Binomial Trees, Part 2

Fixed Income 8 min readModule 2 of 3Save

What you'll learn

LOS 24.d
Describe the process of calibrating a binomial interest rate tree to match a specific term structure.
LOS 24.f
Compare pricing using the zero coupon yield curve with pricing using an arbitrage free binomial lattice.
LOS 24.g
Describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixed income instrument given its cash flows along each path.
LOS 24.h
Describe a Monte Carlo forward rate simulation and its application.

The construction of a binomial interest rate tree is a tedious process. In practice, the interest rate tree is usually generated using specialized computer software. The underlying process conforms to three rules:

The interest rate tree should generate arbitrage free values for the benchmark security. This means that the value of bonds produced by the interest rate tree must be equal to their market price, which excludes arbitrage opportunities.

This requirement is very important because without it, the model will not properly price more complex callable and putable securities, which is the intended purpose of the model.

As stated earlier, adjacent forward rates (for the same period) are two standard deviations apart (calculated as $e^{2 σ}$ ). Hence, knowing one of the forward rates (out of many) for a particular nodal period allows us to compute the other forward rates for that period in the tree.
The middle forward rate (or mid point in case of even number of rates) in a period is approximately equal to the implied (from the benchmark spot rate curve) one period forward rate for that period.

Worked example: Binomial interest rate tree

Xi Nguyen, CFA, has collected the following information on the par rate curve, spot rates, and forward rates. Nguyen had asked a colleague, Alok Nath, to generate a binomial interest rate tree consistent with this data and an assumed volatility of 20%. Nath completed a partial interest rate tree shown as follows:

Binomial Tree With σ = 20% (One Year Forward Rates)

Calculate the forward rate indicated by A.
Estimate the forward rate indicated by C.
Estimate forward rates B and D.

Solution

Forward rate $i_{1, L}$ is indicated by A and is related to forward rate $i_{1, U}$ given as 5.7883%. $i_{1, L}$ = $i_{1, U}$ $e^{- 2 σ}$ = (0.057883) $e^{- (2 \times 0.20)}$ = 0.0388 or 3.88%
Forward rate C is the middle rate for Period 3 and hence the best estimate for that rate is the one year forward rate in two years f(2,1).

Using the spot rates, we can bootstrap the forward rate:

$(1 + S_{3})^{3} = (1 + S_{2})^{2} (1 + f (2, 1)) \to (1.05069)^{3} = (1.0402)^{2} (1 + f (2, 1)) \to f (2, 1) = 7.199%$

The forward rates B and D are related to C as follows (note that C = $i_{2, LU}$ ):

D = $i_{2, LL}$ = $i_{2, LU}$ $e^{- 2 σ}$ = (0.07199) $e^{- 0.40}$ = 0.0483 or 4.83%

B = $i_{2, U U}$ = $i_{2, LU}$ $e^{+ 2 σ}$ = (0.07199) $e^{+ 0.40}$ = 0.1074 or 10.74%

We have already discussed arbitrage free valuation of benchmark option free bonds using the zero coupon yield curve (also known as the spot rate curve). Each known future cash flow is discounted at the underlying spot rate (also known as the zero coupon yield).

Worked example: Valuation of option free bond

Samuel Favre is interested in valuing a three year, 3% annual pay Treasury bond. He has compiled the following information on Treasury spot rates:

Treasury Spot Rate Curve

Compute the value of the $100 face value option free bond.

Solution

For bonds with embedded options, the future cash flows are uncertain as they depend on whether the embedded option will be in the money (and hence exercised). Because the value of the option depends on uncertain future interest rates, the underlying cash flows are also dependent on the same future interest rates. Hence, to value bonds with embedded options, we have to allow for rates to fluctuate.

One way to accomplish this is to use the binomial interest rate tree.

Worked example: Valuation of an option free bond using binomial tree

Samuel Favre is interested in valuing the same three year, 3% annual pay Treasury bond. The spot rate curve is as before, but this time Favre wants to use a binomial interest rate tree with the following rates:

One Period Forward Rate in Year

Compute the value of the $100 par option free bond.

Solution

Note that the underlying interest rate tree in this example was calibrated to generate arbitrage free values consistent with the benchmark spot rate curve and hence produced the same value for the option free bond as in the earlier example.

Another mathematically identical approach to valuation using the backward induction method in a binomial tree is the pathwise valuation approach. For a binomial interest rate tree with n periods, there will be $2^{(n - 1)}$ unique paths. For example, for a three period tree, there will be $2^{2}$ or four paths comprising one known spot rate and varying combinations of two unknown forward rates.

If we use the label of S for the one period spot rate and U and L for upper and lower outcomes, respectively, for forward rates in our binomial framework, the four paths would be SUU, SUL, SLU, and SLL.

Worked example: Valuation of option free bond using pathwise valuation

Samuel Favre wants to value the same three year, 3% annual pay Treasury bond. The interest rate tree (shown next) is the same as before but this time, Favre wants to use a pathwise valuation approach.

One Period Forward Rate in Year

Compute the value of the $100 par option free bond.

Solution

For a three year bond, there are four potential interest rate paths. The value of the bond for each path is computed as the sum of the present values of each cash flow discounted at its respective path specified rate. Pathwise valuation discounts cash flows one year at a time using one year forward rates (similar to backward induction) rather than spot rates.

For example, the value of the bond in Path 1 is computed as:

Path Dependency

Prepayments on underlying residential mortgages affect the cash flows of a mortgage backed security. Prepayment risk is similar to call risk in a callable bond. However, unlike call risk, prepayment risk is affected not only by the level of interest rate at a particular point in time, but also by the path rates took to get there.

Consider a mortgage pool that was formed when rates were 6%, then interest rates dropped to 4%, rose to 6%, and then dropped again to 4%. Many homeowners will have refinanced when interest rates dipped the first time. On the second occurrence of 4% interest rates, most homeowners in the pool who were able to refinance would have already taken advantage of the opportunity, leading to lower prepayments than would be observed had 4% interest rates not occurred previously.

An important assumption of the binomial valuation process is that the value of the cash flows at a given point in time is independent of the path that interest rates followed up to that point. In other words, cash flows are not path dependent; cash flows at any node do not depend on the path rates took to get to that node. Because of path dependency of cash flows of mortgage backed securities, the binomial tree backward induction process cannot be used to value such securities.

We instead use the Monte Carlo simulation method to value mortgage backed securities.

A Monte Carlo forward rate simulation involves randomly generating a large number of interest rate paths, using a model that incorporates a volatility assumption and an assumed probability distribution. A key feature of the Monte Carlo method is that the underlying cash flows can be path dependent.

As with pathwise valuation discussed earlier, the value of the bond is the average of values from the various paths. The simulated paths should be calibrated so benchmark interest rate paths value benchmark securities at their market price (i.e., arbitrage free valuation). The calibration process entails adding (subtracting) a constant to all rates when the value obtained from the simulated paths is too high (too low) relative to market prices.

This calibration process results in a drift adjusted model.

A Monte Carlo simulation may impose upper and lower bounds on interest rates as part of the model generating the simulated paths. These bounds are based on the notion of mean reversion; rates tend to rise when they are too low and fall when they are too high.

Worked example: Valuation of option free bond using Monte Carlo simulation

Samuel Favre is interested in valuing the same three year, 3% annual pay Treasury bond as discussed before. Favre wants to use Monte Carlo simulation and has generated the following rate paths:

Monte Carlo Simulation (Drift Adjusted)

Solution

Valuation using the simulated interest rate paths is shown as follows:

Where, for example:

Note that the interest rates in the example were calibrated to ensure that the valuation obtained was consistent with market prices (i.e., arbitrage free) and hence the value obtained is the same as before.

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