# How to Calculate Option-adjusted Spread (OAS) of a Bond

Option adjusted spread is a measure of the credit risk in option-embedded bonds such as callable and putable bonds. As the name explains, it is the spread after adjusting (removing) the option from the bond. So, from the bond, we remove the value of the embedded option, which gives us the spread of the option-free bond.

Let’s take a Treasury security. We value a Treasury security by discounting its payments with the zero-coupon Treasury rates. The two common curves used for discounting are Treasury curves or LIBOR curves.  The value that we get by discounting with zero-coupon curve is usually very close to the observed market value of the bond.

However, the problem is that we can’t find a zero-coupon Treasure rates such that the value calculated using it matches the observed market price. We can quantify the difference between discounted value and the market price. We can say that how much should the zero-coupon rates be changed/shifted so that the discounted value is equal to the market price. This shift/adjustment to the zero-coupon interest rates is called the Z-spread for that security. A positive Z-spread indicates the security is cheap, while a negative Z-spread indicates the security is expensive.

Instead of Treasury securities, if we look at a corporate bond, we can follow the same approach. Since, corporate bonds have credit risk, they are considered less worthy compared to a similar Treasury bond. This is reflected in the Z-spread of the corporate bond, which will be positive for the corporate bond. Higher the credit risk, the higher is the Z-spread. Also, for corporate bonds, LIBOR curve is considered a better measure to calculate Z-spread. The Z-spread to LIBOR is likely to better reflect the credit risk in a corporate bond.

Now, let’s look at option-embedded bonds, such as callable bonds. For these bonds, Z-spread is not appropriate. The reason is that we can’t just value a callable bond by discounting the scheduled payments. Interest rate volatility plays a huge role here. We will need to use a model that takes into consideration the volatility interest rates which will help us in taking into account the risk of the bond being called. It is possible to use various models. One appropriate model is the stochastic term structure model.

1. We start with a curve of zero-coupon interest rates and also use some parameters for determining the volatility of these interest rates.

2. With these inputs, we can generate a large number of possible scenarios for future interest rates. These future interest rates can be above or below the current spot curve. We can use a rule to determine when the embedded option will be exercised.

3. We calculate the value of the security by discounting the payments in each scenario using the interest rates in that scenario.

4. Then we average over all the scenarios.

So, the calculation of OAS is relatable to the calculation of Z-spread. The OAS is the shift to the zero-coupon interest rates in all scenarios required to ensure that the model value (the average value of all scenarios) equals the market price of the bond.

If the bond did not have an embedded option, the OAS will be exactly the same as Z-spread.

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