Lecture 2 - Calibration of interest rate models and optimization

Transcription

1 - Calibration of interest rate models and optimization Elisabeth Larsson Uppsala University, Uppsala, Sweden March 2015 E. Larsson, March 2015 (1 : 23)

2 Introduction to financial instruments Introduction to interest rate models Nonlinear optimization E. Larsson, March 2015 (2 : 23)

3 Introduction to financial instruments The bank account: The value at time t of the bank account B t is defined by db t = r t B t dt t B t = B 0 e 0 rsds, where r t is a short interest rate, governed by a short rate model. E. Larsson, March 2015 (3 : 23)

4 Zero-coupon bond: A zero-coupon bond Z(t, T ) with time of maturity T is a contract that guarantees the holder 1 unit of currency at time T. The value of this contract at time t T is Z(t, T ) = E Q t (e T t r sds ). (1) Specifically, Z(T, T ) = 1. E. Larsson, March 2015 (4 : 23)

5 t The value of the bank account B t = B 0 e 0 rsds at time t is computed from historical values of the interest rate r s, 0 s t. The value of the zero-coupon bond Z(t, T ) = E Q t (e ) T t r sdsf at time t with time of maturity T is determined by the expectations on the interest rate r s, t s T. We will soon look at some interest rate models. E. Larsson, March 2015 (5 : 23)

6 Day-count convention There are different day-count conventions existing to account for different number of days of the months etc. We will let τ(t 1, t 2 ), t 1 < t 2 represent the time-difference between t 1 and t 2 when the actual day-count convention is taken into account. Example: Actual/360 τ(t, T ) = T t in days 360. E. Larsson, March 2015 (6 : 23)

7 Simply-compunded spot interest rate A simply-compounded spot interest rate L(t, T ) with time of maturity T at time t is defined by Rearranging of (2) gives L(t, T ) = 1 Z(t, T ) τ(t, T )Z(t, T ). (2) Z(t, T )(1 + L(t, T )τ(t, T )) = 1, i.e. the simply-compounded interest rate is the constant interest rate at time t, if the interest rate is paid out at time T. E. Larsson, March 2015 (7 : 23)

8 LIBOR LIBOR The London Interbank Offered Rate is a simply-compunded spot interest rate. LIBOR is the average interest rate estimated by leading banks in London that the average leading bank would be charged if borrowing from other banks. LIBOR rates are calculated for 5 currencies and 7 different maturities ranging from 1 day to 1 year. For most currencies the day-count convention used is Actual/360. E. Larsson, March 2015 (8 : 23)

9 STIBOR and more STIBOR The Stockholm Interbank Offered Rate is the average interest rate that SEB, Nordea, Svenska Handelsbanken, Swedbank och Danske Bank would charge if borrowing from each other. STIBOR rates are calculated for 6 different maturities ranging from 1 day to 6 months and the day-count convention used is Actual/360. Also other IBOR rates exist, e.g. CIBOR (Copenhagen Interbank Offered Rate). LIBOR, STIBOR or other IBOR rates can be used to calibrate the interest models that we shall soon define. E. Larsson, March 2015 (9 : 23)

10 Swap rates Swap rates can also be used in the calibration of interest rate models. The interest rate swap prices are related to the zero-coupon prices as s(t, T ) = 1 Z(t, T ) N j=1 τ(t j 1, T j )Z(t, T j ) where T j, j = 1,..., N are reference dates with T N = T denoting the maturity of the swap. (3) Example: At t = 0, for a swap with maturity T = 1 year, and reference dates T j that are 3 months apart, we have τ(t j 1, T j ) = 0.25, T 0 = t = 0, T 1 = 0.25, T 2 = 0.5, T 3 = 0.75, T 4 = T = 1. E. Larsson, March 2015 (10 : 23)

11 Yield curve The yield curve is showing several yields or interest rates across the time to maturity (the term ). This is also refered to as the term structure of interest rates. By Pcb21 at en.wikipedia [GFDL ( or CC-BY-SA-3.0 ( from Wikimedia Commons E. Larsson, March 2015 (11 : 23)

12 Introduction to interest rate models Vasicek model: dr t = κ(φ r t )dt + σdw t (4) φ is the long term mean of the short rate, κ is the mean reversion rate, σ is the volatility of the interest rate. Note that the volatility is equally large independent of r t. This is a one factor short rate model. E. Larsson, March 2015 (12 : 23)

13 For the Vasicek model it holds Z(t, T ) = E Q t (e T t B(t, T ) = 1 e κτ(t,t ), κ ) r sds = e A(t,T ) B(t,T )rt, A(t, T ) = (φ σ2 2κ 2 ) (B(t, T ) τ(t, T )) σ2 4κ B2 (t, T ). The parameters in this model are θ = {κ, φ, σ, r t }. From e.g. IBOR rates and swap rates, we can calibrate the model-parameters to the observed rates. E. Larsson, March 2015 (13 : 23)

14 Cox Ingersoll Ross (CIR) model: dr t = κ(φ r t )dt + σ r t dw t (5) φ is the long term mean of the short rate, κ is the mean reversion rate, σ is the volatility of the interest rate. If the Feller condition 2κφ > σ 2 holds, then r t > 0 for the CIR model. In the CIR-model, the size of the volatility is scaled with rt, i.e., the smaller r t is, the smaller is the actual volatility. E. Larsson, March 2015 (14 : 23)

15 For the CIR model it holds Z(t, T ) = E Q t B(t, T ) = A(t, T ) = 2κφ σ 2 (e T t ) r sds = e A(t,T ) B(t,T )rt, 2 κ + γ coth(γτ(t, T )/2), ( ) eκ(t t)/2 ln cosh(γτ(t,t )/2)+ κ γ sinh(γτ(t,t )/2), γ = κ 2 + 2σ 2. Again, the parameters are θ = {κ, φ, σ, r t }. For general models, no closed-form solutions exist for Z(t, T ). E. Larsson, March 2015 (15 : 23)

16 Other one-factor models: Ho-Lee Hull-White dr t = φ(t)dt + σdw t (6) dr t = κ(φ(t) r t )dt + σdw t (7) Also multi-factor short rate models exist such as multi-factor Vasicek and multi-factor CIR. E. Larsson, March 2015 (16 : 23)

17 Formulation of the calibration problem When calibrating the short rate models to, e.g., IBOR rates and swap rates, we compute the parameters θ from inf θ Θ ( N1 ) N 2 H(L θ i L M i ) + H(si θ si M ), (8) i=1 i=1 where N 1 is the number of market observations of IBOR rates, and N 2 is the number of market observations of swap rates and Θ is the space for all possible parameter sets. E. Larsson, March 2015 (17 : 23)

18 For the Vasicek model we have L θ i = s θ i = 1 Z(0, T i ) τ(0, T i )Z(0, T i ), 1 Z(0, T i,n ) N j=1 τ(t i,j 1, T i,j )Z(t, T i,j ) Z(0, T i ) = e A(0,T i ) B(0,T i )r 0, B(0, T i ) = 1 e κτ(0,t i ), κ A(0, T i ) = (φ σ2 2κ 2 ) (B(0, T i ) τ(0, T i )) σ 2 4κ B2 (0, T i ), and similarily for the CIR model. E. Larsson, March 2015 (18 : 23)

19 Nonlinear optimization For simplicity of notation we rewrite Equation (8) as inf x f ( x), x = (x 1,..., x N ). (9) We will here describe the Nelder-Mead simplex method Quasi-Newton method to solve (9). E. Larsson, March 2015 (19 : 23)

20 The Nelder-Mead simplex method The algorithm uses a simplex of n + 1 points for an n-dimensional vector x. Let x i denote the list of points in the current simplex, i = 1,..., n Sort the vertex values f (x 1 ) f (x 2 )... f (x n+1 ). 2. Calculate the centroid x 0 of the n best points x 0 = ( n i=1 x i)/n. 3. Generate the reflected point x r = x 0 + α(x 0 x n+1 ). (In the direction away from the worst point.) E. Larsson, March 2015 (20 : 23)

21 4. If f (x 1 ) f (x r ) f (x n ) then replace x n+1 with x r and return to 1. (New point is neither worst nor best.) 5. If f (x r ) < f (x 1 ) then compute the expanded point x e = x 0 + γ(x 0 x n+1 ). (New point best, go further.) If f (x e ) < f (x r ) then replace x n+1 with x e, otherwise replace x n+1 with x r and return to 1. (Choose one.) 6. If f (x r ) f (x n ) then compute the contracted point x c = x 0 + ρ(x 0 x n+1 ). (New point bad, go shorter.) If f (x c ) < f (x n+1 ) then replace x n+1 with x c and return to Else shrink the simplex by x i = x 1 + σ(x i x 1 ), i = 2,..., n + 1 and return to 1. (Shrink towards best corner.) E. Larsson, March 2015 (21 : 23)

22 A quasi-newton method aims at finding the values x for which f = 0. A Taylor-expansion of f ( x) around an iterate x k gives f ( x k + x) f ( x) + x T f ( x k ) x T B x (10) where B is an approximation to the Hessian matrix of f. From (10) we get f ( x k + x) f ( x k ) + B x and setting this gradient to 0 we obtain x = B 1 f (x k ). E. Larsson, March 2015 (22 : 23)

23 Nonlinear optimization in MATLAB A variant of the Nelder-Mead simplex method is implemented in fminsearch. fminsearchcon that allows for bounds on the parameters is available for download at The Broyden Fletcher Goldfarb Shanno (BFGS) algorithm one variant of quasi-newton with cubic line search is implemented in fminunc. E. Larsson, March 2015 (23 : 23)