Financial Engineering

Financial Engineering Interest Rate Models Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 following Hull and White ContTimeFin - 10 Hull and White

Definitions P(t,T) zero-coupon bond price maturing at time T as seen at time t. r - instantaneous short rate v(t,T) volatility of the bond price R(t,T) rate for maturity T as seen at time t F(t,T) instantaneous forward rate as seen at time t for maturity T. ContTimeFin - 10 Hull and White

Modeling Bond Prices Suppose there is only one factor. The process followed by a zero-coupon bond price in a risk-neutral world takes the form: dP(t,T) = rP(t,T)dt + v(t,T)P(t,T)dZ the bond price volatility must satisfy v(t,t)=0, for all t. ContTimeFin - 10 Hull and White

Modeling Bond Prices ContTimeFin - 10 Hull and White

Forward Rates and Bond Prices $1 today grows by time T1 to This amount invested at forward rate F(t,T1,T2) for an additional time T2-T1 will grow to By no arbitrage this must be equal ContTimeFin - 10

Forward Rates and Bond Prices ContTimeFin - 10

Forward Rates and Bond Prices ContTimeFin - 10 Hull and White

Modeling Forward Rates What is the dynamic of F(t,T)? ContTimeFin - 10 Hull and White

Modeling Forward Rates Applying Ito’s lemma we obtain for F(t,T): ContTimeFin - 10 Hull and White

HJM Result Suppose We know that for some v Hence ContTimeFin - 10 Hull and White

Two Factor HJM Result ContTimeFin - 10 Hull and White

Volatility Structure The HJM result shows that once we have identified the forward rate volatilites we have defined the drifts of the forward rates as well. We have therefore fully defined the model! ContTimeFin - 10 Hull and White

Short Rate Non-Markov type of dynamic - path dependence. In a one factor model this process is ContTimeFin - 10 Hull and White

Ho and Lee Model Rates are normally distributed. All rates have the same variability. The model has an analytic solution. ContTimeFin - 10 Hull and White

Ho and Lee Model Where F(t,T) is the instantaneous forward rate as seen at time t for maturity T. ContTimeFin - 10 Hull and White

Bond Prices under Ho and Lee Where ContTimeFin - 10 Hull and White

Option Prices under Ho and Lee A discount bond matures at s, a call option matures at T ContTimeFin - 10 Hull and White

Lognormal Ho and Lee Is like Black-Derman-Toy without mean reversion. Short rate is lognormally distributed. No analytic tractability. ContTimeFin - 10 Hull and White

Black-Derman-Toy Black-Karasinski ContTimeFin - 10 Hull and White

Hull and White Similar to Ho and Lee but with mean reversion or an extension of Vasicek. All rates are normal, but long rates are less variable than short rates. Is analytic tractability. ContTimeFin - 10 Hull and White

Hull and White ContTimeFin - 10 Hull and White

Bond Prices in Hull and White Where ContTimeFin - 10 Hull and White

Option Prices in Hull and White A discount bond matures at s, a call option matures at T ContTimeFin - 10 Hull and White

Generalized Hull and White f(r) follows the same process as r in the HW model. When f(r) is log(r) the model is similar to Black-Karasinski model. Analytic solution is only when f(r)=r. ContTimeFin - 10 Hull and White

Options on coupon bearing bond In a one-factor model an option on a bond can be expressed as a sum of options on the discount bonds that comprise the coupon bearing bond. Let T be the bond’s maturity, s - option’s maturity. Suppose C=Pi - bond’s price. ContTimeFin - 10 Hull and White

Options on coupon bearing bond The first step is to find the critical r at time T for which C=X, where X is the strike price. Suppose this is r*. The correct strike price for each Pi is the value it has at time T when r=r*. Pi(r) is monotonic in r! ContTimeFin - 10 Hull and White

Example Suppose that in HW model a=0.1, =0.015. We wish to value a 3-month European option on a 15-month bond where there is a 12% semiannual coupon. Strike price is =100, bond principal =100. Assume that the yield curve is linear y(t) = 0.09 + 0.02 t ContTimeFin - 10 Hull and White

Example In this case Also ContTimeFin - 10 Hull and White

Example Thus Substituting into the equation for logA(t,T) ContTimeFin - 10 Hull and White

Example The bond price equals the strike price of 100 after 0.25 year when This can be solved, the solution is r = 0.0943. The option is a sum of two options on discount bonds. The first one is on a bond paying 6 at time 0.75 and strike 6x0.9926e-0.4877*0.0943=5.688. ContTimeFin - 10 Hull and White

Example This can be solved, the solution is r = 0.0943. The option is a sum of two options on discount bonds. The first one is on a bond paying 6 at time 0.75 and strike 6x0.9926e-0.4877*0.0943=5.688. The second is on a bond paying 106 at time 1.25 and strike 106x0.9733e-0.9516*0.0943=94.315. ContTimeFin - 10 Hull and White

Example The first option is worth 0.01 The second option is worth 0.41 The value of the put option on the bond is 0.01+0.41=0.42 ContTimeFin - 10 Hull and White

Interest Rates in Two Currencies Model each currency separately (by building a corresponding binomial tree). Combine them into a three-dimensional tree. Include correlations by changing probabilities. ContTimeFin - 10 Hull and White

Two Factor HW Model Where x = f(r) and the correlation between dz1 and dz2 is . ContTimeFin - 10 Hull and White

Discount Bond Prices When f(r) = r, discount bond prices are Where A(t,T), B(t,T), C(t,T) are given in HW paper in Journal of Derivatives, Winter 1994. ContTimeFin - 10 Hull and White

General HJM Model In addition of being functions of t, T and m, the si can depend on past and present term structures. But we must have: ContTimeFin - 10 Hull and White

General HJM Model Once volatilities for all instantaneous forward rates have been specified, their drifts can be calculated and the term structure has been defined. ContTimeFin - 10 Hull and White

One-factor HJM The model is not Markov in r. The behavior of r between times t and t+t depends on the whole history of the term structure prior to time t. ContTimeFin - 10 Hull and White

Specific Cases of One-factor HJM s(t,T) is constant: Ho and Lee s(t,T) = e-a(T-t): Hull and White ContTimeFin - 10 Hull and White

Cheyette Model s(t,T) = (r)e-a(T-t): Cheyette where There are two state variables r and Q. ContTimeFin - 10 Hull and White

Cheyette Model Discount bond prices in the Cheyette model are ContTimeFin - 10 Hull and White

Simualtions P(i,j) price at time it of discount bond maturing at time jt. F(i,j) price at time it of a forward contract lasting between jt and (j+1)t. v(i,j) volatility of P(i,j) s(i,j) standard deviation of F(i,j) m(i,j) drift of F(i,j)  random sample from N(0,1). ContTimeFin - 10 Hull and White

Modeling Bond Prices with One-Factor ContTimeFin - 10 Hull and White

Modeling Bond Prices withTwo-Factors ContTimeFin - 10 Hull and White

Modeling Forward Rates with One-Factor The Heath-Jarrow-Morton result shows that ContTimeFin - 10 Hull and White

Euler Scheme The order of convergence is 0.5 ContTimeFin - 10 Hull and White

Milstein Scheme The order of convergence is 1 ContTimeFin - 10 Hull and White

Trees up down ContTimeFin - 10 Hull and White

Trees  is big   is small ContTimeFin - 10 Hull and White

Financial Engineering