FRM Exam Part II · Arbitrage Pricing with Term Structure Models

Option-Adjusted Spread (OAS) on a Binomial Tree

Updated 11 October 2026 · Fact-checked

Option-adjusted spread (OAS) is the constant spread added to every short rate on an interest rate tree so that the model price of a bond equals its market price. To solve it, discount back through the tree with the spread, apply any call or put rule at each node, then adjust the spread until the prices match.

Understand Option-Adjusted Spread (OAS)

A bond price from a no-arbitrage tree uses risk-free rates only. The market price of a corporate or mortgage bond is usually different. The gap reflects credit risk, liquidity and other compensation. OAS measures that compensation after the value of any embedded option has been taken out.

Here is how it works. You build a tree calibrated to the risk-free curve. You add one constant number, the OAS, to every rate at every node. You then discount the bond's cash flows back through the tree at those higher rates. At each node you apply the option rule: the issuer calls if the bond value would exceed the call price, and the holder puts if it would fall below the put price. The OAS is the spread that makes the resulting model price equal the market price.

The Z-spread is the constant spread over the spot curve that prices the bond's cash flows as if they were fixed. It ignores the option and uses one path. So for an option-free bond, OAS and Z-spread are about the same. For a callable bond, the issuer owns the option, so the holder needs extra yield. The Z-spread is therefore larger than the OAS. The difference is the option cost: option cost = Z-spread − OAS. For a putable bond the holder owns the option, so the OAS is larger than the Z-spread.

Interpretation: OAS is the spread you earn for credit and liquidity risk once option risk is removed. A higher OAS at the same risk means the bond looks cheaper. OAS depends on the model, mainly the volatility input. Higher volatility raises the value of the call option the issuer holds. For a given market price, that lowers the OAS of a callable bond.

Key formulas to remember

OAS definition: Model price (tree rates + OAS) = Market price; OAS is the single constant spread, added at every node, that makes this hold.
Node value with spread: V(node) = [0.5 × V(up) + 0.5 × V(down) + coupon] ÷ (1 + r(node) + OAS); Use the probabilities the tree was built with (0.5 each in the usual exam tree). Work backward from maturity.
Callable bond node rule: V(node) = min[ value from discounting, call price ]; Apply it at each node where the bond is callable, before adding the coupon paid at that date. Follow the question's convention for the coupon.
Putable bond node rule: V(node) = max[ value from discounting, put price ]; The holder exercises when the bond is worth less than the put price.
Option cost: Option cost = Z-spread − OAS; Positive for callable bonds, negative for putable bonds.
Price effect of the option: Callable price = Straight bond price − Call option value; Putable price = Straight bond price + Put option value.

How to solve Option-Adjusted Spread (OAS) questions

Use this order for any OAS question, whether you are given the OAS and asked for a price, or the price and asked for the OAS.

1Write down the tree of short rates, the probabilities, the coupon and face value, and any call or put dates and prices.
2Add the OAS to every rate in the tree. The same number goes at every node.
3Start at maturity. Take face value plus the final coupon.
4Move back one step at a time. Discount the probability-weighted average of the next values, plus the coupon, at the node's rate plus OAS.
5At each exercise date, apply the option rule: cap the value at the call price, or floor it at the put price.
6At the root, you have the model price. If the OAS was given, that is your answer.
7If the market price was given, compare it with the model price. Raise the OAS if the model price is too high, and lower it if too low. Repeat until the prices match, then interpolate.
8Interpret: compare OAS with Z-spread, name the option cost, and say what the OAS says about cheap or rich.

Quickest way: Bracket and interpolate

When to use it: Use when the question gives a market price and asks for the OAS, and the answer options are close together.

Check direction first. Higher OAS means lower model price. If the market price is below the zero-spread model price, the OAS is positive.
Compute the model price at zero spread and at one trial spread, such as 50 bp.
Interpolate linearly between the two prices to estimate the OAS.
Test the nearest answer option once. Stop when it reproduces the market price.
For conceptual questions, skip the tree and use option cost = Z-spread − OAS, plus the sign rule for callable and putable bonds.

Common mistakes in Option-Adjusted Spread (OAS)

Adding the OAS only to the first rate or to the spot curve instead of every tree node.
Students mix up OAS with a yield spread over one curve.
Fix: Add the same spread to every node rate, then discount. The spread goes inside each (1 + r + OAS) factor.
Forgetting to apply the call rule at each node and pricing the callable bond as a straight bond.
Students focus on the spread and treat the option as an afterthought.
Fix: At every call date, replace the node value with the call price if the value is higher. Do this before moving to the earlier node.
Saying the OAS of a callable bond is larger than its Z-spread.
Students think the option adds risk, so more spread is needed.
Fix: The issuer owns the call. Its value is taken out of the bond, so OAS = Z-spread − option cost, which is below the Z-spread. For a putable bond the sign flips.
Treating a higher OAS as always better.
Higher spread looks like higher return.
Fix: OAS is compensation for risk the model cannot remove, mainly credit and liquidity. A higher OAS may signal cheapness or higher risk. Compare bonds of similar quality and the same model.
Raising the spread when the model price is below market.
Students forget the inverse link between spread and price.
Fix: A bigger spread discounts harder and lowers the model price. If the model price is too low, reduce the OAS.
Ignoring the model dependence of OAS.
OAS is presented as one number for each bond.
Fix: OAS changes with the tree, the volatility and the exercise rule. Higher volatility raises the call value and lowers a callable bond's OAS.

Worked examples

Example 1

A 2-year bond pays an annual coupon of 5 on a face value of 100 and has no options. The current one-year rate is 4%. The one-year rate one year from now is 5% in the up state and 3% in the down state, each with probability 0.5. The OAS is 50 bp. What is the model price?

Show the solution

Add 0.50% to each rate. The first-year rate becomes 4.50%. The up node becomes 5.50% and the down node 3.50%.
At year 1 the remaining payment is 105 at year 2. Up node: 105 ÷ 1.055 = 99.5261.
Down node: 105 ÷ 1.035 = 101.4493.
Average the two values: (99.5261 + 101.4493) ÷ 2 = 100.4877.
Add the year-1 coupon of 5: 100.4877 + 5 = 105.4877.
Discount one year at 4.50%: 105.4877 ÷ 1.045 = 100.9452.

Answer: The model price is about 100.95.

Example 2

A callable bond trades at a market price that gives an OAS of 80 bp. Its Z-spread is 140 bp. A comparable option-free bond from the same issuer has an OAS of 70 bp. (a) What is the option cost? (b) Which bond looks cheaper? (c) If the volatility input is raised and the market price stays the same, what happens to the callable bond's OAS?

Show the solution

(a) Option cost = Z-spread − OAS = 140 − 80 = 60 bp.
(b) Compare OAS values, because both are stated after removing option effects. The callable bond has 80 bp and the option-free bond has 70 bp. The callable bond offers 10 bp more spread, so it looks cheaper, assuming similar credit and liquidity risk.
(c) Higher volatility raises the value of the call option that the issuer holds. The model price of the callable bond falls at any given spread. To match the unchanged market price, the spread must fall.

Answer: (a) 60 bp. (b) The callable bond looks cheaper by about 10 bp of OAS. (c) Its OAS decreases.

Exam tips

Memorise the sign rule: callable means OAS < Z-spread, putable means OAS > Z-spread, option-free means they are about equal.
In tree questions, check whether the call price caps the value before or after the coupon is added, and follow the question's wording.
Expect conceptual questions on volatility: higher volatility raises the call option's value, lowers the callable bond's model price at any given spread, and so lowers its OAS at a fixed market price. An option-free bond's value is unaffected by volatility.
If the options are close, estimate the model price at two spreads and interpolate. Do not solve it algebraically.
Always state what the OAS represents: spread for credit and liquidity risk after the option is removed. Examiners reward that wording.

Practice questions from Arbitrage Pricing with Term Structure Models

Option-Adjusted Spread (OAS) in other exams

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

Option-Adjusted Spread (OAS): frequently asked questions

What is the difference between OAS and Z-spread?

Z-spread is the constant spread over the spot curve that prices the bond's cash flows as fixed, so it ignores embedded options. OAS is the constant spread over the tree rates after the option is valued. The difference is the option cost.

How do I calculate OAS using a binomial tree?

Add a trial spread to every node rate, discount back through the tree, and apply any call or put rule at each exercise node. Compare the root price with the market price. Adjust the spread and interpolate until the two match.

How do I interpret OAS for a callable bond?

It is the spread you earn for credit and liquidity risk once the issuer's call option has been priced. It is smaller than the Z-spread. A higher OAS than similar bonds suggests the bond is cheaper, assuming similar risk.

Does OAS depend on the interest rate model?

Yes. OAS depends on the tree, the volatility assumption and the exercise rule. Different models or volatilities give different OAS values for the same market price, so compare OAS only when the model is consistent.