Good evening everybody,

Tonight’s focus will be dedicated to valuing bonds with embedded options. I chose this topic because it involves **binomial trees**, which is an important an very testable point of the Level II curriculum. Binomial trees are encountered in two different topics: Fixed Income and Derivatives. Although the concept is the same for the two instances, the main difference is the fact that the *interest rate tree* is **given**, whereas you have to construct the tree for stock values yourself. Besides, in interest rate trees, the probability of going to up or down nodes is 0.5. For options on stocks, you have to determine the risk-neutral probability value.

For simple bonds with no embedded options, in each node, you write:

- The current price of the bond for the remaining payments (including face value)
- The coupon at that node
- The interest rate (more specifically the forward rate) at that not (which is
*given*).

You use the interest rate tree using backward induction, i.e. you start by the node at the right of the tree (the ones the further away from now) where you know the price: the face value. It’s the only place where you know what the price will be. At a given node before the final layer, you have to compute the price of the bond which is given by computing the average (because the probability of ups and downs is 0.5) of the present value of the prices at the two following nodes. To compute the present value, you use the forward rate given at the current nodes. Therefore, there are no forward rates given for nodes at the final layer. Using this process sequentially for each node from the right to the left of the tree, you end up computing the initial node’s value: the price of the bond today.

Because the interest rate trees are built to be arbitrage trees, they are computed such that the benchmark security’s price you get using the tree is the market price.

So if you compute the price of a bond which is different from the benchmark, you will get a different value. This is where **spreads** come into play. Nominal spreads are computed using YTM. You compute the constant yield implied by the market price of the bond and its benchmark, and you compute the difference between them. However, this is not a good measure as it assumes the yield curve is flat. The **Z-spread**, for zero-volatility spread, is the spread that is added to each of the term structure’s rate to make its price equal to the benchmark’s price. You can do exactly this by adding the Z-spread to each of the nodes’ rates (you use a value to add to each node’s rate until you find the right value, by trial and error).

For bonds with embedded options, you do exactly the same as for the option-free bond, but the price at the nodes where the call can be exercised you compute the price using the option’s criteria. For example, for a bond callable at 100 in 2 years, if you compute that the price at the second node is 101.5, then you have to use the price which is equal to $\min(101.5,100)=100$.

Recall the following for bonds with embedded options:

$$V_\text{callable}=V_\text{option-free bond} – V_\text{call}$$

$$V_\text{putable}=V_\text{option-free bond} + V_\text{put}$$

The thing is, *the market value of the bond with the embedded option will often be different than is theoretical, arbitrage-free, value*. The reason for this comes mainly from the fact that the interest rate tree assumes some interest rate volatility. Remember the following very important facts:

**Option-free bonds prices are unaffected by interest rate volatility**, they are priced as of today and that’s it.
**The price of the embedded ****option** (call or put) is positively related to interest rate volatility.

When we make the valuation of the bond with the embedded option, we essentially compute the value of the cash flow assuming the interest rate volatility. In a sense, we price the option-feature of the bond, assuming the interest rate volatility of the tree. However, market can change its expectation on the volatility of the bond, and hence, the price is not the same anymore, because although the option-free bond value has not changed, the value of the embedded option is different.

Similarly to the Z-spread, if you find the value you have to add to each node’s rate to get the same value from the tree as the market value, you get the **OAS**, the option-adjusted spread, which is the spread of the bond “removing” its option feature. You have to use this spread to compare bonds with embedded option with bonds without embedded options (such as their relative benchmark) or even bonds with embedded options with each other. The price of the option is relative to the volatility of interest rate, not to the credit risk and liquidity risk of the bond; it shall hence be remove to compare inherent quality of the fixed income security.

That’s it for this post.