Determining Yield Volatilities and the Basics of Constructing Binomial Trees in the Black-Derman-Toy (BDT) Interest Rate Model: Practice Problems and Solutions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov. This is Section 79 of the Study Guide. See an index of all sections by following the link in this paragraph.

This section will focus on determining yield volatilities in the BDT model. Some of the following information and formulas were already given in Section 77.

We can let P[h, T, r(h)] be the time-h price of a zero-coupon bond maturing at T. Here, r(h) is is the time-t short-term interest rate. Then the yield of the bond is

y[h, T, r(h)] = P[h, T, r(h)]^-1/(T-h) - 1.

The yield volatility is Yield volatility = 0.5ln(y[h, T, r_u]/y[h, T, r_d]) =

0.5ln([P[h, T, r_u]^-1/(T-h) - 1]/[P[h, T, r_d]^-1/(T-h) - 1])

It may be possible for one to be asked on the exam to determine the Black-Derman-Toy binomial interest rate tree, given data on bonds with various times to maturity - including their yields to maturity, current bond prices, and volatility in year 1 (or time period t = h, where h is one period in the BDT binomial model). This section will begin introduce students to this process for finding r₀, r_u, and r_d.

If P_h is the price of an h-year bond, where h is one time period in the BDT model, then

P_h = 1/(1 + r₀) and thus

r₀ = 1/P_h - 1

If P_2h is the price of an 2h-year bond, then R_h = r_d and σ_h meet the following conditions:

P_2h = 0.5P_h(1/(1 + R_hexp[2σ_h]) + 1/(1 + R_h))

[To memorize this formula, think of it as P_2h = 0.5P_h(1/(1 + r_u) + 1/(1 + r_d)]

r₀ = 0.5ln[R_hexp[2σ_h]/R_h]

That is,

r₀/0.5 = ln[R_hexp[2σ_h]/R_h]

exp[r₀/0.5] = R_hexp[2σ_h]/R_h

exp[r₀/0.5] = exp[2σ_h]

r₀/0.5 = 2σ_h

r₀ = σ_h

Now we can make the substitution r₀ = σ_h into the expression for P_2h:

P_2h = 0.5P_h(1/(1 + R_hexp[2r₀]) + 1/(1 + R_h))

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 24, pp. 804-805.