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.
The first two blogs in this series implemented the work of Heath Jarrow and Morton (1990, 1990, and 1992) using one random risk factor to drive interest rate movements under two different volatility assumptions.
van Deventer, Donald R. “Heath Jarrow and Morton Example One:
van Deventer, Donald R. “Heath Jarrow and Morton Example Two:
Modeling Interest Rates with One Factor and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 2, 2012.
Modeling Interest Rates with One Factor and Rate and Maturity-Dependent Volatility,” Kamakura blog, www.kamakuraco.com, March 6, 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 assumption that that used by Ho and Lee  (constant volatility) or by Vasicek  and Hull and White  (declining volatility).
In this blog series, we use data from the Federal Reserve statistical release H15 published on April 1, 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
The table below is taken from the blog entry
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.
It shows that, for 12,386 days of movements in U.S. Treasury forward rates, yield curve twists occurred on 94.3% of the observations.
In order to incorporate yield curve twists in interest rate simulations under the Heath Jarrow and Morton framework, we introduce a second risk factor driving interest rates in this blog. The single factor yield curve models of Ho and Lee, Vasicek, Hull and White, Black Derman and Toy, and Black and Karasinski are unable to model yield curve twists and therefore they do not provide a sufficient basis for interest rate risk management.
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 two 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
The HJM framework is usually implemented using Monte Carlo simulation or a “bushy tree” approach where a lattice of interest rates and forward rates is constructed. This lattice, in the general case, does not “recombine” like the popular “binomial” or “trinomial” trees used to replicate Black-Scholes options valuation or simple 1 factor term structure models. In general, the bushy tree does not recombine because the interest rate volatility assumptions imply a path-dependent interest rate model, not one that is path independent like the simplest one factor term structure model implementations. 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 was driving interest rates. In this blog, with two random factors, we move to a bushy tree that has “up shifts,” “mid shifts,” and “down shifts” at each node in the tree. Please consider these terms as labels only, since forward rates and zero coupon bond prices, of course, move in opposite directions:
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 three sets of data, the “up set,” “the mid set” and the “down set.” At time two, there are nine sets of data (up up, up mid, up down, mid up, mid mid, mid down, down up, down mid, and down down), and at time three there are 27=33 sets of data.
The 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.
The results also confirm that the level of interest rates does impact the level of forward rate volatility but in a more complex way than the lognormal interest rate movements assumed by Fischer Black with co-authors Derman and Toy (1990), and Karasinski (1991).
The table below shows the actual volatilities for the 1 year changes in continuously compounded 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 two uncorrelated risk factors.
We will use the zero coupon bond prices prevailing on March 31, 2011 as our other inputs:
Introducing a Second Risk Factor Driving Interest Rates
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. In this blog, we take a cue from the popular academic models of the term structure of interest rates and postulate that one of the two factors driving changes in forward rates is the change in the short run rate of interest. In the context of our annual model, the “short rate” is the one year U.S. Treasury yield. For each of the 1 year U.S. Treasury forward rates fk(t), we run the regression
where the change in continuously compounded yields is measured over annual intervals from 1963 to 2011. The coefficients of the regressions for the 1 year forward rates maturing in years 2, 3, …, 10 are as follows:
Graphically, it is easy to see that the historical response of forward rates to the spot rate of interest is neither constant (as assumed by Ho and Lee ) nor declining, as assumed by Vasicek  and Hull and White . Indeed, the response of the 1 year forwards maturing in years 8, 9, and 10 year to changes in the 1 year spot rate is larger than the response of the 1 year forward maturing in year 7:
We use these regression coefficients to separate the total volatility (shown in the table above) of each forward rate between two risk factors. The first risk factor is changes in the one year spot U.S. Treasury. The second factor represents all other sources of shocks to forward rates.
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 risk factor 2 are uncorrelated. Therefore total volatility for the forward rate maturing in T-t=k years can be divided as follows between the two 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 αk of changes in forward rate maturing in year k on changes in the 1 year spot rate. We denote the total volatility of the 1 year U.S. Treasury spot rate by the subscript “1,total”:
This allows us to solve for the volatility of risk factor 2 using this equation:
Because the total volatility for each forward rate varies by the level of the 1 year U.S. Treasury spot rate, so will the values of σk,1 and σk,2. In one case (data group 6 for the forward rate maturing at time T=3), we set the sigma for the second risk factor to zero and ascribed all of the total volatility to risk factor one because the calculated contribution of risk factor 1 was greater than the total volatility. In the worked example, we will select the volatilities from look up tables for the appropriate risk factor volatility. 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 the general “all other” risk factor 2 is as follows:
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 constraints that there be no arbitrage in the economy. Modelers who are unaware of this insight would choose means of the distributions for forward rates such that valuation would provide different prices for the zero coupon bonds on March 31, 2011 than those used as input. This would create the appearance of an arbitrage opportunity, but it is in fact a big error that calls into question the validity of the calculation, as it should.
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 two 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:
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:
The mapping of the sequence of up and down states is shown here, consistent with the stretched tree above:
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.
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 two previous blogs in this series, If the computed probabilities of an up shift, a “mid shift” and a down 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 an up shift to ¼, the probability of a mid shift to ¼, and the probability of a down shift 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 Two Risk Factors
In the first two blogs in this series, we used equation 15.17 in Jarrow (2002, page 286) to calculate the no arbitrage shifts in zero coupon bond prices. Alternatively, when there is one risk factor, we could have used equation 15.19 in Jarrow (2002, page 287) to shift forward rates and derive zero coupon bond prices from the forward rates.
Now that we have two risk factors, it is convenient to calculate the forward rates first. We do this using equations 15.32, 15.39a, and 15.39b in Jarrow (2002, pages 293 and 296). We use this equation for the shift in forward rates:
The values for the pseudo probabilities and Index(1) and Index (2) 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). When t=0 and T=2, we know Δ=1 and
P(0,2,st) = 0.98411015.
The one period risk free rate is again
The 1 period spot rate for U.S. Treasuries is r(0, st) =R(0,st)-1=0.3003206%. At this level of the spot rate for U.S. Treasuries, volatilities for risk factor 1 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 0.003810177, 0.006694414, and 0.008636852.
and therefore K(1,0,T,st) =(√1)( 0.000492746) = 0.000492746. Similarly, K(2,0,T, st)= 0.003810177. We also can calculate that [μ(t,t+Δ)Δ]Δ = 0.00000738.
Using formula 1 with these inputs and the fact that the variable Index(1)=-1 and Index(2)=-1 for an up shift gives us the forward returns for an up shift, mid shift and down shift as follows: 1.007170561, 1.018083343, and 1.013610665. From these we calculate the zero coupon bond prices:
P(1,2,st = up) = 0.992880 =1/F(1,2,st = up)
For a mid shift, we set Index(1)=-1 and Index(2)=+1 and calculate
P(1,2,st = mid) = 0.982238=1/F(1,2,st = mid)
For a down shift we set Index(1) = 1 and Index(2)= 0 and recalculate formula 1 to get
P(1,2,st = down) = 0.986572 =1/F(1,2,st = down)
We have fully populated the bushy tree for the zero coupon bond maturing at T=2 (note values have been rounded to six decimal places above for display only), since all of the up, mid and down states 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 and 2 to calculate
K(1,0,3,st) = 0.00080617
K(2,0,3,st) = 0.010504592
[μ(t,T)Δ]Δ = 0.00004816
The resulting forward returns for an up shift, mid shift and down shift are 1.013189027, 1.032556197, and 1.023468132. Zero coupon bond prices are calculated from the 1 period forward returns, so
P(1,3,st = up) =1/[F(1,2,st = up)F(1,3,st = up)]
The zero coupon bond prices for an up shift, mid shift, and down shift are 0.979956, 0.951268, and 0.963950. To eight decimal places, we have 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 for both risk factors 1 and 2, we find that
K(1,0,4,st) = 0.000975351
K(2,0,4,st) = 0.019141444
[μ(t,T)Δ]Δ = 0.00012830
Using formula (1) with the correct values for Index(1) and Index (2) leads to the following forward returns for an up shift, mid shift and down shift: 1.020756042, 1.045998862, and 1.033650059. The zero coupon bond price is calculated as follows:
P(1,4,st = up) =1/[F(1,2,st = up)F(1,3,st = up) F(1,4,st = up)]
This gives us the three zero coupon bond prices of the column labeled 1 in this table for up, mid and down shifts: 0.960029, 0.909435, and 0.932569.
Now we move to the third column, which displays the outcome of the T=4 zero coupon bond price after 9 scenarios: up-up, up-mid, up-down, mid-up, mid-mid, mid-down, down-up, down-mid, and down-down. We calculate P(2,4,st = up), P(2,4,st=mid) and P(2,4,st = down) after the initial “down” state as follows. 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 6 for risk factor 1: 0.007871303 , and 0.006312739. For risk factor 2, the volatilities for the two remaining 1 period forward rates are also chosen from data group 6: 0.003086931 , and 0. The zero value was described above-the implied volatility for risk factor 1 was greater than measured total volatility for the forward rate maturing at T=3 in data group 6, so the risk factor 1 volatility was set to total volatility and risk factor 2 volatility was set to zero.
At time 1 in the down state, the zero coupon bond prices for maturities at T=2, 3 and 4 are 0.986572, 0.963950, and 0.932569. We make the intermediate calculations as above for the zero coupon bond maturing at T=4:
K(1,1,4,st) = 0.014184042
K(2,1,4,st) = 0.003086931
[μ(t,T)Δ]Δ = 0.00006964
We can calculate the values of the 1 period forward return maturing at time T=4 in the up state, mid state, and down state as follows: 1.027216983, 1.027216983, and 1.040268304. Similarly, using the appropriate intermediate calculations, we can calculate the forward returns for maturity at T=3: 1.011056564, 1.01992291, and 1.031592858. Since
P(2,4,st = down up) =1/[F(1,3,st = down up) F(1,4,st = down up)]
the zero coupon bond prices for maturity at T=4 in the down up, down mid, and down down states are as follows:
We have correctly populated the seventh, eighth and ninth 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 to six decimal places for display only). The remaining calculations are left to the reader.
If we combine all of these tables, we can create a table of the term structure of zero coupon bond prices in each scenario as in examples one and two. The shading highlights two nodes of the bushy tree where values are identical because of the occurrence of σ2 = 0 at one point on the bushy tree:
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 no negative rates on this bushy tree and that yield curve shifts are much more complex than in the two prior examples using one risk factor:
We can graph yield curve movements as shown below at time t=1. We plot yield curves for the up, mid and down shifts. These shifts are relative to the 1 period forward rates prevailing at time zero for maturity at time T=2 and T=3. Because these 1 period forward rates were much higher than yields as of time t=0, all three shifts produce yields higher than time zero yields.
When we add 9 yield curves prevailing at time t=3 and 27 “single point” yield curves prevailing at time t=4, two things are very clear. First, yield curve movements in a two factor model are much more complex and much more realistic than what we saw in the two one-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.
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 scenario number 39 (three consecutive down shifts in zero coupon bond prices), 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 scenario 3 (“down”), by the one year spot rate in scenario 12 (“down down”) at time t=2, and by the one year spot rate at time t=3 in scenario 39 (“down down down”). The discount factor is
Discount Factor (0,4, down down down) =1/(1.003003)(1.013611)(1.031593)(1.050231)
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 an up shift, mid shift, and down shift are ¼, 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:
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 8 nodes of the bushy tree that prevail at time T=4:
When we multiply this vector of 1s times the probability weighted discount factors in the time T=4 column in the previous table and add them, we get a zero coupon bond price of 0.93085510, 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 104.70709974. It will surprise many that this is the same value that we arrived at in examples one and two, 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 104.70709974 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 4%. If we look at the table of the term structure of one year spot rates over time, this happens at the seven shaded scenarios out of 27 possibilities at time t=3.
The evolution of the spot rate can be displayed graphically as follows:
The cash flow payoffs in the 7 relevant scenarios can be input in the table below and multiplied by the probability weighted discount factors to find that this option has a value of 0.29701554:
Replication of HJM Example 3 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 email@example.com.
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 third in a series, shows how to simulate zero coupon bond prices, forward rates and zero coupon bond yields in an HJM framework with two 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.
In the next blog in this series, we introduce a third risk factor to further advance the realism of the model.
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Donald R. van Deventer
Honolulu, March 13, 2012