Valuing Callable and Putable Bonds on a Binomial Tree

Understand Valuing Callable and Putable Bonds

Key formulas to remember

Callable bond value

V(callable) = V(straight) − V(call option)

The investor is short the call, so the call value is subtracted. The callable bond value is never above the straight bond value.

Putable bond value

V(putable) = V(straight) + V(put option)

The investor is long the put, so the put value is added. The putable bond value is never below the straight bond value.

Option value from the tree

V(call) = V(straight) − V(callable); V(put) = V(putable) − V(straight)

Value both bonds on the same tree and take the difference. Use this form when the question gives you a tree.

Backward induction at a node

V = 0.5 × [(V_up + C) ÷ (1 + i)] + 0.5 × [(V_down + C) ÷ (1 + i)]

V_up and V_down are next-period node values after any exercise adjustment. C is the coupon paid at the next date. i is the one-period rate at this node. The 0.5 weights assume equal up and down probabilities.

Call exercise rule

Node value = min(computed value, call price)

Apply at each node on or after a date when the bond is callable. Use the value excluding the coupon paid at that date.

Put exercise rule

Node value = max(computed value, put price)

Apply at each node on or after a date when the bond is putable.

How to solve Valuing Callable and Putable Bonds questions

1. 1Pull out the data from the vignette: par value, coupon rate, maturity, the one-period rates at each node, and the dates and prices at which the option can be exercised.
2. 2Write the cash flow at maturity: par plus the final coupon. Place it on every final node.
3. 3Work backward one date at a time. At each node compute V = 0.5 × [(V_up + C) ÷ (1 + i)] + 0.5 × [(V_down + C) ÷ (1 + i)] using that node's rate i.
4. 4For the straight bond, do nothing more at the nodes. Keep going back to time 0 to get the straight bond value.
5. 5For a callable bond, redo the tree. At each node where the bond is callable, replace the value with the call price if the value is higher. For a putable bond, replace the value with the put price if the value is lower.
6. 6Carry the adjusted values back to time 0 to get the callable or putable bond value.
7. 7Take the difference from the straight bond value to get the option value. Callable: straight minus callable. Putable: putable minus straight.
8. 8Check the sign. The call value and put value must both be zero or positive.

Quickest way: Check the exercise nodes first

Common mistakes in Valuing Callable and Putable Bonds

• Applying the call price as a floor and the put price as a cap

  Students remember that the option 'sets a price' but not which direction it limits.

  Fix: The issuer calls when the bond is worth more than the call price, so the call price is a cap. The investor puts when the bond is worth less than the put price, so the put price is a floor. Callable uses min, putable uses max.

• Adding the call option value to the straight bond price

  Options usually have positive value, so students add them without asking who holds the option.

  Fix: The investor is short the call. Callable value = straight value − call value. Only the put is added.

• Forgetting to add the coupon when discounting from a node

  Students discount only the next node value and forget the coupon paid on that date.

  Fix: Always use (V_next + C) ÷ (1 + i). Check that the maturity nodes include par plus the final coupon.

• Applying the exercise rule after the coupon is added, or at the wrong date

  Students compare the value including the coupon with the call price, or apply the option at dates when it is not exercisable.

  Fix: Compare the value at the node, excluding the coupon paid that date, with the call or put price. Apply it only at the dates in the vignette.

• Using different trees for the straight bond and the optioned bond

  Students rebuild the tree or change rates when moving to the second bond.

  Fix: Use the same rates and the same tree for both. Only the exercise adjustment differs.

• Dropping the 0.5 weights or using the wrong rate at a node

  Students rush and discount both branches at the time-0 rate.

  Fix: Each node is discounted at its own rate. The two branches from a node are averaged with equal weights unless the question gives probabilities.

Worked examples