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State what the option-adjusted spread (OAS) is and use it when pricing securities.

FRM® Part II · Market Risk Measurement and Management · Arbitrage Pricing with Term Structure Models · FRM2.MR.12.f

The idea

The option-adjusted spread, written OAS, is the constant spread added to every short rate on the tree so that the model price of the security equals its market price. It is the compensation for credit, liquidity and model risk that remains after the cost of embedded options has been valued with the model. OAS is below the static Z-spread for a callable bond and above it for a putable bond, since the option has been valued. The option cost, written Cost, is the gap between the two spreads for a callable bond. For a callable bond, Cost = Z-spread − OAS. Let P0 be the model price at zero spread, P the market price and D the effective duration. Approximately, OAS = (P0 − P) ÷ (P × D). A positive OAS means the bond looks cheap versus the model. Because OAS depends on the volatility assumption, a higher volatility lowers the OAS of a callable bond. Compare OAS only across bonds valued with the same model.

Formulas

Option cost of a callable bond

Cost = Z-spread − OAS

Z-spread
the static spread over the spot curve on contractual cash flows
OAS
the option-adjusted spread from the tree

For a callable bond the option cost is subtracted, so OAS lies below the Z-spread. For a putable bond OAS lies above it.

Approximate option-adjusted spread from prices

OAS = (P0 − P) ÷ (P × D)

P0
the model price at zero spread
P
the market price
D
the effective duration

An approximation. A market price below the model price gives a positive OAS; divide by duration, do not multiply.

A practice question

From the question bank, with the reason each wrong option is wrong.

A callable bond has a Z-spread of 140 bp. A tree-based valuation puts the cost of the embedded call at 55 bp. The bond's OAS is closest to:

  1. 195 bpWhy not: Adds the option cost to the Z-spread; for a callable bond the option cost is subtracted.
  2. 55 bpWhy not: Reports the option cost itself as the OAS.
  3. 85 bp - correct
  4. 140 bpWhy not: Reports the Z-spread: the OAS differs from it by the option cost.

Option cost = Z-spread - OAS for a callable bond, so OAS = 140 - 55 = 85 bp.

A harder one

A tree has a six-month rate of 5.00% today and 7.00% or 3.00% after six months (probability 0.5 each, semiannual compounding). A one-year zero (face 100) has a model price of 95.19 at zero spread. If the bond's OAS is 50 bp, its market price is closest to:

  1. 95.19Why not: Ignores the spread: this is the model price at zero spread.
  2. 95.66Why not: Subtracts the spread from every rate; a positive OAS must lower the price.
  3. 94.96Why not: Adds the 50 bp only to the first-period rate rather than to every node.
  4. 94.73 - correct

Add 0.50% to every node: root 5.50%, nodes 7.50% and 3.50%. Price = [0.5x100/1.0375 + 0.5x100/1.0175]/1.0275 = 94.73.

The hardest tier

An analyst values a callable bond on a tree and computes its OAS from its market price. She then raises the interest-rate volatility input and recalibrates the tree to today's curve. Holding the market price fixed, the callable bond's OAS will most likely:

  1. rise, because higher volatility means more rate risk to be compensated.Why not: Confuses OAS with a risk premium for volatility; higher volatility lowers the bondholder's option value (the issuer owns the call).
  2. fall, because higher volatility raises the call's value, lowers the model price at zero spread, and less spread is needed to reach the market price. - correct
  3. stay the same, because OAS is a spread and volatility only changes the tree's shape.Why not: The OAS is solved so that model price equals market price, and the model price depends on the volatility input.
  4. rise, because a higher volatility makes the model price of the callable bond higher.Why not: Higher volatility raises the value of the embedded call, which lowers the callable bond's model price.

A callable bond equals an option-free bond minus the call. Higher volatility increases the call, cutting the model price at zero spread, so a lower spread is needed to hit the unchanged market price: OAS falls. A putable bond behaves oppositely.

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