Callable Bond and OAS: Method

Callable Bond and OAS: Method

What it does

A callable bond lets the issuer repay the bond early at a stated call price on stated dates. The issuer will do so when rates have fallen and the bond can be replaced more cheaply, so the holder's payments depend on the path of interest rates and a single discount rate cannot value them. The tool builds a tree of possible future rates that agrees with today's yield curve, values the bond backwards through it with the issuer's choice at every call date, and reports the value of the bond with and without the call, the value of the call option, the usual yields, and effective duration and convexity. With a market price it also finds the option-adjusted spread (OAS).

The method

1. Discount factors. Coupons are paid m times a year (m = 1 or 2) on dates k/m, k = 1 … N. A flat yield y gives the discount factor d_k = (1 + y/m)^−k. Par yields entered at a few maturities are interpolated linearly to every coupon date and bootstrapped as in the Yield Curve and Forward Rates tool: d_k = (1 − (c_k/m) A_k−1) / (1 + c_k/m), where c_k is the par yield and A_k−1 the sum of the earlier factors.

2. Straight bond. With face value F and coupon C = F c/m per period, P_s = C A_N + F d_N.

3. The rate tree. The tree takes one step per coupon period. At step i there are i + 1 possible one-period rates, r(i, j) = a_i u^j for j = 0 … i up-moves, with u = exp(2σ√Δt), Δt = 1/m and σ the volatility as a decimal. Each rate is an annual rate compounded m times a year and applies from date i to date i + 1; an up-move and a down-move each have probability ½. This is the Black-Derman-Toy lognormal model with one constant volatility, in the form used by Kalotay, Williams and Fabozzi. The level a_i of each step is solved, one step after another, so that 1 paid at date i + 1 has the tree value d_i+1:

d_i+1 = Σ_j Q(i, j) / (1 + r(i, j)/m)

where Q(i, j) is the value today of 1 paid at node (i, j), carried forward by Q(i+1, j) = ½ Q(i, j)/(1 + r(i, j)/m) + ½ Q(i, j−1)/(1 + r(i, j−1)/m). The tree therefore reprices every zero-coupon bond, and with it any bond without an option.

4. Backward induction. At maturity V(N, j) = F. Earlier, the continuation value is

W(i, j) = (½ [V(i+1, j) + V(i+1, j+1)] + C) / (1 + (r(i, j) + s)/m)

with s = 0 unless a spread is being solved. On a call date V = min(W, K_i); on other dates V = W. Values are taken just after the coupon of their own date, so the holder of a called bond receives the coupon and the call price. The bond is callable on every coupon date from the first date of the schedule; each later row of the schedule changes the price from its date on. A putable bond uses the mirror rule V = max(W, K_i), the choice being the holder's.

5. Option value. Call: P_s − P_callable. Put: P_putable − P_s.

6. Option-adjusted spread. For a market price P, the OAS is the constant s for which the induction of step 4 returns V(0, 0) = P. The value falls as s rises, so s is found by Newton steps kept inside a bracket.

7. Effective duration and convexity. Every par yield is moved by h = 25 basis points up and down, the tree is fitted again with the same volatility (and the same s), and the bond revalued: D = (P_dn − P_up) / (2 P₀ h), convexity = (P_dn + P_up − 2 P₀) / (P₀ h²).

8. Yields. Yield to maturity, yield to the first call date and yield to worst (the lowest over maturity and every call date) are each the single rate that discounts one fixed set of payments to the price.

What it assumes

One source of uncertainty (the short rate) with a volatility that is the same at every date and proportional to the level of the rate; rates never turn negative. The issuer calls exactly when the continuation value exceeds the call price: no refinancing cost, no notice period, no delay. The entered curve is the curve for the issuer's bonds without a call, so credit risk enters only through that curve or through the spread. The bond is valued on a coupon date, the call can be exercised only on coupon dates, and steps are as long as coupon periods.

When it misleads

The option value is only as good as the volatility, which is an estimate and not an observable fact; the OAS inherits that. Two models, or one model with two volatilities, give two different OAS for the same bond and price, so OAS figures from different sources are not comparable without their assumptions. Annual or semi-annual steps are coarse: the value moves a little when the step is changed, and a bond callable at any time on notice is approximated by one callable on coupon dates. Issuers do not call the moment the model says so, which makes real callable bonds worth somewhat more than the model value. A constant-volatility lognormal tree cannot match the prices of interest-rate options of different maturities, and it needs positive forward rates. Effective duration describes a parallel move of the par curve only.

Worked example

The default case: face value 100, coupon 6 % paid twice a year, 10 years, callable at 100 from year 3, flat yield 5 %, volatility 15 %.

Discount factor of the first date: d₁ = 1/1.025 = 0.975610; A₂₀ = 15.589162, d₂₀ = 0.610271. Straight bond: P_s = 3 × 15.589162 + 100 × 0.610271 = 107.795.

Tree: u = exp(2 × 0.15 × √0.5) = 1.23631. Step 0: r = 5.000 %. Step 1: the lower rate 4.473 % and the upper rate 1.23631 × 4.473 = 5.530 % give ½ (1/1.022365 + 1/1.027650) / 1.025 = 0.951814 = d₂ = 1/1.025². Step 2: 4.002 %, 4.948 %, 6.118 %.

Induction: on the first call date (step 6, year 3) at the lowest rate, 2.573 %, the continuation value is 101.692, above the call price, so the value there is 100. Working back to today, the two values of step 1 are 102.940 and 98.703, and P_callable = (½ (98.703 + 102.940) + 3) / 1.025 = 101.289.

Call option: 107.795 − 101.289 = 6.505. Yield to maturity at 101.289: 5.828 %; yield to call: 5.528 %, which is also the yield to worst.

Curve moved by 25 basis points: 100.146 (up) and 102.357 (down). Effective duration (102.357 − 100.146) / (2 × 101.289 × 0.0025) = 4.37 years against 7.57 for the straight bond; effective convexity −118 against +71.

A calculator for education and analysis, not investment advice.