How do you calculate the cost of an embedded option using OAS and Z-spread?

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AcadiFi · Accepted Answer

The option cost embedded in a bond is the spread difference between the Z-spread (which ignores optionality) and the OAS (which adjusts for optionality). This difference represents the yield the investor gives up (callable) or gains (putable) due to the embedded option. **Key Relationships:** | Bond Type | OAS vs Z-spread | Option Cost | Who Benefits? | |---|---|---|---| | Callable | OAS < Z-spread | Positive (investor pays) | Issuer (right to call) | | Putable | OAS > Z-spread | Negative (investor receives) | Investor (right to put) | | Option-free | OAS = Z-spread | Zero | No optionality | **Formula:** Option Cost = Z-spread - OAS **Worked Example -- Three Bonds from Sentinel Corp (fictional)** All three bonds are 10-year maturity, 5% coupon, same credit quality, priced by the same dealer: | Bond | Type | Z-spread (bps) | OAS (bps) | Option Cost (bps) | |---|---|---|---|---| | Sentinel A | Option-free | 150 | 150 | 0 | | Sentinel B | Callable at par in 3yr | 185 | 150 | +35 | | Sentinel C | Putable at par in 5yr | 130 | 150 | -20 | \`\`\`mermaid flowchart LR A[Z-Spread] --> B{Embedded Option?} B -->|None| C[OAS = Z-Spread] B -->|Call Option| D[OAS < Z-Spread] B -->|Put Option| E[OAS > Z-Spread] D --> F[Option Cost = Z - OAS = Positive] E --> G[Option Cost = Z - OAS = Negative] F --> H[Investor Compensated for Giving Up Upside] G --> I[Investor Pays for Downside Protection] \`\`\` **Interpretation:** - Sentinel B's investors demand a 185 bps Z-spread because 35 bps compensates for the call risk. After removing the option effect, the pure credit spread (OAS) is 150 bps -- the same as the option-free bond. - Sentinel C's investors accept only a 130 bps Z-spread because they hold a valuable put option worth 20 bps. The OAS is still 150 bps once the put's value is removed. **Analytical Use:** OAS is the appropriate measure for comparing bonds with different embedded options because it strips out the option effect and isolates the credit component. A bond with a higher OAS than its comparable peers offers better compensation for credit risk regardless of its option structure.

How do you calculate the cost of an embedded option using OAS and Z-spread?

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