CFA Level II Exam · The Arbitrage-Free Valuation Framework

Valuing Option-Free, Callable and Putable Bonds with a Binomial Tree

Updated 7 October 2026 · Fact-checked

Valuing these bonds means working backward through an interest rate tree, discounting each node's expected value plus coupon at that node's rate. For a callable bond, cap the node value at the call price. For a putable bond, floor it at the put price. The embedded option's value is the difference from the straight bond.

Understand Valuing Option-Free, Callable and Putable Bonds

A straight (option-free) bond is worth the present value of its cash flows. In an arbitrage-free tree, you find that value by working backward from maturity. At each node you take the average of the two possible next-period values, add the coupon, and discount at that node's one-period rate.

An embedded option changes who controls the bond's future. A callable bond lets the issuer buy the bond back at a set call price. The issuer will do this when rates fall and the bond is worth more than the call price. So the investor has sold a call option to the issuer, and the bond is worth less than a straight bond.

A putable bond lets the investor sell the bond back at a set put price. The investor will do this when rates rise and the bond is worth less than the put price. So the investor owns a put option, and the bond is worth more than a straight bond.

In the tree, the option is exercised only at nodes on exercise dates. At each such node you compare the computed value of the bond with the exercise price. The issuer calls if the value is above the call price, so the value becomes the call price. The investor puts if the value is below the put price, so the value becomes the put price. Then you keep working backward with the adjusted value.

Volatility matters because options gain value when the rate paths spread out. A wider tree means more nodes where exercise pays off. Higher volatility raises the value of both the call and the put. It lowers the callable bond's value and raises the putable bond's value.

Key formulas to remember

Node value in the tree: V = [0.5 × (V_up + C) + 0.5 × (V_down + C)] ÷ (1 + r); V_up and V_down are next-period values at the up and down nodes. C is the coupon paid at that next date. r is the one-period rate at the current node. Probabilities are 0.5 each.
Callable bond value: V_callable = V_straight − V_call; The investor is short the call, so the call value is subtracted.
Putable bond value: V_putable = V_straight + V_put; The investor is long the put, so the put value is added.
Embedded option value from bond values: V_call = V_straight − V_callable; V_put = V_putable − V_straight; Build the straight-bond tree and the option-bond tree on the same rates, then subtract.
Exercise rule at a node (callable): Value used = min(computed value, call price); Apply on call dates, using the value before that date's coupon is added.
Exercise rule at a node (putable): Value used = max(computed value, put price); Apply on put dates, using the value before that date's coupon is added.
Effect of volatility: Higher volatility: V_call ↑, V_put ↑, V_callable ↓, V_putable ↑; The straight bond's value is roughly unaffected in a tree calibrated to the same curve.

How to solve Valuing Option-Free, Callable and Putable Bonds questions

Use this order for any item-set question on the binomial valuation of straight, callable or putable bonds.

1Read the vignette and list the bond terms: maturity, coupon, par, call or put price, and which dates the option can be exercised.
2Locate the interest rate tree in the exhibit. Note the rate at each node and confirm the time steps match the coupon dates.
3Start at the last date. Each node's value is the final par plus the final coupon, discounted at that node's rate.
4Move back one date at a time. At each node compute 0.5 × (V_up + C) + 0.5 × (V_down + C), then divide by (1 + r) at that node.
5On an exercise date, compare the computed node value with the exercise price. Cap at the call price for a callable bond. Floor at the put price for a putable bond. Use the adjusted value in the next step back.
6Keep working back to time 0 to get the bond value. Do the same without the option to get the straight bond value.
7Find the option value by subtraction: V_call = V_straight − V_callable, or V_put = V_putable − V_straight.
8For volatility questions, decide the direction from the option's value, then translate it to the bond: callable falls when volatility rises, putable rises.

Quickest way: Two-pass check with the straight bond

When to use it: Use when the question asks for an option value or a comparison, and you have little time.

Value the straight bond first. You need it anyway for the option value.
Scan the nodes on the exercise date. For a callable bond, check whether any node value exceeds the call price. For a putable bond, check whether any node value is below the put price.
If no node is affected, the option is worthless in the tree and the option bond equals the straight bond.
If nodes are affected, replace only those node values with the exercise price and redo the backward steps from that date. Do not rebuild the whole tree.
Check the sign: the callable bond must be at or below the straight bond, and the putable bond at or above it. If not, you used the wrong rule.

Common mistakes in Valuing Option-Free, Callable and Putable Bonds

Adding the call option value to the straight bond value for a callable bond.
Students remember that options have positive value and forget who holds the option.
Fix: The issuer holds the call, so the investor's bond is worth less. Callable = straight − call. Putable = straight + put.
Applying the call price cap to the wrong value, such as after adding that date's coupon.
The coupon and the exercise price sit at the same date, so the order gets mixed up.
Fix: Compute the node value, apply min or max with the exercise price, then add the coupon when you step back to the previous node. Follow the convention in the vignette if it states one.
Using the callable bond's tree values when valuing the straight bond, or the reverse.
Students edit the tree in place and lose the original values.
Fix: Keep two columns: one for the straight bond and one for the option bond. Only the option bond gets capped or floored.
Stating that higher volatility raises the value of a callable bond.
Students link volatility with option value and stop before the next step.
Fix: Higher volatility raises the call's value. The callable bond is straight minus call, so it falls.
Calling a call or put exercised at a node where it does not pay off.
Students call whenever rates fall, without comparing the value with the call price.
Fix: Always compare the node's computed value with the exercise price. Exercise only if the value is above the call price, or below the put price.
Forgetting the probabilities are 0.5 and averaging only values, not the coupons.
The formula is applied from memory, and the coupon is dropped from one branch.
Fix: Add the coupon to both up and down values before averaging, then discount once.

Worked examples

Example 1

A 2-year, 5% annual-coupon bond has par 100. The one-year rate today is 3.00%. One year from now the one-year rate is 4.00% at the up node and 2.00% at the down node, each with probability 0.5. The issuer can call the bond at 100 at the end of year 1, just after the year-1 coupon is paid. Q1: What is the straight bond value? Q2: What is the callable bond value? Q3: What is the value of the embedded call?

Show the solution

Year-1 node values before the year-1 coupon: up = 105 ÷ 1.04 = 100.9615; down = 105 ÷ 1.02 = 102.9412.
Straight bond: add the year-1 coupon of 5 to each: 105.9615 and 107.9412. Average = (105.9615 + 107.9412) ÷ 2 = 106.9514.
Discount at 3.00%: 106.9514 ÷ 1.03 = 103.84.
Callable bond: both node values (100.9615 and 102.9412) exceed the call price of 100, so the issuer calls at both nodes. Both values become 100.
Add the coupon: 105 at each node. Average = 105. Discount: 105 ÷ 1.03 = 101.94.
Call value = 103.84 − 101.94 = 1.89 (using unrounded values, 1.8945).

Answer: Q1: 103.84. Q2: 101.94. Q3: 1.89.

Example 2

Use the same tree and bond as the previous example, but now the bond is putable at 102 at the end of year 1, just after the year-1 coupon is paid. The straight bond value is 103.84. Q1: What is the putable bond value? Q2: What is the value of the embedded put? Q3: If volatility rises, what happens to the putable bond's value, other things equal?

Show the solution

Node values before the year-1 coupon: up = 100.9615; down = 102.9412.
Up node: 100.9615 is below the put price of 102, so the investor puts. Value becomes 102.
Down node: 102.9412 is above 102, so no exercise. Value stays 102.9412.
Add the coupon of 5: 107 and 107.9412. Average = (107 + 107.9412) ÷ 2 = 107.4706.
Discount at 3.00%: 107.4706 ÷ 1.03 = 104.34.
Put value = 104.34 − 103.84 = 0.50 (unrounded 0.5042).
Higher volatility raises the put's value. The putable bond is straight plus put, so its value rises.

Answer: Q1: 104.34. Q2: 0.50. Q3: The putable bond's value rises.

Exam tips

Always check the exhibit for the exercise dates and prices. Many wrong answers come from applying the cap or floor at the wrong date.
Use the sign check: callable ≤ straight ≤ putable. If your answer breaks this order, recheck your work.
For qualitative questions on volatility, answer in two steps: the option's value, then the bond's value. Do not skip to the bond.
Carry at least four decimals through the tree and round only at the end, because the option value is a small difference of two large numbers.
If a vignette gives the callable bond value and the option value, you can recover the straight bond value by adding them. This saves building the tree.

Valuing Option-Free, Callable and Putable Bonds in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IBond Price Behavior: Callable, Putable and Option-Free

Valuing Option-Free, Callable and Putable Bonds: frequently asked questions

How do I value a callable bond with a binomial tree?

Work backward from maturity, averaging the next values plus coupon and discounting at each node's rate. On each call date, replace any node value above the call price with the call price. Continue back to time 0 to get the callable bond's value.

Why is a callable bond worth less than a straight bond?

The issuer holds the right to call the bond when rates fall, which caps the bond's upside. The investor has effectively sold a call option. So the callable bond equals the straight bond minus the call option value.

How does volatility affect callable and putable bond values?

Higher volatility raises the value of both the call and put options. That lowers the value of a callable bond, because you subtract a bigger call value. It raises the value of a putable bond, because you add a bigger put value.

How do I find the value of the embedded option?

Value the option-free bond and the bond with the option on the same tree. For a call, subtract the callable bond value from the straight bond value. For a put, subtract the straight bond value from the putable bond value.