional actuarial education - finance and investment courses for Actuaries Institute in Australia, co-author of text for SOA Financial Economics, author of text Money and Capital Markets: Pricing, Yields and Analysis. Since 1985: Academic teaching and research - quantitative finance, financial economics, insurance, longevity risk. Sherris, M. (1994), A One Factor Interest Rate Model and the Valuation of Loans with Prepayment Provisions, Transactions of The Society of Actuaries, Vol 46, Sherris, M. and A. Ang, (1997), Interest Rate Risk Management: Developments in Interest Rate Term Structure Modeling for Risk Management and Valuation of Interest-Rate-Dependent Cash Flows, North American Actuarial Journal, Vol 1, No 2, Michael Sherris (UNSW) AFIR ERM Workshop August / 137

5 Assumed knowledge Actuarial students studying actuarial professional courses with financial mathematics, probability and statistics background. Practitioners with foundation background in interest rate models seeking a deeper understanding of different model approaches. Practitioners seeking an overview of interest rate models. Practitioners seeking a refresher of interest rate models. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

6 Actuarial Applications Actuarial Applications Michael Sherris (UNSW) AFIR ERM Workshop August / 137

7 Actuarial Applications Term structure models are used in many actuarial applications. Valuation of interest dependent cash flows - expected claims at different future dates using a spot rate yield curve. Interest rate risk management using the models to quantify cash flow duration and convexity using a spot rate yield curve Valuation of option and guarantee features in insurance products that depend on uncertain future yield curves. Determination of solvency requirements for economic and regulatory capital purposes. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

8 Yield Curves and Market Instruments Yield Curves and Market Instruments Michael Sherris (UNSW) AFIR ERM Workshop August / 137

9 Term Structure Models Yields for differing maturities for: coupon paying government bonds, par coupon bonds, zero coupon bonds, FRA s, interest rate futures, interest rate swaps. Default-free interest rates (yields to maturity, spot interest rates, forward interest rates) and bond prices for different maturities. Market instruments and methods to construct the interest rate term structure to be modelled. Model for random changes in (risk-free) interest rates and prices of (default-free) discount bonds. Historical yield curve time series (P measure) and market pricing (Q measure). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

10 Spot Rates Zero coupon bond prices P (t, T ) = exp [ (T t) R (t, T )] = (T t) L (t, T ) where P (t, T ) is the price at time t of $1 payable at time T (zero coupon bond), R (t, T ) is the continuous compounding yield to maturity and L (t, T ) simple interest yield to maturity (similar to LIBOR rate). Spot interest rates (spot rates - continuous compounding) log P (t, T ) R (t, T ) = T t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

11 Forward Rates Forward rate agreement (FRA) - agreement to fix an interest rate for a future time period (time T to S) at time t P (t, T ) = P (t, S) exp [(S T ) F (t, T, S)] = P (t, S) [1 + (S T ) L (t, T, S)] Forward rate at time t (continuous compounding) for time T to S (S > T ) F (t, T, S) = 1 P (t, T ) log S T P (t, S) and simple forward rate (similar to LIBOR forward rate) L (t, T, S) = 1 [ ] P (t, T ) S T P (t, S) 1 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

12 Coupon Paying Bonds Most bonds traded are coupon paying bonds including government issued bonds (regarded often as default free) Price of coupon paying bond paying N coupons (in arrears) with C i paid at time T i and nominal (face value) N paid at maturity T N P TN (t) = = N P (t, T i ) C i + P (t, T N ) N i=1 N i=1 1 [1 + Y (T N )] t i C i + 1 [1 + Y (T N )] T Ni N where Y (T N ) is the yield to maturity for the T N maturity coupon bond. Market conventions - most government coupon bonds pay semi-annual coupons and are quoted as semi-annual compounding yields. Times for price quotes are usually between coupon dates (broken periods) and sometimes quoted with accrued interest and sometimes without (dirty and clean prices). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

13 Par Coupon Bonds Yields to maturity for coupon paying bonds often used to construct default free term structure using bootstrapping. Quoted market yields are impacted by many factors such as different size of coupons, tax, liquidity. Smoothing is usually used to extract a par yield curve - the yield curve for coupons bonds with coupons equal to the yield to maturity payable on the same coupon dates and all priced on a coupon date, hence priced at par (face value usually taken as N = 100). Par yield to maturity (equal to par coupon rate) given by N = c TN N c TN = 1 P (t, T N) N P (t, T i ) i=1 N P (t, T i ) + P (t, T N ) N i=1 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

14 Futures and Swaps Interest rate futures (Eurodollar futures, futures on bank bills and bonds) are similar to forwards and have mark to market and deposit requirements Futures rates require a convexity adjustment Forward rate = Futures rate 1 2 σ2 (T t) 2 Interest rate swaps - swap fixed for floating coupons, value of payer swap is: ( [ ]) Π p (t) = N P (t, T 0 ) δk n i=1 P (t, T i) + P (t, T n ) ( ) = N δ n i=1 P (t, T i) [F (t, T i 1, T i ) K ] where N is nominal value, K is fixed rate, δ is time between swap payments. Par swap rates have Π p (t) = 0, similar to par coupon bond rates. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

15 Bootstrapping Par Yields Given the par yield to maturity curve for coupon paying bonds then spot interest rates and forward interest rates can be determined. All of these are equivalent but spot rates and forward rates can be used more flexibly to value more general cash flows. Spot rates are determined working forward by maturity using bootstrapping. The first spot rate (continuous compounding) is given using the first par coupon rate: 1 + c T1 = exp [ R (t, T 1 )] R (t, T 1 ) = log [1 + c T1 ] Then having solved for R (t, T j ) for j = 1 to i 1, solve period by period, for R (t, T i ) using i 1 = c Ti exp [ R (t, T j )] + 1 exp [ R (t, T i )] j=1 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

16 Bootstrapping Par Yields Rearranging gives spot rates (continuous compounding) R (t, T i ) = log 1 + c Ti i 1 1 c Ti exp [ R (t, T j )] j=1 i = 2,..., N Then forward rates (continuous compounding) are derived from the spot rates F (t, T i 1, T i ) = 1 T i T i 1 [R (t, T i ) R (t, T i 1 )] Michael Sherris (UNSW) AFIR ERM Workshop August / 137

17 Yield Curve Smoothing Methods used to smooth market bond price/yield curve data include: Weighted Least Squares or Regression, MLE or Bayesian Fit discount factors, spot rates or forward rates with: Splines, Nelson-Siegel or Svensson Parametric Curve, and similar exponential-polynomial class (see Cairns (2004) Chapter 12), Smith-Wilson Kernel method Michael Sherris (UNSW) AFIR ERM Workshop August / 137

18 Yield Curve Smoothing General approach where Issues: P = Cd + ɛ P is (column) vector of market prices, C is the cash flow matrix with cash flows for the market instruments used to fit term structure, d discount factors for the term structure (determined from smoothed yield curve in terms of yields to maturity, spot rates or forward rates), ɛ vector of errors to be minimised in the smoothing of market data smoothing spot rates results in saw tooth forward rates so better to smooth forward rates, market instruments include coupon bonds (short maturities, illiquidity) or futures and swaps (longer maturities, more active trading), need for interpolation and extrapolation. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

19 Term Structure Models - R packages R packages for Term Structure termstrc cubic splines approach of McCulloch (1971, 1975), the Nelson and Siegel (1987) method with extensions by Svensson (1994),Diebold and Li(2006) and DePooter(2007). Weighted constrained optimization procedure with analytical gradients and a globally optimal start parameter search algorithm. YieldCurve fits Nelson-Siegel, Diebold-Li and Svensson. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

20 Instantaneous Forward Rates and the Short Rate Instantaneous forward rate at time t f (t, T ) = lim F (t, T, S) = P (t, T ) / T log P (t, T ) = S T T P (t, T ) which gives P (t, T ) = exp [ T ] f (t, u) du = exp [ R (t, T ) (T t)] t and f (τ) = R (τ) + τ τ R (τ) where τ = T t. Short rate at time t is : r (t) = lim T t R (t, T ) = R (t, t) = f (t, t) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

21 Static Nelson-Siegel Parametric formula for instantaneous forward yield curve (cross sectional) f (τ) = β 1 + β 2 e λτ + β 3 λτe λτ Nelson-Siegel spot rate yield curve (integrate forward yield curve) ( ) ( ) 1 e λτ 1 e λτ y (τ) = β 1 + β 2 + β λτ 3 e λτ λτ Desirable features: P (0) = 0 and lim τ P (τ) 0 instantaneous short rate lim τ 0 y (τ) = f (0) = r (equals β 1 + β 2 ) lim τ y (τ) = β 1 (the long term interest rate) flexible shapes - flat, increasing, decreasing, humped, U-shaped (β 3 determines size and shape of hump) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

22 Dynamic Nelson-Siegel Static Nelson Siegel fits constant parameters β 1, β 2, β 3, λ to produce a best fit term structure Dynamic Nelson Siegel treats parameters β 1, β 2, β 3 as variables, or latent factors, and the coefficients as factor loadings ( ) ( ) 1 e λτ 1 e λτ y t (τ) = β 1t + β 2t + β λτ 3t e λτ λτ Factor loadings 1, the loading on β 1t, impacts on all yields, but impacts long term yields ( relatively ) more compared to other factors 1 e λτ λτ, the loading on β 2t, starts at 1 and decreases quickly to zero -( impacts short term yields most, a short term factor, 1 e λτ λτ e λτ), the loading on β 3t, starts at 0, increases and then decreases to zero - impacts medium term yields most, a medium term factor Michael Sherris (UNSW) AFIR ERM Workshop August / 137

23 Svensson Yield Curve Parametric formula for instantaneous forward yield curve (cross sectional) f (τ) = β 1 + (β 2 + β 3 λ 1 τ) e λ 1τ + β 4 λ 2 τe λ 2τ Svensson spot rate yield curve (integrate forward yield curve) ( 1 e λ 1 ) ( τ 1 e λ 1 τ y (τ) = β 1 + β 2 + β λ 1 τ 3 λ 1 τ ) e λ 2τ +β 4 ( 1 e λ 2 τ λ 2 τ allows for second hump in the yield curve. ) e λ 1τ Michael Sherris (UNSW) AFIR ERM Workshop August / 137

24 Discussion - Static Term Structure Term structure from a static perspsective (P and Q measures). Different ways of representing the yield curve (level, slope, curvature). Different market instruments used to fit the yield curve (bonds, futures, forwards, swaps). Methods of fitting term structure yield curves (yield curve smoothing). Parametric form of cross sectional yield curve and links to dynamic models (Nelson-Siegerl). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

25 Concepts of no-arbitrage and risk neutral versus real world probabilities Concepts of no-arbitrage and risk neutral versus real world probabilities Michael Sherris (UNSW) AFIR ERM Workshop August / 137

26 PDE Approach - Short Rate Models Start from stochastic process for interest rates (short interest rate), Use Ito formula to derive stochastic process for bond prices (bonds are contingent claims or derivatives on interest rates), Require no-arbitrage and derive partial differential equation (PDE) for (zero coupon) bond prices, Solve PDE subject to boundary conditions. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

27 Brownian Motion Sometimes called a Wiener process A stochastic process {W (t), t 0} is said to be a standard Brownian motion process, or simply Brownian motion, if: W (0) = W 0 = 0; W (t) has stationary and independent increments; and for every t > 0, W (t) N (0, t). A stochastic process {X (t) t 0} is said to be a Brownian motion process with drift coefficient µ and variance parameter σ 2 if: X (0) = X 0 = 0; {X (t) t 0} has stationary and independent increments; and for every t > 0, X (t) N ( µt, σ 2 t ). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

28 Standard Brownian motion Integral from Properties t W (t) = W 0 + dw (u) 0 W (t) is continuous W (t) is nowhere differentiable W (t) is process of unbounded variation W (t) is process of bounded quadratic variation Conditional distribution of W (u) W (t) for u > t is normal with mean W (t) and variance (u t) Variance of a forecast of W (u) increases indefinitely as u Michael Sherris (UNSW) AFIR ERM Workshop August / 137

29 Generalized univariate Wiener process Drift and volatility depend on X (t) and t dx (t) = α (X (t), t) dt + σ (X (t), t) dw (t) Special cases: Arithmetic Brownian motion Geometric Brownian motion Mean reverting process Dynamics for function of Brownian motion f (X (t), t). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

30 Arithmetic Brownian Motion The process defined by dx = αdt + σdw α (X (t), t) = α σ (X (t), t) = σ W (t) = X (t) αt, σ is a standard Brownian motion. If {W (t), t 0} is a standard Brownian motion, then by defining X (t) = αt + σw (t), the process {X (t), t 0} is a Brownian motion with a drift α and volatility σ. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

31 Arithmetic Brownian Motion X (t) may be positive or negative distribution of X (u) given X (t) for u > t is normal with mean X (t) + α (u t) and standard deviation σ (u t) variance tends to infinity as u goes to infinity (variance grows linearly with time) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

32 Geometric Brownian Motion If X (t) starts at positive value then it remains positive X (t) has an absorbing barrier at 0 Conditional distribution of X (u) given X (t) for u > t is lognormal. Conditional mean of ln X (u) given by ln X (t) + α (u t) 1 2 σ2 (u t) and conditional standard deviation of ln X (u) is σ (u t). Often used to model values - positive and increases at exponential rate. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

33 Geometric Brownian Motion E (X (t) X (u), 0 u s ) ( ) = E e Y (t) X (u), 0 u s ( ) = E e Y (s)+y (t) X (s) X (u), 0 u s ( ) = e Y (s) E e Y (t) Y (s) X (u), 0 u s = Y (s) exp [α (t s)]. since (Y (t) Y (s)) N (( α 1 2 σ2) (t s), σ 2 (t s) ), then E ( e Y (t) Y (s)) = e [(α 1 2 σ2 )(t s)+ 1 2 σ2 (t s)] = e [α(t s)] Michael Sherris (UNSW) AFIR ERM Workshop August / 137

34 Mean Reverting Process dx = κ (µ X ) dt + σx γ dw α (X (t), t) = κ (µ X ) κ > 0 σ (X (t), t) = σx γ κ is speed of adjustment parameter, µ long run mean and σ is volatility parameter X (t) is positive as long as X (t) starts positive As X (t) approaches zero, the drift is positive and volatility vanishes As u the variance of X (u) is finite Michael Sherris (UNSW) AFIR ERM Workshop August / 137

35 Mean Reverting Process If γ = 1 2 the conditional distribution of X (u) given X (t) for u > t is non-central chi-squared with mean (X (t) µ) exp [ κ (u t)] + µ and variance ( σ 2 X (t) κ ( σ 2 +µ 2κ ) ( exp [ κ (u t)] exp [ 2κ (u t)] ) (1 exp [ κ (u t)]) 2 ) CIR or square root process Michael Sherris (UNSW) AFIR ERM Workshop August / 137

36 One-Dimensional Ito s Formula Consider the SDE dx = α(x (t), t)dt + σ(x (t), t)dw (t) = αdt + σdw and suppose that we have another stochastic process f (X (t), t) = f t which depends on X (t) and t. Itô s formula tells us this process satisfies the following: df t = f f dx + X t dt f 2 X 2 σ2 t dt. This formula is also known as Itô s formula (stochastic Taylor series). df = f X = f dx + t dt [ 2 f X 2 dx f X t dxdt + 2 f [ α f X + f t σ2 2 f X 2 t 2 dt2 ] dt + σ f X dw ] + o (dt) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

37 Bivariate Ito s Formula Consider the SDE s dx = α (X, Y, t) dt + σ (X, Y, t) dw 1 dy = β (X, Y, t) dt + ν (X, Y, t) dw 2 dw 1 dw 2 = E [dw 1 dw 2 ] = ρdt Probabilistically, dw 2 = ρdw 1 + (1 ρ 2 )dz = df (X, Y, t) = f X f f dx + dy + Y t dt [ 2 f X 2 dx f X Y dxdy + 2 f [ α f X + β f Y + f t σ f X dw 1 + ν f Y dw 2 ] Y 2 dy 2 + o (dt) [ σ 2 2 f X 2 + 2ρσν 2 f X Y + ν2 2 f Y 2 ]] dt Michael Sherris (UNSW) AFIR ERM Workshop August / 137

38 Covariation Process and Quadratic Variation Consider dx i (t) = a i (t) dt + b i (t) dw i (t) then quadratic covariation process for X i (t) and X j (t) is X i, X j (t) = t 0 b i (u) b j (u) du d X i, X j (t) = b i (t) b j (t) dt sometimes written as dx i (t) dx j (t) Quadratic variation is X, X (t) = t 0 b (u)2 du. Product rule d (X (t) Y (t)) = X (t) dy (t) + Y (t) dx (t) + d X, Y (t) Integration by parts formula t t X (t) Y (t) = X (0) Y (0) + X (s) dy (s) + Y (s) dx (s) + X, Y (t) 0 0 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

39 Feynman-Kac formula Consider the pde [ α (X, t) f X + f t + 1 ] 2 σ2 (X, t) 2 f X 2 rf + c (X, t) = 0 defined on interaval [0, T ] with terminal condition f (X, T ) = Ψ (X ) Then Feynman-Kac formula is solution in terms of conditional expectation [ T exp ( s t t f (X, t) = E rdτ) ] ( c (X, s) ds + exp ) T rdτ Ψ t (X ) X t = x where X is a diffusion process dx = α (X, t) dt + σ (X, t) dw Michael Sherris (UNSW) AFIR ERM Workshop August / 137

40 PDE for Bond Pricing Model short interest rate with a diffusion process (single factor) dr (t) = α (r, t) dt + σ (r, t) dz A bond has a value P (r, t, T ) = P (r, τ) with dynamics (from Itô s formula) dp (r, t, T ) = where P (r, t, T ) dr P (r, t, T ) r 2 r 2 dr P = P r (αdt + σdz ) r 2 σ2 dt + P t dt [ = α P r P 2 r 2 σ2 + P ] dt + P t r σdz = P [mdt + sdz ] m = 1 P ( α P r P 2 r 2 σ2 + P ) t and s = 1 P P (r, t, T ) dt t P r σ Michael Sherris (UNSW) AFIR ERM Workshop August / 137

41 PDE for Bond Pricing Construct a portfolio, V, of two bonds, V 1 and V 2, with different maturities τ 1 > τ 2 V = V 1 (r, τ 1 ) + V 2 (r, τ 2 ) dv = dv 1 + dv 2 = V 1 [m 1 dt + s 1 dz ] + V 2 [m 2 dt + s 2 dz ] = [V 1 m 1 + V 2 m 2 ] dt + [V 1 s 1 + V 2 s 2 ] dz Now if we choose V 1 and V 2 so that the volatility of the portfolio is zero and [V 1 s 1 + V 2 s 2 ] = 0 V 1 V 2 = s 2 s 1 (V V 2 ) s 1 + V 2 s 2 = 0 V 2 = Vs 1 and V 1 = Vs 2 s 1 s 2 s 1 s 2 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

42 PDE for Bond Pricing We then have, since the portfolio is now riskless, or, rearranging, [V 1 m 1 + V 2 m 2 ] = rv = r (V 1 + V 2 ) Vs 2 s 1 s 2 m 1 + Vs 1 s 1 s 2 m 2 = rv (m 1 r) = (m 2 r) = λ s 1 s 2 where λ is the market price of (interest rate) risk. The excess return per unit of risk is the same for all maturity bonds. We vary V 1 so the portfolio is self financing ( ) m2 s 1 m 1 s 2 dv = V dt s 1 s 2 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

43 PDE for Bond Pricing Substituting expressions for m and s we obtain the PDE for bond prices ( 1 α P P r P 2 r 2 σ2 + P ) r = 1 P t P r σλ P t P + (α σλ) r P 2 r 2 σ2 rp = 0 Then solve subject to boundary conditions. Solving PDEs using known solutions for particular cases, guesses or Feynman-Kac. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

44 PDE for Bond Pricing - Example Consider random walk for yield (y (0) = r) P (τ) = e yτ and dy = σdz From Ito formula Hence P t = yp dp = P y = τp 2 P y 2 = τ2 P (y + 12 σ2 τ 2 ) Pdt στpdz m = y σ2 τ 2 and s = στ and risk premium for interest rate risk gives ( y σ2 τ 2) r στ = λ Michael Sherris (UNSW) AFIR ERM Workshop August / 137

45 PDE for Bond Pricing - Example In this case y (τ) = r + λστ 1 2 σ2 τ 2 Note that = risk free rate + price of risk risk convexity adjustment yield curve is quadratic in maturity maximum at λ σ, so humped shaped can have negative yields instantaneous forward rates are quadratic f (τ) = log P (τ) = ( y (τ) τ) = y (τ) + τ (y (τ)) τ τ τ = r + λστ 1 2 σ2 τ 2 τ ( λσ σ 2 τ ) = r + 2λστ 3 2 σ2 τ 2 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

46 Martingale Approach for Bond Pricing Background on Martingales Change of Measure Fundamental Theorem of Asset Pricing Bond Pricing as an Expectation under Risk Neutral Measure Michael Sherris (UNSW) AFIR ERM Workshop August / 137

47 Continuous-Time Martingales We denote the filtration generated by the process at time t by F t and is interpreted as the information (or history) of the process up until time t. We shall assume the process {W (t)} is F t -adapted, which means that given this history F t, the value of the process at t, W (t), is known. The stochastic process {W (t), t 0} is said to be a martingale if: E [ W (t) ] <, i.e. finite expectation of the absolute value, and for all s < t, we have E (W (t) F s ) = W (s). The best forecast of unobserved future values is the most recently available observed value of the process. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

48 Continuous-Time Martingales Example (Brownian Motion) Consider a Brownian motion {X t, t 0} with drift parameter α and variance parameter σ 2. Then, for s < t, we have E (X t F s ) = E [X s + (X t X s ) F s ] = E [X s F s ] + E [X t X s F s ] = X s + α (t s). since (X t X s ) N ( α (t s), σ 2 (t s) ). Thus, we see that E (X t F s ) = X s only if α = 0. A B.M. with a zero drift is a martingale. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

49 Continuous-Time Martingales Example (Geometric Brownian Motion) Consider a geometric Brownian motion {Y t, t 0} defined as Y t = exp (X t ), where X t is a B.M. with drift parameter µ and variance parameter σ 2. Then, for s < t, we have E (Y t F s ) = E [exp (X t ) F s ] = E [exp (X s + (X t X s )) F s ] = E [exp (X s ) F s ] E [exp (X t X s ) F s ] = Y s E [exp (X t X s F s ] = Y s exp [(µ + 12 ) ] σ2 (t s) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

50 Continuous-Time Martingales Recalling the moment generating function of a normal distribution, we have [ E (Y t F s ) = Y s exp α (t s) + 1 ] 2 σ2 (t s) = Y s exp [(α + 12 ) ] σ2 (t s). Thus, we see that the geometric Brownian motion is not a martingale unless of course if α = 1 2 σ2. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

51 Continuous-Time Martingales We can transform the geometric B.M. to form a martingale by defining the process ( Z t = exp W (t) 1 ) 2 σ2 t. By following the same argument as above, it is left as an exercise to prove that the process {Z t, t 0} is a martingale if the Brownian motion has zero drift. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

52 Stochastic Differential Equations Definition: A stochastic differential equation, or simply SDE, has the following form: dx t = µ (X t, t) dt + σ (X t, t) db t, where B t is the standard Brownian motion process. The drift µ (X t, t) and σ (X t, t) can be functions of the process X t and time t. A stochastic process X t that satisfy the above SDE is sometimes called a diffusion process. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

53 Stochastic Integration {W (t), t 0} is a Brownian motion with zero drift and variance parameter σ 2 Let f be a function with continuous derivative on the interval [a, b]. b a f (t) dw (t) = n lim n k=1 max(t k t k 1 ) 0 f (t k 1 ) (W (k) W (k 1)) where a = t 0 < t 1 < < t n = b is a partition of [a, b]. This definition of a stochastic integral is sometimes called the Ito integral. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

54 Stochastic Integration To simplify notations, W (k) is the value of the stochastic process at time t k. The integral is itself another stochastic process which we can label as Y t = b a f (t) dw (t). The integration by parts formula applied to sums is given by n k=1 n f (t k 1 ) (W (k) W (k 1)) = [f (b) W (b) f (a) W (a)] k=1 [f (t k ) f (t k 1 )] W (k). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

55 Stochastic Integration Now, we take appropriate limits on both sides, we see that b a b f (t) dw (t) = [f (b) W (b) f (a) W (a)] W (t) df (t) Take the expectation of both sides, = [f (b) W (b) f (a) W (a)] a b E (Y t ) = [f (b) E (W (b)) f (a) E (W (a))] }{{} b E (W (t)) df (t) a } {{ } 0 0 = 0. a W (t) f (t) dt. No matter what the form of the function is, the mean of the Ito integral is always zero. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

56 Stochastic Integration Its variance [ can also be computed as follows. n ] Var f (t k 1 ) (W (k) W (k 1)) k=1 = n k=1 f 2 (t k 1 ) Var (W (k) W (k 1)) = σ 2 n f 2 (t k 1 ) (t k t k 1 ), k=1 Var (Y t ) = lim n max(t k t k 1 ) 0 b = σ 2 f 2 (t) dt. a σ 2 n k=1 [ f 2 (t k 1 ) (t k t k 1 ) ] Michael Sherris (UNSW) AFIR ERM Workshop August / 137

57 Stochastic Integration - Example Let the process {Y t, t 0} be defined by Y t = t 0 e α(t u) dw (u), for t 0, where α is some constant parameter. Find the mean and variance of this process. Solution: The process has zero mean (Ito integral). Variance is t Var [Y t ] = σ 2 = σ 2 = σ2 2α 0 t 0 [ e α(t u)] 2 du e 2α(t u) du ( e 2αt 1 ). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

58 Cameron-Martin-Girsanov Theorem - Change of Measure If W (t) is a Brownian motion ( under measure P and λ (t) is a (previsible) process satisfying E P exp 1 ) T 2 0 λ (t)2 dt <, the Novikov condition, then there exists a measure Q such that Q is equivalent to P ( dq dp = exp T 0 λ (t) dw (t) ) 1 T 2 0 λ (t)2 dt W (t) = W (t) + t λ 0 (t) dt is a Q Brownian motion This can be applied to change measure as follows [ ] dq E Q [X ] = E P dp X ( T = E P [exp λ (t) dw (t) T 0 ) ] λ (t) 2 dt X Michael Sherris (UNSW) AFIR ERM Workshop August / 137

59 Fundamental Theorem of Asset Pricing Define the money market account (or cash account) as ( t ) B (t) = B (0) exp r (s) ds The Fundamental Theorem of Asset Pricing as applied to Bond Prices states that Bond prices are arbitrage free if and only if there exists a measure Q, equivalent to P, under which, for each T, the discounted bond price process P (t, T ) /B (t) is a martingale for all t : 0 < t < T. The market is complete if and only if Q is the unique measure for which P (t, T ) /B (t) are martingales. Q is referred to as the risk-neutral measure with the cash account as numeraire since the expected return on a bond under this measure is the risk-free rate. 0 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

60 Martingale Approach for Bond Pricing Assume SDEs for the risk free short rate and the bond price are (note change in notation from earlier) dr (t) = a (t) dt + b (t) dw (t) dp (t, T ) = P (t, T ) [m (t, T ) dt + s (t, T ) dw (t)] Money market account Market price of risk db (t) = r (t) B (t) dt λ (t) = Discounted bond price process Z (t, T ) = P (t, T ) B (t) m (t, T ) r (t) s (t, T ) ( t ) = P (t, T ) exp r (s) ds 0 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

61 Martingale Approach for Bond Pricing We have (Ito product rule) dz (t, T ) = 1 dp (t, T ) + P (t, T ) d B (t) ( ) d B (t) B, P (t) and using Ito formula ( ) 1 d = 1 B (t) B (t) 2 db (t) (t) dt d 2 3 B (t) = r B (t) B (t) Hence dz (t, T ) = P (t, T ) (m (t, T ) dt + s (t, T ) dw (t)) r (t) P (t, T ) dt B (t) B (t) = Z (t, T ) [(m (t, T ) r (t)) dt + s (t, T ) dw (t)] [ ] (m (t, T ) r (t) λ (t) s (t, T )) dt = Z (t, T ) +s (t, T ) (dw (t) + λ (t)) = Z (t, T ) s (t, T ) d W (t) and Z (t, T ) is a martingale under measure Q. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

62 Martingale Approach for Bond Pricing Discounted bond price P(t,T ) B(t) and for 0 < S < T P (t, T ) B (t) is a martingale under risk neutral measure = E Q [ P (T, T ) B (T ) P (t, T ) = E Q [ B (t) B (T ) F s P (t, S) = E Q [exp F s ] ] ( S ) ] r (u) du F s t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

63 Discussion - Bond Pricing Use of PDE approach versus Martingale Approach. Difference between P and Q measure for interest rate dynamics versus bond prices. Importance of market price of risk. Single factor versus multiple factor models. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

64 Main Continuous Time Single Factor Short Rate Models Main Continuous Time Single Factor Short Rate Models Michael Sherris (UNSW) AFIR ERM Workshop August / 137

65 Main continuous time single factor short rate models One factor models with constant interest rate volatility (affine models) Vasicek (1977) dr (t) = α (µ r (t)) dt + σd W (t) Ho and Lee (1986) dr (t) = θ (t) dt + σd W (t) Extended Vasicek or Hull and White Model (1990, 1993) dr (t) = α (µ (t) r (t)) dt + σd W (t) One factor models with rate-dependent interest rate volatility; Cox, Ingersoll and Ross (1985) - dr (t) = α (µ r (t)) dt + σ r (t)d W (t) Black, Derman ( and Toy (1990) - log-normal ) short rate - d log r (t) = θ (t) σ (t) σ(t) log r (t) dt + σ (t) d W (t) or with constant volatility d log r (t) = θ (t) dt + σd W (t) Black and Karasinski (1991) - log-normal short rate - d log r (t) = α (t) (log µ (t) log r (t)) dt + σ (t) d W (t) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

66 Vasicek model Risk neutral dynamics dr (t) = α (µ r (t)) dt + σd W (t) mean reverting process µ long term mean risk free rate under risk neutral measure, α rate r (t) reverts to long term mean σ local volatility of short term rate r (t + s) given r (t) is normally distributed under Q with mean µ + (r (t) µ) e αs and variance σ 2 [ 1 e 2αs] 2α Michael Sherris (UNSW) AFIR ERM Workshop August / 137

67 Vasicek model Bond prices determined by: using PDE for bond price of zero coupon bonds, guess form of solution and derive form of bond price by solving PDE, or using martingale approach and the dynamics for r (t) to evaluate dscounted expected value under risk neutral measure (often solved using Laplace transforms). Model does not give a perfect fit to an initial yield curve. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

68 Vasicek model Bond price takes the form P (t, T ) = exp [A (t, T ) B (t, T ) r (t)] where B (t, T ) = 1 e α(t t) α ) A (t, T ) = (B (t, T ) (T t)) (µ σ2 2α 2 σ2 B (t, T )2 4α Use this to construct the zero coupon bond yield curve and derive spot rates and forward rates. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

69 Ho-Lee model Gives a perfect fit to an initial yield curve. Risk neutral dynamics dr (t) = θ (t) dt + σd W (t) time varying drift allowing to fit initial yield curve short rate normally distributed. Bond price for zero coupon bonds P (t, T ) = exp [A (t, T ) B (t, T ) r (t)] where B (t, T ) = (T t) A (t, T ) = σ2 T 6 (T t)3 θ (s) (T s) ds t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

70 Ho-Lee model Setting where We have θ (t) = T f (0, T ) + σ2 T f (0, T ) = log P (0, T ) T r (t) = t r (0) + θ (s) ds + σ W (t) 0 = f (0, t) + σ2 t 2 + σ W (t) 2 which allows a perfect fit to the initial yield curve. We then have f (t, T ) = T log P (t, T ) = r (t) + f (0, T ) f (0, t) + σ2 t (T t) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

71 Extended Vasicek or Hull and White Model Risk neutral dynamics dr (t) = α (µ (t) r (t)) dt + σd W (t) Bond price for zero coupon bonds, matching initial term structure, where µ (t) = 1 α σ2 ( f (0, t) + f (0, t) + 1 e 2αt ) t 2α 2 P (t, T ) = exp [A (t, T ) B (t, T ) r (t)] 1 e α(t t) B (t, T ) = α A (t, T ) = log P (0, T ) P (0, t) + B (t, T ) f (0, t) σ2 4α 3 ( 1 e α(t t)) 2 ( 1 e 2αt ) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

72 Extended Vasicek or Hull and White Model We also have t t r (t) = e αt r (0) + α e α(t s) µ (s) ds + σ e α(t s) d W (s) 0 0 = f (0, t) + σ2 ( 1 e αt ) t 2 + σ 2α 2 e α(t s) d W (s) Special cases µ (t) = µ, a constant, gives Vasicek if α 0 and αµ (t) θ (t) as α 0 then gives Ho-Lee. 0 Michael Sherris (UNSW) AFIR ERM Workshop August / 137

73 Cox-Ingersoll-Ross Model Instantaneous interest rate r(t) dynamics under the risk neutral Q measure given by: dr(t) = α(µ r(t))dt + σ r(t)d W (s) where α > 0 is the speed of mean reversion of r(t), µ > 0 is the long-run mean of r(t), σ r(t) 0 is the volatility of the short rate process. Require 2αµ σ 2 to ensure the process is positive. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

74 Cox-Ingersoll-Ross Model Zero coupon bond price is given by: P(t, T ) = E Q t [e T t r(u)du ] = exp [A (t, T ) B (t, T ) r (t)] where A(t, T ) and B(t, T ) are given by: ( ) A(t, T ) = 2αµ σ 2 log 2γe (γ+α)(t t)/2 (γ + α)(e γ(t t) 1) + 2γ with B(t, T ) = 2(e γ(t t) 1) (γ + α)(e γ(t t) 1) + 2γ γ = α 2 + 2σ 2 with boundary conditions A(T, T ) = 0 and B(T, T ) = 0. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

75 Other Short Rate Models Many other models with differing assumptions, for example: Brennan and Schwartz (1979) - two-factor model with short rate and long term rate on a consol bond. Longstaff and Schwartz (1992) - two-factor version of CIR with short rate and instantaneous volatility of short rate as factors Michael Sherris (UNSW) AFIR ERM Workshop August / 137

76 Affine Term Structure Models - Short Rate Models Affine short rate term structure models have bond prices of affine form in the short rate P (t, T ) = exp [A (t, T ) B (t, T ) r (t)] and for bond prices to have this form the risk neutral drift and volatility of the short rate take the form m (t, r (t)) = a (t) + b (t) r (t) s (t, r (t)) = δ (t) + γ (t) r (t) where a (t), b (t), δ (t), and γ (t) are deterministic functions. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

77 Affine Term Structure Models - Short Rate Models Consider the risk neutral dynamics for the short rate dr (t) = m (t, r (t)) dt + s (t, r (t)) d W (t) Then an application of Ito s formula gives [( A dp (t, T ) = P (t, T ) t A t r (t) Bm + 1 ) ] 2 (Bs)2 dt Bsd W (t) Under the risk neutral dynamics all bonds have instantaneous expected returns equal to the short rate so that [ ] dp (t, T ) = P (t, T ) r (t) dt + S (t, T, r (t)) d W (t) where S (t, T, r (t)) is the volatility of the bond price P (t, T ). Michael Sherris (UNSW) AFIR ERM Workshop August / 137

78 Affine Term Structure Models - Short Rate Models Define g (t, r) = ( A t A t r (t) Bm (t, r (t)) + 1 ) [Bs (t, r (t))]2 r 2 then this must be zero for all t and r for the bond prices to be arbitrage-free. Differentiating twice with respect to r gives and for this to hold 2 g (t, r) r 2 = B 2 m (t, r) r B2 2 s (t, r (t)) 2 r 2 = 0 2 m (t, r) r 2 = 0 and 2 s (t, r (t)) 2 r 2 = 0 which gives the affine form for the short rate dynamics. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

79 Affine Term Structure Models - Short Rate Models Special cases include a number of the spot rate models already covered. Vasicek Cox, Ingersoll and Ross a = αµ, b = α, δ = σ 2, γ = 0 dr (t) = α (µ r (t)) dt + σd W (t) a = αµ, b = α, δ = 0, γ = σ 2 dr (t) = α (µ r (t)) dt + σ r (t)d W (t) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

80 Discussion - Short Rate Models Some models were developed as equilibrium models (Vasicek and CIR) and others as arbitrage-free or relative valuation models. Models with constant parameters, drift and volatility, do not provide an exact fit to the current yield curve, so models with time varying drift and volatility were developed to do this. Model fitting requires both the current yield curve and a volatility structure for time varying volatility models. Gaussian models have geneally small chance of negative interest rates. Gaussian models have closed form expressions for bond prices (and options on bonds) because of log-normal distrubution of bond prices. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

81 Discussion - Short Rate Models Log-normal short rate models require numerical implementation to compute bond prices usually with a lattice. Many models developed as discrete time lattice models have continuous time limits (Ho-Lee, Black-Derman-Toy). Many are single factor models, with perfect correlation of instantaneous bond returns across maturities, but can be extended to multiple factors. Matching volatility and correlations between different maturity bond returns and option prices requires time varying volatility and multiple factor models. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

82 Affine Term Structure Models - Multi-Factor Models More generally affine term structure (ATS) models are arbitrage-free multifactor model of interest rates in which the yield on any risk-free zero-coupon bond is an affine function of a set of unobserved latent factors. Affine term structure (ATS) models have bond prices of affine form P (t, T ) = exp [ A (t, T ) + n j=1 B j (t, T ) X j (t) [ = exp A (t, T ) + B (t, T ) X (t) where X (t) = (X 1 (t), X 2 (t),..., X n (t)) is a vector of state variables and B (t, T ) = (B 1 (t, T ), B 2 (t, T ),..., B n (t, T )). We will only consider time homogeneous models where A (t, T ) and B (t, T ) are functions of (T t) only and the state variables X (t) are time homogeneous. Michael Sherris (UNSW) AFIR ERM Workshop August / 137 ] ]

83 Affine Term Structure Models - Multi-Factor Models If bond prices have the form P (t, t + τ) = exp [ ] A (τ) + B (τ) X (t) then X (t) must have SDE s dx (t) = (α + β) X (t) + SD (X (t)) d W (t) where W (t) is an n-dimensional Brownian motion ( under ) Q, α = (α 1, α 2,..., α n ) n is a constant vector, β = β ij is a constant i.j=1 matrix, S = (σ ij ) n i.j=1 is a constant matrix and D (X (t)) a diagonal matrix γ 1 X (t) + δ γ 2 X (t) + δ γ n X (t) + δ n where δ 1, δ 2,..., δ n are constants and γ = (γ 1, γ 2,..., γ n ) is a constant vector. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

84 Affine Term Structure Models - Gaussian Model As an example, a 3-factor Gaussian ATS model has r (t) = µ + 1 X (t) where 1 = (1, 1, 1) and X (t) = (X 1 (t), X 2 (t), X 3 (t)) follows a zero-mean Ornstein-Uhlenbeck process under the real world probability measure dx (t) = KX (t) dt + ΣdW (t) with K is lower triangular and Σ is diagonal, both 3 3 matrices. Market prices of risk for the latent factors are given by λ (t) = λ (0) + ΛX (t) where λ (t) = (λ 1 (t), λ 2 (t), λ 3 (t)) and Λ a 3 3 matrix. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

85 Affine Term Structure Models - Gaussian Model Bond prices for a τ maturity bond given by ( t+τ ) ] P (t, τ) = E Q [exp r (u) du F s t and the risk neutral dynamics are gven by dx (t) = ( K λ (t) Σ) X (t) dt + Σd W (t) = ((K + ΣΛ) X (t) + Σλ (0)) dt + Σd W (t) Zero coupon bond prices can be written as P (t, τ) = exp [A (τ) + B (τ) X (t)] where expressions for A (τ) and B (τ) are factor loadings that are functions of the parameters of the model solved from a set of differential equations. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

86 Dynamic Nelson-Siegel Model The Dynamic Nelson Siegel (DNS) model is effectively a 3-factor Gaussian model, although not arbitrage-free. Recognising the 3 factors as level, slope and curvature of the yield curve, we can write it as ( ) ( ) 1 e λτ 1 e λτ y t (τ) = l t + s t + c t e λτ λτ λτ and can be written in state-space form as a measurement equation y t = Λf t + ε t where y t = y t (τ 1 ) y t (τ 2 ). y t (τ N ), f t = l t s t c t, ε t = ε t (τ 1 ) ε t (τ 2 ). ε t (τ N ) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

87 Dynamic Nelson-Siegel Model The parameter matrix is 1 e 1 λτ 1 1 e λτ 1 λτ 1 λτ 1 e λτ 1 1 e 1 λτ 2 1 e λτ 2 Λ = λτ 2 λτ 2 e λτ 2... e λτ N 1 1 e λτ N λτ N 1 e λτ N λτ N where we have observable yields at times t = 1,..., T, and N maturities for zero coupon bond yields at each time. The l t, s t, and c t are common factors with dynamics given by the transition equation. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

88 Dynamic Nelson-Siegel Model The transition equation assumes a first-order vector autoregression (f t µ) = A (f t 1 µ) + η t where µ = µ l µ s µ c, A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33, η t = η l t η s t η c t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

89 Dynamic Nelson-Siegel Model Finally the state space formulation requires assumptions for the covariance structure of the measurement and transition errors. Here we assume white noise and orthogonal errors so that ( εt η t ) WN ( 0 0 Q 0 0 H This formulation allows the use of Kalman filter for optimal filtering and smoothing as well as prediction of yield factors and observed yields. Estimation is then by maximum likelihood with the state space representation and the Kalman filter. ) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

90 Dynamic Nelson-Siegel Model Kalman filter provides one-step-ahead prediction errors for which the Gaussian pseudo likelihood can be evaluated for any set of parameters. The parameter configuration that maximises the likleihood is found numerically using some form of either gradient based methods or analytic score functions. Other model estimation procedures are possible including an EM based optimization or a Bayesian approach using Markov-chain Monte Carlo. There are a large number of parameters so that traditonal gradient based numerical optimization methods may be intractable. If there are N maturities at each time point to fit then the measurement equation requires one-parameter λ, unless a fixed value is used, the transition equation has 12 parameters (3 means and 9 dynamic parameters), the measurement error covariance matrix has ( N 2 + N ) /2 parameters and the transition disturbance covariance marix has 6 parameters. So if there were 15 yield maturities to be fitted then there would be 139 parameters. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

91 Arbitrage-Free Nelson-Siegel Model The Dynamic Nelson-Seigel Model is flexible. It can fit cross section and time series of yields well. However it does not impose restrictions to make the model arbitrage-free. The DNS model can be made arbitrage-free and formulated as an affine term structure model. The arbitrage free Nelson Seigel (AFNS) maintains the DNS factor-loading structure and is an affine arbitrage-free term structure model. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

92 Arbitrage-Free Nelson-Siegel Model Consider a three-factor affine model with X t = ( Xt 1, Xt 2, Xt 3 ) then we require a yield function of the form ( ) y (t, T ) = A (t, T ) + X 1 1 e λ(t t) t + Xt 2 T t λ (T t) ( ) 1 e λ(t t) + e λ(t t) Xt 3 λ (T t) so the factor loadings in the affine model need to be B 1 (t, T ) = (T t), B 2 1 e λ(t t) (t, T ) =, λ and B 3 1 e λ(t t) λ(t t) (t, T ) = + (T t) e λ Michael Sherris (UNSW) AFIR ERM Workshop August / 137

93 Arbitrage-Free Nelson-Siegel Model The required model has r (t) = X 1 t + X 2 t and risk neutral dynamics for the state variables dx 1 t dx 2 t dx 3 t = λ λ 0 0 λ + σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 θ Q 1 θ Q 2 θ Q 3 d W 1 t d W 2 t d W 3 t X 1 t X 2 t X 3 t dt Michael Sherris (UNSW) AFIR ERM Workshop August / 137

94 Arbitrage-Free Nelson-Siegel Model Zero coupon bond prices are P (t, T ) = exp ( C (t, T ) + B 1 (t, T ) Xt 1 + B 2 (t, T ) Xt 2 + B 3 (t, T ) Xt 3 ) and the ODEs for the factor loadings are db 1 (t,t ) dt db 2 (t,t ) dt db 3 (t,t ) dt = λ 0 0 λ λ B 1 (t, T ) B 2 (t, T ) B 3 (t, T ) and dc (t, T ) dt = B (t, T ) θ Q ( ) Σ B (t, T ) B (t, T ) Σ j=1 j.,j with boundary conditions C (T, T ) = B 1 (T, T ) = B 2 (T, T ) = B 3 (T, T ) = 0. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

95 Arbitrage-Free Nelson-Siegel Model The relationship between the real-world dynamics and the risk-neutral dynamics is d W t = dw t + Γ t dt and use the essentially affine risk premium specification in which Γ t is affine in the state variables so that γ 0 1 γ 1 Γ t = γ 0 11 γ 1 12 γ 1 13 X γ 1 γ 0 21 γ 1 22 γ 1 t 23 X 2 3 γ 1 31 γ 1 32 γ 1 t 33 Xt 3 The P measure dynamics are dx t = K P [ θ P X t ] dt + ΣdW t and we are free to choose the mean vector θ P and the mean reversion matrix K P under the P measure and preserve the Q dynamics. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

96 Arbitrage-Free Nelson-Siegel Model The independent-factor AFNS model has independent state variables under the P measure dx 1 t κ P dxt θ P = 0 κ P 1 X 1 dxt θ P t 0 0 κ P 2 Xt 2 dt 33 Xt 3 + σ σ σ 33 θ P 3 dw 1 t dw 2 t dw 3 t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

97 Arbitrage-Free Nelson-Siegel Model The correlated-factor AFNS model has independent state variables under the P measure dx 1 t κ P dxt 2 11 κ P 12 κ P 13 θ P = κ P dxt 3 21 κ P 22 κ P 1 X 1 23 θ P t κ P 31 κ P 32 κ P 2 Xt 2 dt 33 Xt 3 + σ σ 21 σ 22 0 σ 31 σ 32 σ 33 θ P 3 dw 1 t dw 2 t dw 3 t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

98 Arbitrage-Free Nelson-Siegel Model The measurement equation is y t (τ 1 ) y t (τ 2 ). y t (τ N ) = 1 e 1 λτ 1 1 e λτ 1 λτ 1 λτ 1 e λτ 1 1 e 1 λτ 2 1 e λτ 2 λτ 2 λτ 2 e λτ e 1 λτ N 1 e λτ N λτ N λτ N e λτ N ε t (τ 1 ) ε t (τ 2 ) C (τ 1 ) τ 1 C (τ 2 ) τ 2. C (τ N ) τ N +. ε t (τ N ) X 1 t X 2 t X 3 t Michael Sherris (UNSW) AFIR ERM Workshop August / 137

99 Arbitrage-Free Nelson-Siegel Model Estimation of the AFNS uses a Kalman filter maximum likelihood approach. The model needs to be converted from continuous to discrete time. The conditional mean vector is E P [X T F s ] = ( I exp and the conditional variance is V P [X T F s ] = ( K P (T t) (T t) 0 )) ( ) θ P + exp K P (T t) X t e K Ps ΣΣ (K P ) s ds The state transition equation is ( ( )) ( ) X t = I exp K P t θ P + exp K P t X t 1 + η t where t is the time between observations. The variance of η t is Q = t 0 e K Ps ΣΣ (K P ) s ds Michael Sherris (UNSW) AFIR ERM Workshop August / 137

100 Arbitrage-Free Nelson-Siegel Model The AFNS measurement equation is y t = BX t + C + ε t and the stochastic error term is ( ) εt N η t ( 0 0 Q 0 0 H ) For the Kalman filter we start with unconditional mean and variance of state variables under the P measure X 0 = θ P, and Σ 0 = 0 e K Ps ΣΣ (K P ) s ds Michael Sherris (UNSW) AFIR ERM Workshop August / 137

101 Arbitrage-Free Nelson-Siegel Model Denote model parameters by ψ and the information at time t by Y t = (y 1, y 2,..., y T ), then at time t 1 assume we have the state update X t 1 and the mean square error matrix Σ t 1. The prediction step is X t t 1 = E P [X t Y t 1 ] = Φ X t,0 (ψ) + Φt X,1 (ψ) X t 1 Σ t t 1 = Φ X t,1 (ψ) Σ t 1 Φ X,1 (ψ) + Q t (ψ) t where ( ( )) Φ X t,0 (ψ) = I exp K P t θ P ( ) Φ X t,1 (ψ) = exp K P t Q t (ψ) = t 0 e K Ps ΣΣ (K P ) s ds Michael Sherris (UNSW) AFIR ERM Workshop August / 137

102 Arbitrage-Free Nelson-Siegel Model The time t update step improves X t t 1 using the additional information contained in Y t. where X t = E [X t Y t ] = X t t 1 + Σ t t 1 B (ψ) Ft 1 ν t Σ t = Σ t t 1 Σ t t 1 B (ψ) Ft 1 B (ψ) Σ t t 1 ν t = y t E [y t Y t 1 ] = y t B (ψ) X t t 1 C (ψ) F t = cov (ν t ) = B (ψ) Σ t t 1 B (ψ) + H (ψ) H (ψ) = diag ( σ 2 ε (τ 1 ),..., σ 2 ε (τ N ) ) Michael Sherris (UNSW) AFIR ERM Workshop August / 137

103 Arbitrage-Free Nelson-Siegel Model A single pass of the Kalman filter allows us to evaluate the Gaussian log likelihood N ( log l (y 1, y 2,..., y T ψ) = 1 t=1 2 N log (2π) 1 2 log F t 1 ) 2 ν tft 1 ν t where N is the number of observed yields. This is then maximised with respect to ψ. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

104 Discussion - Affine Arbitrage-Free and Nelson-Siegel Model Gaussian versus other affine models. Fitting current yield curve. Number of factors. Benefits of closed form solutions. Michael Sherris (UNSW) AFIR ERM Workshop August / 137

105 HJM Framework HJM (Heath, Jarrow, Morton) Michael Sherris (UNSW) AFIR ERM Workshop August / 137