Zvi WienerContTimeFin - 8 slide 1 Financial Engineering Term Structure Models Zvi Wiener tel:

Zvi WienerContTimeFin - 8 slide 2 Interest Rates F Dynamic of IR is more complicated. F Power of central banks. F Dynamic of a curve, not a point. F Volatilities are different along the curve. F IR are used for both discounting and defining the payoff. Source: Hull and White seminar

Zvi WienerContTimeFin - 8 slide 3 Main Approaches F Black’s Model (modification of BS). F No-Arbitrage Model.

Zvi WienerContTimeFin - 8 slide 4 Notations D - face value (notional amount) C - coupon payments (as % of par), yearly N - maturity See Benninga, Wiener, MMA in Education, vol. 7, No. 2, 1998

Zvi WienerContTimeFin - 8 slide 5 Continuous Version Denote by C t dt the payment between t and t+dt, then the bond price is given by: Principal should be written as Dirac’s delta.

Zvi WienerContTimeFin - 8 slide 6 Forward IR The Forward interest rate is a rate which investor can promise today, given the term structure. Suppose that the interest rate for a maturity of 3 years is r 3 =10%, and the interest rate for 5 years is r 5 =11%. No borrowing-lending restrictions.

Zvi WienerContTimeFin - 8 slide 7 Forward IR r 3 =10%, r 5 =11%. Lend $1,000 for 3 years at 10%. Borrow $1,000 for 5 years at 11%. Year 0 -$1,000+$1,000= $0 Year 3 $1,000(1.1) 3 = $1331 Year 5 -$1,000(1.11) 5 = -$1658 Is identical to a 2-year loan starting at year 3.

Zvi WienerContTimeFin - 8 slide 8 Forward IR Forward interest rate from t to t+n.

Zvi WienerContTimeFin - 8 slide 9 Forward IR Continuous compounding

Zvi WienerContTimeFin - 8 slide 10 Forward IR

Zvi WienerContTimeFin - 8 slide 11 Estimating TS from bond data Idea - to take a set of simple bonds and to derive the current TS. F Too many bonds. F Too few zero coupons. F Non simultaneous pricing. F Very unstable!

Zvi WienerContTimeFin - 8 slide 12 Estimating TS from bond data Assume that r 1 =5.5%, r 2 =5.55%, r 3 =5.6%, r 4 =5.65%, r 5 =5.7%. Bond prices 1 year 3% years5% years3% years7% years0%

Zvi WienerContTimeFin - 8 slide 13 Estimating the TS We can easily extract the interest rates from the prices of bonds. However if the bond prices are rounded to a dollar, the resulting TS looks weird. Conclusion: TS is very sensitive to small errors. Instead of solving the system of equations defining a unique TS it is recommended to fit the set of points by a reasonable curve representing TS. Another problem - time instability.

Zvi WienerContTimeFin - 8 slide 14 Is flat TS possible? Why can not IR be the same for different times to maturity? Arbitrage: F Zero investment. F Zero probability of a loss. F Positive probability of a gain.

Zvi WienerContTimeFin - 8 slide 15 Is flat TS possible? Form a portfolio consisting of 3 bonds maturing in one, two, and three years and without coupons. Choose a, b, c units of these bonds. Zero investment: ae -r + be -2r + ce -3r = 0 Zero duration: -ae -r - 2be -2r - 3ce -3r = 0

Zvi WienerContTimeFin - 8 slide 16 Is flat TS possible? Two equations, three unknowns ae -r + be -2r + ce -3r = 0 -ae -r - 2be -2r - 3ce -3r = 0 Possible solution (r=10%): a = 1,b = ,c=1.2214

Zvi WienerContTimeFin - 8 slide 17 Arbitrage in a flat TS

Zvi WienerContTimeFin - 8 slide 18 Arbitrage in a flat TS However even a small costs destroy this arbitrage. In many cases the assumption that TS is flat can be used.

Zvi WienerContTimeFin - 8 slide 19 Yield Denote by P(r,t,t+T) the price at time t of a pure discount bond maturing at time t+T > t. Define yield to maturity R(r, t,T) as the internal rate of return at time t on a bond maturing at t+T.

Zvi WienerContTimeFin - 8 slide 20 Yield The relation between forward rates and yield: When interest are continuously compounded the average of forward rates gives the yield.

Zvi WienerContTimeFin - 8 slide 21 TS model Assume that interest rates follow a diffusion process. What is the price of a pure discount bond P(r,t,T)? Implicit one factor assumption!

Zvi WienerContTimeFin - 8 slide 22 TS model Substituting dr we obtain: Taking expectation and dividing by dt we get:

Zvi WienerContTimeFin - 8 slide 23 TS model Using equilibrium pricing models assume that Here is the risk premium. The basic bond pricing equation is (Merton 1971,1973):

Zvi WienerContTimeFin - 8 slide 24 TS model Merton has shown that in a continuous-time CAPM framework, the ration of risk premium to the standard deviation is constant (over different assets) when the utility function is logarithmic. Sharpe ratio

Zvi WienerContTimeFin - 8 slide 25 TS model For a pure discount bond we have: Thus by Ito’s lemma

Zvi WienerContTimeFin - 8 slide 26 TS model Hence for the risk premium we have The basic equation becomes

Zvi WienerContTimeFin - 8 slide 27 Vasicek’s model Ornstein-Uhlenbeck process

Zvi WienerContTimeFin - 8 slide 28 Vasicek’s model Discrete modeling Negative interest rates. Can be used for example for real interest rates.

Zvi WienerContTimeFin - 8 slide 29 Shapes of Vasicek’s model All three standard shapes are possible in Vasicek’s model. Disadvantages: calibration, negative IR, one factor only. There is an analytical formula for pricing options, see Jamshidian 1989.

Zvi WienerContTimeFin - 8 slide 30 Extension of Vasicek Hull, White

Zvi WienerContTimeFin - 8 slide 31 CIR model Precludes negative IR, but under some conditions zero can be reached.

Zvi WienerContTimeFin - 8 slide 32 CIR model

Zvi WienerContTimeFin - 8 slide 33 CIR model

Zvi WienerContTimeFin - 8 slide 34 CIR model Bond prices are lognormally distributed with parameters:

Zvi WienerContTimeFin - 8 slide 35 CIR model As the time to maturity lengthens, the yield tends to the limit: Different types of possible shapes.

Zvi WienerContTimeFin - 8 slide 36 One Factor TS Models

Zvi WienerContTimeFin - 8 slide 37  1  2  3  1  2 Cox-Ingersoll-Ross ***0.5 Pearson-Sun ****0.5 Dothan *1.0 Brennan-Schwartz ***1.0 Merton (Ho-Lee) **1.0 Vasicek ***1.0 Black-Karasinski ***1.0 Constantinides-Ingersoll*1.5

Zvi WienerContTimeFin - 8 slide 38 Black-Derman-Toy The BDT model is given by for some functions U and . Find conditions on  2,  3, and  2 under which the Black-Karasinski model specializes to the BDT model.

Zvi WienerContTimeFin - 8 slide 39 The Gaussian One-Factor Models For  3 =  2 = 0 we get a Gaussian model, in which the short rates r(t 1 ), r(t 2 ), …,r(t k ) are jointly normally distributed (under the risk- neutral measure). Special cases: Vasicek and Merton models. In this case a negative  2 is mean reversion.

Zvi WienerContTimeFin - 8 slide 40 The Gaussian One-Factor Models For a Gaussian model the bond-price process is lognormal. An undesirable feature of the Gaussian model is that the short rate and yields on bonds are negative with positive probability at any future date.

Zvi WienerContTimeFin - 8 slide 41 The Affine One-Factor Models The Gaussian and CIR models are special cases of single factor models with the property that the solution has the form:

Zvi WienerContTimeFin - 8 slide 42 The Affine One-Factor Models The yield for all t is affine in r: Vasicek, CIR, Merton (Ho-Lee), Pearson-Sun.

Zvi WienerContTimeFin - 8 slide 43 TS Derivatives Suppose a derivative has a payoff h(r,t) prior to maturity, and a terminal payoff g(r,  ) when exercised (  <T). Then by the definition of the equivalent martingale measure, the price at time t is defined by:

Zvi WienerContTimeFin - 8 slide 44 TS Derivatives

Zvi WienerContTimeFin - 8 slide 45 TS Derivatives By Feynman-Kac theorem it can be equivalently written as a solution of PDE: With boundary conditions:

Zvi WienerContTimeFin - 8 slide 46 Bond Option A European option on a bond is described by setting h(x, t) = 0, g(x,  ) = Max( f(x,  ) - K, 0).

Zvi WienerContTimeFin - 8 slide 47 Interest Rate Swap Can be approximated as a contract paying the dividend rate h(r, t) = r t - r*, where r* is the fixed leg g(r,  ) = 0.

Zvi WienerContTimeFin - 8 slide 48 Cap Is a loan at variable rate that is capped at some level r*. Per unit of the principal amount of the loan, the value of the cap is defined when h(r t, t) = Min(r t,r*) g(r ,  ) = 1 (sometimes 0)

Zvi WienerContTimeFin - 8 slide 49 Floor Similar to a cap, but with maximal rate instead of minimal: h(r t, t) = Max(r t,r*) g(r ,  ) = 1 (sometimes 0)

Zvi WienerContTimeFin - 8 slide 50 MBS Mortgage Backed Securities Sinking fund bond. At origination a sinking fund bond is defined in terms of a coupon rate, a scheduled maturity date, and an initial principle. At each time prior to maturity there is an associated scheduled principle.

Zvi WienerContTimeFin - 8 slide 51 MBS Assume that the coupon rate is  and principal repayment is at a constant rate h. For a given initial principal p 0. The schedule is chosen so that at time T the loan is repaid.

Zvi WienerContTimeFin - 8 slide 52 MBS Home mortgages can be prepaid. This is typically done when interest rates decline. Unscheduled amortization process should be defined. It has psychological and economical factors. Standard solution - Monte Carlo simulation.

Zvi WienerContTimeFin - 8 slide 53 Monte Carlo X(  ) - random variable Let Y be a similar variable, which is correlated with X but for which we have an analytic formula.

Zvi WienerContTimeFin - 8 slide 54 Monte Carlo Introduce a new random variable (here Y* is the analytic value of the mean of Y(  ) and  - is a free parameter which we fix later)

Zvi WienerContTimeFin - 8 slide 55 Monte Carlo Calculate the variance of the new variable:

Zvi WienerContTimeFin - 8 slide 56 Monte Carlo If we can reduced variance! The optimal value of the parameter  is

Zvi WienerContTimeFin - 8 slide 57 Monte Carlo This choice leads to the variance of the estimator where  is the correlation coefficient between X and Y.