1. © 2011 Neil D. Pearson A Simulation Implementation of the Hull- White Model Neil D. Pearson

2. Hull-White Model In the Hull-White model, under the risk-neutral probability the “short rate” follows the process dr(t) = (  (t) – ar(t))dt +  dB Q (t), or equivalently where B Q is a Brownian motion under the risk-neutral probability. Note that one models the movement of the short-rate under the risk-neutral probability, because this is what matters for pricing derivatives and other securities. The model does not specify the movement of the interest rate under the original probability. © 2011 Neil D. Pearson

3. Hull-White Model Hull-White model: Some properties: the short rate tends to revert to the level  (t)/a parameter a determines the strength of mean reversion  is the local volatility typically  (t) is chosen to match the term structure of zero- coupon bond prices (or equivalently, yields) Often a and  are chosen to provide an approximate match to the prices (implied volatilities) of interest rate caps/floors or swaptions. © 2011 Neil D. Pearson

4. Partial Differential Equation If the interest rate follows the process dr(t) = (  (t) – ar(t))dt +  dB Q (t), then the value V of a security satisfies the pde Boundary conditions are determined by the security. Note that, as usual, the differential equation has the interpretation that drift under the risk-neutral probability (computed using Ito’s formula) equals the riskless return rV

5. © 2011 Neil D. Pearson Partial Differential Equation for a Zero- Coupon Bond If the security is a zero-coupon bond paying $1 at maturity T with current price P(t, T), then the pde is with terminal boundary condition P(r, T,T) = 1. The solution can be characterized as

6. Calibration to Bond Prices It can be shown that the function  can be calculated from the initial (time 0) term structure* Here F(0,t) is the instantaneous forward rate, observed at time 0, for a forward loan from time t to the next instant. If we let f(0,t, t+dt) be the forward rate, observed at time 0, for a loan from t to t + dt, then Instantaneous forward rates are related to the bond price through © 2011 Neil D. Pearson *This is a difficult argument. If you are interested, see Hull, J. and A. White, “Pricing Interest-Rate Derivative Securities,” Review of Financial Studies, 1990, Wilmott IQF, sections 17.2 and 17.3

7. Calibration to Bond Prices The term  F(0,t)/  t is the slope of the initial forward curve. The term (  2 /2a)(1  e  2at ) is a “convexity adjustment,” and is usually small. For the time being, let’s assume  = 0 so that the convexity adjustment is zero. In this case the drift of the short rate r is And we conclude that r follows the forward curve. If  > 0 then:

8. Aside: Why Is There a “Convexity Adjustment”? Consider the probabilistic characterization of the bond price We also have the characterization in terms of forward rates: If  = 0 then r(t) = F(0,t). But if  > 0 then we cannot have E[r(t) | r(0)] = F(0,t), because due to the convexity of the function e  x we have © 2011 Neil D. Pearson

9. Bond Prices in the Hull-White Model Bond prices in the Hull-White model are given by where and If we know bond prices at time t, then we can compute all zero-coupon rates, forward rates, swap rates, and any quantity (e.g., the payoff of an option on a bond) that depends on bond prices or interest rates. © 2011 Neil D. Pearson

10. Remark: Key Limitation of 1-Factor Models Bond prices in the Hull-White model are given by where and All bond prices and yields, regardless of maturity, depend only on the short-rate r, which driven by a single Brownian motion. An implication of this is that changes in the prices/yields of all bonds are (locally) perfectly correlated. © 2011 Neil D. Pearson

11. Implementing the Hull-White Model Using Simulation Given the interest rate process dr(t) = (  (t) – ar(t))dt +  dB Q (t) and the function we can implement the Hull-White model using simulation. If we consider a discrete time-step  t (e.g.,  t = 1/12 year), then we can approximate the interest rate process as r(t+  t)  r(t) = (  (t) – ar(t))  t +  t    t+  t where  t+  t ~ N(0,1) is a standard Normal r.v. © 2011 Neil D. Pearson

12. Implementing the Hull-White Model Using Simulation There is a slightly better approximatio: r(t+  t)  r(t) = (  (t) – ar(t))  t +  t    t+  t Due to the mean reversion, the variance of the short-rate process is not var[r(t+  t)  r(t)] =  2  t. Instead, it is which for small  t is approximately equal to but smaller than  2  t. This leads us to use © 2011 Neil D. Pearson

13. Implementing the Hull-White Model: Estimate  We will simulate: where An open issue is how to estimate the forward curve slope  F(0,t)/  t. Should we estimate it as a forward difference  F(0,t)/  t  (F(0,t+  t)  F(0,t))/  t? Or a backward difference (F(0,t)  F(0,t  t))/  t? Or a central difference? Your first thought might be that it doesn’t matter; but, if we want to use a long time-step, say  t = 1/12, then this issue can be important © 2011 Neil D. Pearson

14. “Reprice” Bonds We want the simulation to reprice the bonds, that is if we simulate N interest rate paths and let r n denote an interest rate on the nth path, we want We want this to be true for all parameters a and . All values of a and  include the choice a,  = 0, and we will focus on this choice. © 2011 Neil D. Pearson

15. “Reprice” Bonds: a,  = 0 If a,  = 0, we have that is the interest rate r follows the forward curve. With a discrete time-step dt, we need to reinterpret the interest rates. Now r(t) means a rate that is quoted at t, and covers the period from t to t + dt, and r(t+dt) is quoted at t+dt, and covers the period from t+dt to t+2dt. Similarly, F(0,t) is the forward rate, quoted at 0, covering the period from t to t + dt, and F(0,t+dt) covers the period from t+dt to t+2dt. © 2011 Neil D. Pearson

16. “Reprice” Bonds: a,  = 0 The continuous-time expression suggests which in turn suggests using a forward difference which is what we want. © 2011 Neil D. Pearson

17. “Reprice” Bonds: a,  = 0 To recap, we want to estimate the derivative of the forward curve using a forward difference over the same period that we are simulating the interest rates: If we do this then the simulated interest rates will be consistent with the forward curve. In the limit dt  0 the choice of how to estimate the derivative doesn’t matter, but we want to be able to work with relatively large dt. For large dt and rapidly changing yield curves, this issue can be important.* *It is not unusual for the short end of the yield curve to display rapid changes in rates. © 2011 Neil D. Pearson

18. Implementing the Hull-White Model Using Simulation To summarize, we will simulate interest rates using where  (t) is given by © 2011 Neil D. Pearson

19. Spreadsheet Implementation Spreadsheet PRDC_Simulation.xlsm implements 1-factor Hull- White models of the USD and JPY term structures (it also simulates the USD/JPY exchange rate). We focus on the USD model Column J of the tab InitialTermStructure contains zero-coupon rates, estimated from swap rates. Columns K, L, and M contain forward rates, the forward derivative  F(0,t)/  t  (F(0,t+  t)  F(0,t))/  t, and  (t). Column L of the tab Simulation_Crystal ” simulates the short- rate process. The simulated sample paths (up to 5,000) are stored in the tab r_USD (and also r_JPY and FX). These simulated paths can be used to estimate the values of securities. © 2011 Neil D. Pearson