2. The Arbitrage-Free Valuation Framework

Question 1

Explain the two types of arbitrage opportunities.

Answer 1

There are two types of arbitrage opportunities:

• Value Additivity: when the value of whole differs from the sum of the values of parts
• Dominance: when one asset trades at a lower price than another asset with identical characteristics

Question 2

Explain the Binomial Interest Rate Tree Framework.

Answer 2

While we can value option-free bonds with a simple spot rate curve, for bonds with embedded options, changes in future rates will affect the probability of the options being exercised and the underlying future cash flows. Thus, a model that allows both rates and the underlying cash flows to vary should be used to value bonds with embedded options. One such model is the binomial interest rate framework.

The binomial interest rate tree framework assumes that interest rates have an equal probability of taking one of two possible values in the next period (hence the term binomial).

• RATES

Consider the node on the right side where the interest rate i(2,LU) appears. This is the rate that will occur if the initial rate, i(0), follows the lower path from node 0 to node 1 to become i(1,L), then follows the upper of the two possible paths to node 2, where it takes on the value i(2,LU). At the risk of stating the obvious, the upper path from given node leads to a higher rate than the lower path.
In fact:

i(2,LU) = i(2,LL) * exp(2,sigma)

exp = 2,7183
sigma = standar deviation of interest rates (i.e. the interest rate volatility used in the model)

Note that an upward move followed by a downward move, or a down-then-up move, produces the same result: i(2,LU) = i(2,UL)

The interest rates at each node in this interest rate tree are one-period forward rates corresponding to the nodal period. Each forward rate is related to the other forward rates in the same nodal period.

Adjacent forward rates are two standard deviations apart. For the first period, there are two forward rates and hence:

i(1,U) = i(1,L) * exp(2,sigma) 
I(1,L) = i(1,U)* -exp(2,sigma)

Beyond the first nodal period, non-adjacent forward rates are a multiple of the two standard deviations (depending on how separated the forward rates are). For example:

i(2,UU) = i(2,LL)*exp(4,sigma)

The binomial interest rate tree framework is a lognormal random walk model with two desirable properties: a) higher volatility at higher rates b) non-negative interest rates

• If you have i(1,U) and i(1,L), i(2,LU) 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)
• VALUING AN OPTION-FREE BOND WITH THE BINOMIAL MODEL

V(1,U) = 1/2 * [V(2,UU)/i(1,U) + V(2,UL)/i(1,U)]

V(1,L) = 1/2 * [V(2,LU)/i(1,L) + V(2,LL)/i(1,L)]

V0 = 1/2* [V(1,U)/i(0) + V(1,L)/i(0)]

Question 3

Describe the process of calibrating binomial interest rate tree to match a specific term structure.

Answer 3

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:

1) 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 purtpose of the model.
2) Adjacent forward rates (for the same period) are two standard deviations apart (calculated as exp(2*sigma). 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.
3) 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.

Question 4

Describe ‘Pathwise Valuation’ in a binomial interest rate framework.

Answer 4

Another mathematically identical approach to valiation 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, the four paths would be: SUU, SUL, SLU and SLL.

Value = average(value of each patch)

Question 5

If the binomial lattice is correctly calibrated, it should give the same value for an option-free bond as USING THE PAR CURVE used to calibrate the tree.

True or false?

Answer 5

True.

Question 6

Backward Induction Method as opposed to the Pathwise Valuation Method produce the same value? If not, explain the expected results.

Answer 6

The two methods are identical and will always give the same result.