**All of us at Kamakura would like to thank our readers for the very high level of interest in this blog. This version, updated April 19, 2012, incorporates some important revisions in previously published work by Robert A. Jarrow. We would like to add a special thanks to Professor Jarrow for his contributions to this revision.**

In the first three blogs in this series, we provided worked examples of how to use the yield curve simulation framework of Heath, Jarrow and Morton using two different assumptions about the volatility of forward rates and one or two independent risk factors. The first volatility assumption was that volatility was dependent on the maturity of the forward rate and nothing else. The second volatility assumption was that the volatility of forward rates was dependent on both the level of rates and the maturity of the forward rates being modeled. The first two models were one factor models, implying that random rate shifts are either all positive, all negative, or zero. Our third example postulated that the change in 1 year spot rates, along with an unspecified second risk factor, where in fact driving interest rate movements. In this blog we generalize the model to include three risk factors in order to increase further the realism of the simulated yield curve. Consistent with the research of Dickler, Jarrow and van Deventer, we explain why even three risk factors understate the number of risk factors driving the U.S. Treasury yield curve.

The author wishes to thank Robert A. Jarrow for his encouragement and advice on this series of worked examples of the HJM approach. What follows is based heavily on Prof. Jarrow’s *Modeling Fixed Income Securities and Interest Rate Options* (second edition, 2002), particularly chapters 4, 6, 8, 9 and 15, along with selected enhancements to the formulas in chapter 15 by Professor Jarrow.

The first three blogs in this series implemented the work of Heath Jarrow and Morton (1990, 1990, and 1992) with one and two driving risk factors and two assumptions about the volatility of interest rates:

van Deventer, Donald R. “Heath Jarrow and Morton Example One: Modeling Interest Rates with One Factor and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 2, 2012.

van Deventer, Donald R. “Heath Jarrow and Morton Example Two: Modeling Interest Rates with One Factor and Rate and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 6, 2012.

van Deventer, Donald R. “Heath Jarrow and Morton Example Three: Modeling Interest Rates with Two Factors and Rate and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 13, 2012.

The volatility assumptions used so far include (a) interest rate volatility that is a function of the maturity of the forward rate and (b) interest rate volatility that depends both on the maturity of the forward rate and the level of the one year spot rate of interest. Both assumptions are much more general assumptions that those used by Ho and Lee [1986] (constant volatility) or by Vasicek [1977] and Hull and White [1990] (declining volatility).

In this blog series, we use data from the Federal Reserve statistical release H15 published on April 1, 2011 for yields prevailing on March 31, 2011. U.S. Treasury yield curve data was smoothed using Kamakura Risk Manager version 7.3 to create zero coupon bonds via the maximum smoothness forward rate technique of Adams and van Deventer as documented in these two recent blog issues:

van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 10: Maximum Smoothness Forward Rates and Related Yields versus Nelson-Siegel,” Kamakura blog, www.kamakuraco.com, January 5, 2010. Redistributed on www.riskcenter.com on January 7, 2010.

van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 12: Smoothing with Bond Prices as Inputs,” Kamakura blog, www.kamakuraco.com, January 20, 2010. Redistributed on www.riskcenter.com

The smoothed U.S. Treasury yield curve and the implied forward yield curves monthly for ten years looks like this:

The continuous forward rate curve and zero coupon bond yield curve that prevailed as of the close of business on March 31, 2011 were as follows:

**Probability of Yield Curve Twists in the U.S. Treasury Market**

In our March 13, 2012 blog, we explained that one factor models imply no twists in the yield curve. Random yield shocks drive yields either all up or all down together. Yield curve twists, where some rates rise and others fall, are assumed never to occur. Yet during 12,386 days of movements in U.S. Treasury forward rates, yield curve twists occurred on 94.3% of the observations. We now ask a simple question. To what extent did the two factor model succeed explaining the correlations between changes in forward rates? We answer that question by analyzing the correlation between the residuals of the regression we used to determine the impact of our first risk factor, the annual change in the one year spot U.S. Treasury rate, on changes in forward rates at various maturities:

If the two factor model is correct, all forward rate movements would be explained by our two factors: the 1 year change in the one year spot rate and the unnamed second factor. What would the correlations remaining after use of the first risk factor show if our hypothesis was correct? The correlation of the residuals at any two maturities would show a 100% correlation, with all residuals either moving up together or down together after eliminating the common impact of changes in the 1 year spot rate. Unfortunately the correlation of the residuals shows that the 2 factor model is not rich enough to achieve that objective:

The correlations are positive and far from 100%, which leads to an immediate conclusion. We need at least one more risk factor to realistic mimic real yield curve movements. We incorporate a third factor in this blog for that reason.

**Objectives of the Example and Key Input Data**

Following Jarrow (2002), we make the same modeling assumptions for our worked example as in the first blog in this series:

- Zero coupon bond prices for the U.S. Treasury curve on March 31, 2011 are the basic inputs.
- Interest rate volatility assumptions are based on the Dickler, Jarrow and van Deventer blog series on daily U.S. Treasury yields and forward rates from 1962 to 2011. In this blog, we retain the volatility assumptions used in the second blog but expand the number of random risk factors driving interest rates to three factors
- The modeling period is 4 equal length periods of one year each.
- The HJM implementation is that of a “bushy tree” which we describe below

We can use either Monte Carlo simulation or the “bushy tree” approach of the first three blogs in this series to implement HJM. In the first two blogs in this series, the bushy tree consisted solely of “up shifts” and “down shifts” because we were modeling as if only one random factor were driving interest rates. With two factors, the bushy tree has “up shifts,” “mid shifts,” and “down shifts” at each node in the tree. With three risk factors, there are 4 branches at each node of the tree. For simplicity and consistency with an n-factor HJM implementation, we abandon the terms up and down and instead label the shifts “shift 1,” “shift 2,” “shift 3” and “shift 4.” The bushy tree starts with 1 branch at time zero and moves to 4 at time 1, 16 at time 2, and 64 at time 3. We split the tree into its upper and lower halves for better visibility:

At each of the points in time on the lattice (time 0, 1, 2, 3 and 4) there are sets of zero coupon bond prices and forward rates. At time 0, there is one set of data. At time one, there are four sets of data, the “shift 1 set,” the “shift 2 set, ” the “shift 3 set” and the “shift 4 set.” At time two, there are sixteen sets of data, and at time three there are 64=4^{3} sets of data.

As shown in previous blogs, volatilities of the U.S. Treasury one year spot rate and 1 year forward rates with maturities in years 2, 3, …,10 depend dramatically on the starting level of the one year U.S. Treasury spot rate. The graph below shows the standard deviation in the annual changes in the one year U.S. Treasury forward rates maturing in years 2, 3 and 4 as a function of the starting U.S. Treasury 1 year spot rate. Volatilities are reported for spot rates between 0.2498% and 0.50%, 0.50% and 0.75%, 0.75% and 1.00%, and then in single percent increments up to 10%. Spot rates over 10% made up the final grouping:

Forward rate volatility rises in a smooth but complex way as the level of interest rates rises.

As before, we use the following table to measure total volatility, measured as the standard deviation of 1 year changes in continuously compounded spot and forward rates from 1963 to 2011. There were no observations at the time this data was compiled for starting 1 year U.S. Treasury yields below 0.002499%, so we have set those volatilities by assumption. With the passage of time, these assumptions can be replaced with facts or with data from other low rate counties like Japan.

We use this table later to divide total volatility between three uncorrelated risk factors.

We will use the zero coupon bond prices prevailing on March 31, 2011 as our other inputs:

**Risk Factor 1:
Annual Changes in the 1 Year U.S. Treasury Spot Rate**

In the first two worked examples of the Heath Jarrow and Morton approach, the nature of the single factor shocking 1 year spot and forward rates was not specified. As in the March 13 blog, we postulate that the first of the three factors driving changes in forward rates is the change in the 1 year spot rate of interest. For each of the 1 year U.S. Treasury forward rates f_{k}(t), we run the regression

where the change in continuously compounded yields is measured over annual intervals from 1963 to 2011. We add an additional assumption that the second risk factor is the 1 year change in the 1 year forward rate maturing in year 10. To be consistent with the HJM assumptions that the individual risk factors are uncorrelated, we estimate the incremental explanatory power of this factor by adding the 1 year change in the 1 year forward rate maturing in year 10 as an explanatory variable in a regression where the dependent variable is the appropriate series of residuals from the first set of regressions, which we label the “Regression 1” set.

Because of the nature of the linear regression of changes in 1 year forward rates on the first risk factor (changes in the 1 year spot U.S. Treasury rate), we know that risk factor 1 and the residuals e_{k}(i) are uncorrelated. The regression above will pick up the residual influence of the 1 year changes in the 1 year forward rate maturing in year 10.

Our hypothesis that the 1 year change in the 1 year forward rate maturing in year 10 was important was right on the mark. The coefficients, shown below, are very statistically significant with t statistics ranging from 68 to 175. This graph shows that the impact of the change of the 1 year forward rate maturing in year 10 generally increases for rates maturing closest to year 10:

The third risk factor is “everything else” not explained by the changes in the 1 year spot rate or the changes in the 1 year forward rate maturing in year 10. Because of the sequential nature of the regression 1 set and the regression 2 set (which means correlation among the risk factors is zero), total volatility for the forward rate maturing in T-t=k years can be divided as follows between the three risk factors:

We also know that the risk contribution of the first risk factor, the change in the spot 1 year U.S. treasury rate, is proportional to the regression coefficient α_{k} of changes in forward rate maturing in year k on changes in the 1 year spot rate. Because regression set 2 is done on the residuals of regression set 1, the risk contribution of risk factor 2 (the change in the 1 year forward rate maturing in year 10) is also proportional to the relevant regression coefficient. Note again that, by construction, risk factor 2 is not correlated with risk factor 1. We denote the total volatility of the 1 year U.S. Treasury spot rate by the subscript “1,total” and the total volatility of the 1 year forward rate maturing in year 10 by “10, total.” Then the contribution of risk factors 1 and 2 to the volatility of the 1 year forward rate maturing in year k is

This allows us to solve for the volatility of risk factor 3 using this equation:

If ever the quantity in brackets is negative, σ_{k,3} is set to zero and σ_{k,2} is set to the value which makes the quantity in brackets exactly zero. Because the total volatility for each forward rate varies by the level of the 1 year spot U.S. Treasury spot rate, so will the values of σ_{k,1}, σ_{k,2}, and σ_{k,3} . The regression coefficients used in this process are summarized in this table:

Using the equations above, the look up table for risk factor 1 (changes in the 1 year spot rate) volatility is given here:

The look up table for risk factor 2 (changes in the 1 year forward rate maturing in year 10) are given in this table:

The look up table for the general “all other” risk factor 3 is as follows:

When we graph the relative contributions of each risk factor to the volatility of the forward rate maturing in year 2, we see an interesting pattern as the level of the 1 year spot Treasury rate rises:

The contribution of risk factor 1 rises steadily as the level of interest rates rises. The impact of risk factor 2 rises more gradually as the level interest rates rises. The “all other risk factor,” risk factor 3, is volatile and spikes a number of times, due in part to the spike in total volatility when the 1 year spot U.S. Treasury rate is between 75 and 100 basis points. We will see below that this has an impact on our simulations and that this is a value for total volatility that one may want to override as a matter of professional judgment.

**Key Implications and Notation of the HJM Approach**

The Heath Jarrow and Morton conclusions are very complex to derive but their implications are very straightforward. Once the zero coupon bond prices and volatility assumptions are made, the mean of the distribution of forward rates (in a Monte Carlo simulation) and the structure of a bushy tree are completely determined by the constraint that there be no arbitrage in the economy. As in prior blogs, we show in this example that the zero coupon bond valuations are 100% consistent with the inputs. We now introduce the same notation used in the first blog in this series:

Note that this is a slightly different definition of K than we used in the first three blogs in this series.

We will also see the rare appearance of a trigonometric function in finance, one found in common spreadsheet software:

Note that the current times that will be relevant in building a bushy tree of zero coupon bond prices are current times t=0, 1, 2, and 3. We’ll be interested in maturity dates T=2, 3, and 4. We know that at time zero, there are 4 zero coupon bonds outstanding. At time 1, only the bonds maturing at T = 2, 3, and 4 will remain outstanding. At time 2, only the bonds maturing at times T = 3 and 4 will remain, and at time 3 only the bond maturing at time 4 will remain. For each of the boxes below, we need to fill in the relevant bushy tree (one for each of the four zero coupon bonds) with each up shift and down shift of the zero coupon bond price as we step forward one more period (by Δ = 1) on the tree. In the interests of saving space, we’ll again arrange the tree to look like a table by stretching the bushy tree as follows. Note that S-1 means “shift 1,” S-2 means “shift 2,” and so on. The “stretched” bushy tree is rearranged from the graph above as shown below:

A completely populated zero coupon bond price tree would then be summarized like this; prices shown are for the zero coupon bond price maturing at time T=4 at times 0, 1, 2, and 3:

In order to populate the trees with zero coupon bond prices and forward rates, there is one more piece of information which we need to supply.

**Pseudo Probabilities**

In Chapter 7 of Jarrow (2002), Prof. Jarrow shows that a necessary and sufficient condition for no arbitrage is that, at every node in the tree, the one period return on a zero coupon bond neither dominates nor is dominated by a one period investment in the risk free rate. As explained in the three previous blogs in this series, if the computed probabilities of every type of shift are between 0 and 1 everywhere on the bushy tree, then the tree is arbitrage free. Without loss of generality, we set the probability of shift 1 to 1/8, the probability of shift 2 to 1/8, the probability of shift 3 to ¼, and the probability of shift 4 to ½. The no arbitrage restrictions that stem from this set of pseudo probabilities are given below.

Prof. Jarrow goes on to explain on page 129 that “risk neutral valuation” is computed by “taking the expected cash flow, using the pseudo probabilities, and discounting at the spot rate of interest.” He adds “this is called risk neutral valuation because it is the value that the random cash flow ‘x’ would have in an economy populated by risk-neutral investors, having the pseudo probabilities as their beliefs.”

We now demonstrate how to construct the bushy tree and use it for risk-neutral valuation.

**The Formula for Zero Coupon Bond Price Shifts with Three Risk Factors**

Like the two risk factor case, with three risk factors it is convenient to calculate the forward rates first and then derive zero coupon bond prices from them. We do this using slightly modified versions (developed with Prof. Jarrow) of equations 15.40 and 15.42 in Jarrow (2002, page 297). We use this equation for the shift in forward rates:

(1)

where when T=t+Δ, the expression μ becomes

We use this expression to evaluate formula (1) when T>t+Δ:

The values for the pseudo probabilities and Index (1), Index (2), and Index (3) are set as follows:

**Building the Bushy Tree for Zero Coupon Bonds Maturing at Time T=2**

We now populate the bushy tree for the 2 year zero coupon bond. We calculate each element of equation (1). Note that, even as the bushy tree grows more complex, accuracy of valuation to 10 decimal places in common spreadsheet software is maintained. When t=0 and T=2, we know Δ=1 and

P(0,2,s_{t}) = 0.9841101497

The one period risk free rate is again

R(0,s_{t})=1/P(0,1,s_{t})=1/0.997005786=1.003003206

The 1 period spot rate for U.S. Treasuries is r(0, s_{t}) =R(0,s_{t})-1=0.3003206%. At this level of the spot rate for U.S. Treasuries, volatilities for risk factors 1, 2 and 3 are selected from data group 3 in the look up table above. The volatilities for risk factor 1 for the 1 year forward rates maturing in years 2, 3 and 4 are 0.000492746, 0.000313424 , and 0.00016918. The volatilities for risk factor 2 for the 1 year forward rates maturing in years 2, 3 and 4 are 0.001146096, 0.001545761, and 0.00185028. Finally, the volatilities for risk factor 3 are also taken from the appropriate risk factor table for data group 3: 0.003633719, 0.00651351, and 0.008436331.

The scaled sum of sigmas K(1,t,T,s_{t}) for t=0 and T=2 becomes

and therefore K(1,0,T,s_{t}) =(√1)( 0.000492746) = 0.000492746. Similarly, remembering the minus sign, K(2,0,T,s_{t})= -0.001146096 and K(3,0,T, s_{t}) = -0.003633719 . We also can calculate that μ(t,t+Δ) = 0.0000073730.

Using formula 1 with these inputs and the fact that the variable Index(1)=-1, Index(2)=1, and Index(3)=1 for shift 1 gives us the forward returns for shift 1 of 1.021652722. For shifts 2, 3, and 4, the values of the three index variables change. Using these new index values, the forward returns in shifts 2, 3, and 4 are 1.006910523, 1.010972304, and 1.013610654. Note that the forward rates (as opposed to forward returns) are equal to these values minus 1. From the forward returns, we calculate the zero coupon bond prices:

P(1,2,s_{t} = shift 1) = 0.9788061814 =1/F(1,2,s_{t} = shift 1)

P(1,2,s_{t} = shift 2) = 0.9931369048 =1/F(1,2,s_{t} = shift 2)

P(1,2,s_{t} = shift 3) = 0.9891467808 =1/F(1,2,s_{t} = shift 3)

P(1,2,s_{t} = shift 4) = 0.9865721079 =1/F(1,2,s_{t} = shift 4)

We have fully populated the bushy tree for the zero coupon bond maturing at T=2), since all of the 4 shifts at time t=2 result in a riskless pay-off of the zero coupon bond at its face value of 1.

**Building the Bushy Tree for Zero Coupon Bonds Maturing at Time T=3**

For the zero coupon bonds and 1 period forward returns ( = 1 + forward rate) maturing at time T=3, we use the same volatilities listed above for risk factors 1, 2 and 3 to calculate

K(1,0,3,s_{t}) = 0.000806170

Remembering the minus sign for risk factors 2 and 3 gives us

K(2,0,3,s_{t}) = -0.002691857

K(3,0,3,s_{t}) = -0.010147229

μ(t,T) = 0.0000479070

The resulting forward returns for shifts 1, 2, 3, and 4 are 1.038505795, 1.011797959, 1.020593023, and 1.023467874. Zero coupon bond prices are calculated from the 1 period forward returns, so

P(1,3,s_{t} = Shift 1 Shift 1) =1/[F(1,2,s_{t} = Shift 1)F(1,3,s_{t} = Shift 1)]

The zero coupon bond prices for maturity in time T=3 for the four shifts are 0.9425139335, 0.9815565410, 0.9691882649, and 0.9639502449. We have now populated the second column of the zero coupon bond price table for the zero coupon bond maturing at T=3.

**Building the Bushy Tree for Zero Coupon Bonds Maturing at Time T = 4**

We now populate the bushy tree for the zero coupon bond maturing at T=4. Using the same volatilities as before, we find that

K(1,0,4,s_{t}) = 0.000975351

Again, remembering the minus sign, we calculate

K(2,0,4,s_{t}) = -0.004542136

K(3,0,4,s_{t}) = -0.018583560

μ(t,T) = 0.0001272605

Using formula (1) with the correct values for Index(1), Index(2) and Index (3) leads to the following forward returns for Shifts 1, 2, 3, and 4: 1.0536351507, 1.0186731103, 1.0305990034, and 1.0336489801. The zero coupon bond price for Shift 1 is calculated as follows:

P(1,4,s_{t} = Shift 1) =1/[F(1,2,s_{t} = Shift 1)F(1,3,s_{t} = Shift 1) F(1,4,s_{t} = Shift 1)]

A similar calculation gives us the zero coupon bond prices for all 4 shifts 1, 2, 3, and 4: 0.8945353929, 0.9635638078, 0.9404125772, and 0.9325702085.

We have now populated the column labeled 1 for T=1 for the zero coupon bond price maturing at time T=4:

Now we move to the third column, which displays the outcome of the T=4 zero coupon bond price after 16 scenarios: Shift 1 followed by Shifts 1, 2, 3, or 4; Shift 2 followed by Shifts 1, 2, 3 or 4 and so on. We calculate P(2,4,s_{t} = shift 1), P(2,4,s_{t}=shift 2), P(2,4,s_{t}=shift 3) and P(2,4,s_{t} = shift 4) after the initial state of “shift 2” as follows. After a shift = shift 2, the prevailing 1 period zero coupon bond price at time t=1 is 0.9931369048, which implies a 1 period (1 year) U.S. Treasury spot rate of 0.0069105228 (i.e. 69 basis points). This is data group 4. When t=1, T=4, and Δ=1 then the volatilities for the two remaining 1 period forward rates that are relevant are taken from the lookup table for data group 4 for risk factor 1: 0.0006425743, and 0.0004087264. For risk factor 2, the volatilities for the two remaining 1 period forward rates are also chosen from data group 4: 0.0021277322, and 0.0028697122. Finally, the volatilities for risk factor 3 are also taken from data group 4: 0.0040702120 and 0.0054595415.

At time 1 in the shift 2 state, the zero coupon bond prices for maturities at T=2, 3 and 4 are 0.9931369048, 0.9815565410, and 0.9635638078. We make the intermediate calculations as above for the zero coupon bond maturing at T=4:

K(1,1,4,s_{t}) = 0.0010513007

As before, remembering the minus sign,

K(2,1,4,s_{t}) = -0.0049974444

K(3,1,4,s_{t}) = -0.0095297535

μ(t,T) = 0.0000474532

We can calculate the values of the 1 period forward return maturing at time T=4 in shifts 1, 2, 3, and 4 as follows: 1.0336715790, 1.0113427849, 1.0141808568, and 1.0191379141. Similarly, using the appropriate intermediate calculations, we can calculate the forward returns for maturity at T=3: 1.0224958583, 1.0059835405, 1.0081207950, and 1.0124591923. Since

P(2,4,s_{t} = Shift 2 Shift 1) =1/[F(1,3,s_{t} = Shift 2 Shift 1) F(1,4,s_{t} = Shift 2 Shift 1)]

the zero coupon bond prices for maturity at T=4 after an initial shift 2 for subsequent shifts 1, 2, 3, and 4 are as follows:

0.9461410097

0.9829031898

0.9780746840

0.9691466829

We have correctly populated the fifth, sixth, seventh, and eighth rows of column 3 (t=2) of the bushy tree above for the zero coupon bond maturing at T=4 (note values have been rounded for display only). The remaining calculations are left to the reader.

If we combine all of these tables, we can create a consolidated table of the term structure of zero coupon bond prices in each scenario as in examples one, two and three.

At any point in time t, the continuously compounded yield to maturity at time T can be calculated as y(T-t)=-ln[P(t,T)]/(T-t). Note that we have one incidence of negative rates (in state 46, shaded) on this bushy tree. This can be corrected by overriding the spike in volatility we discussed above in the data group for the 1 year spot U.S. Treasury rate levels between 0.75% and 1.00%. Note that yield curve curve shifts are much more complex than in the three prior examples using one and two risk factors:

We can graph yield curve movements as shown below at time t=1. We plot the original spot Treasury yield curve and then shifted yield curves for shifts 1, 2, 3, and 4. These shifts are relative to the 1 period forward rates prevailing at time zero for maturity at times T=2, 3 and 4. Because these 1 period forward rates were much higher than yields as of time t=0, all four shifts produce yields higher than time zero yields with the exception of one yield for maturity at T=4.

When we add 16 yield curves prevailing at time t=3 and 64 “single point” yield curves prevailing at time t=4, two things are very clear. First, yield curve movements in a three factor model are much more complex and much more realistic than what we saw in the one-factor and two-factor examples. Second, in the low yield environment prevailing as of March 31, 2011, no arbitrage yield curve simulation shows “there is nowhere to go but up” from a yield curve perspective, with just a few exceptions.

Finally, we can display the 1 year U.S. Treasury spot rates and the associated term structure of 1 year forward rates in each scenario.

**Valuation in the Heath Jarrow and Morton Framework**

Prof. Jarrow, in a quote above, described valuation as the expected value of cash flows using the risk neutral probabilities. Note that column 1 denotes the riskless 1 period interest rate in each scenario. For the state number 84 (three consecutive shift 4’s), cash flows at time T=4 would be discounted by the one year spot rates at time t=0, by the one year spot rate at time t=1 in state 4 (“shift 4”), by the one year spot rate in state 20 (“shift 4 shift 4”) at time t=2, and by the one year spot rate at time t=3 in state 84 (“shift 4 shift 4 shift 4”). The discount factor is

Discount Factor (0,4, Shift 4-Shift 4-Shift 4)

=1/(1.003003)(1.013519)(1.031104)(1.049009)

These discount factors are displayed here for each potential cash flow date:

When taking expected values, we can calculate the probability of each scenario coming about since the probabilities of shifts 1, 2, 3, and 4 are 1/8, 1/8, ¼ and 1/2:

It is convenient to calculate the “probability weighted discount factors” for use in calculating the expected present value of cash flows. The values shown are simply the probability of occurrence of that scenario times the corresponding discount factor:

We now use the HJM bushy trees we have generated to value representative securities.

**Valuation of a Zero Coupon Bond Maturing at Time T=4**

A riskless zero coupon bond pays $1 in each of the 64 nodes of the bushy tree that prevail at time T=4. Its present value is simply the sum of the probability weighted discount factors in the last column on the right hand side. Their sum is 0.9308550992, which is the value we should get in a no-arbitrage economy, the value observable in the market and used as an input to create the tree.

**Valuation of a Coupon-Bearing Bond Paying Annual Interest**

Next we value a bond with no credit risk that pays $3 in interest at every scenario at times T=1, 2, 3, and 4 plus principal of 100 at time T=4. The valuation is calculated by multiplying each cash flow by the matching probability weighted discount factor, to get a value of 104.7070997370 . It will surprise many that this is the same value that we arrived at in examples one, two, and three, even though the volatilities used and number of risk factors used are different. The values are the same because, by construction, our valuations for the zero coupon bond prices at time zero for maturities at T = 1, 2, 3, and 4 continue to match the inputs. Multiplying these zero coupon bond prices times 3, 3, 3, and 103 also leads to a value of 104.7070997370 as it should.

**Valuation of a Digital Option on the 1 Year U.S. Treasury Rate**

Now we value a digital option that pays $1 at time T=3 if (at that time) the one year U.S. Treasury rate (for maturity at T=4) is over 8%. If we look at the table of the term structure of one year spot rates over time, this happens in only one scenario, in row 4:

The evolution of the spot rate can be displayed graphically as follows:

The cash flow payoff in the one relevant scenario can be input in the table above and multiplied by the appropriate probability weighted discount factor to find that this option has a value of only $ 0.007286. That is because the probability of row 12 occurring is only 0.7813% and the payoff is discounted by the simulated spot rates prior to time T=3.

**Replication of HJM Example 4 in Excel**

Kamakura Risk Manager and Kamakura Risk Information Services clients may request a copy of the Excel spreadsheet supporting this blog after signing a supplemental confidentiality agreement. Please request a copy of the spreadsheet from your Kamakura representative or from info@kamakuraco.com.

**Conclusion**

As we have noted in prior blogs, the Dickler, Jarrow and van Deventer studies of movements in U.S. Treasury yields and forward rates from 1962 to 2011 confirm that 5-10 factors are needed to accurately model interest rate movements. Popular one factor models (Ho and Lee, Vasicek, Hull and White, Black Derman and Toy) cannot replicate the actual movements in yields that have occurred. The interest rate volatility assumptions in these models (constant, constant proportion, declining, etc.) are also inconsistent with observed volatility.

In order to handle a large number of driving factors and complex interest rate volatility structures, the Heath Jarrow and Morton framework is necessary. This blog, the fourth in a series, shows how to simulate zero coupon bond prices, forward rates and zero coupon bond yields in an HJM framework with three risk factors and rate-dependent and maturity-dependent interest rate volatility. The results show a rich twist in simulated yield curves and a pull of rates upward from a very low rate environment. Monte Carlo simulation, an alternative to the bushy tree framework, can be done in a fully consistent way. The technique can be generalized to any user-specified number of risk factors, a long-standing feature of the Kamakura Risk Manager enterprise risk management system.

In the next blog in this series, we discuss the implications of this type of analysis for the Basel-specified Internal Capital Adequacy Assessment Process (“ICAAP”) for interest rate risk.

**Other References**

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates,” Kamakura Corporation memorandum, September 13, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Forward Rates,” Kamakura blog, www.kamakuraco.com, September 14, 2011.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Zero Coupon Bond Yields,” Kamakura Corporation memorandum, September 26, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Zero Coupon Bond Yields,” Kamakura blog, www.kamakuraco.com, September 26, 2011.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Par Coupon Bond Yields,” Kamakura Corporation memorandum, October 5, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Par Coupon Bond Yields,” Kamakura blog, www.kamakuraco.com, October 6, 2011.

Heath, David, Robert A. Jarrow and Andrew Morton, “Contingent Claims Valuation with a Random Evolution of Interest Rates,” *The Review of Futures Markets*, 9 (1), 1990.

Heath, David, Robert A. Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation,” *Journal of Financial and Quantitative Analysis*, December 1990.

Heath, David, Robert A. Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation,” *Econometrica*, 60(1), January 1992.

Jarrow, Robert A. *Modeling Fixed Income Securities and Interest Rate Options*, second edition, Stanford Economics and Finance, Stanford University Press, Stanford, California, 2002.

Jarrow, Robert A. and Stuart Turnbull. *Derivative Securities*, 1996, Southwestern Publishing Co., second edition, fall 2000.

van Deventer, Donald R. “Pitfalls in Asset and Liability Management: One Factor Term Structure Models,” Kamakura blog, www.kamakuraco.com, November 7, 2011. Reprinted in *Bank Asset and Liability Management Newsletter*, January, 2012.

van Deventer, Donald R. “Pitfalls in Asset and Liability Management: One Factor Term Structure Models and the Libor-Swap Curve,” Kamakura blog, www.kamakuraco.com, November 23, 2011. Reprinted in *Bank Asset and Liability Management Newsletter*, February, 2012.

Slattery, Mark and Donald R. van Deventer, “Model Risk in Mortgage Servicing Rights,” Kamakura blog, www.kamakuraco.com, December 5, 2011.

van Deventer, Donald R., Kenji Imai, and Mark Mesler, *Advanced Financial Risk Management*, John Wiley & Sons, 2004. Translated into modern Chinese and published by China Renmin University Press, Beijing, 2007. Second edition forthcoming in 2012.

Donald R. van Deventer

Kamakura Corporation

Honolulu, March 28, 2012

© Copyright 2012 by Donald R. van Deventer. All rights reserved.