Binomial models are usually constructed to value bonds with embedded options. The process of valuing a bond with embedded options is only slightly more complicated than the one that we have already described. The only notable difference is that we now have to make so-called "nodal decisions" at each node.

Zero Volatility

In the absence of default and interest rate volatility, the bond's future cash flows are certain.

Let's assume that a Bermudan-style bond can be called by the issuer since the end of year 1 at par (price of 100). We use the same backward induction approach to value such bond, but at each node we make a nodal decision, specified by the following formula:

Value at the Node = Min {Value from backward induction, Call price}

This formula is an example of a decision rule. In our case, the rule is very simple - the bond is called whenever the intrinsic value of the call becomes positive (value exceeds call price). So, at each node we compare the value that we get from backward induction and the call price, which is assumed to be par in this example. Whenever the value is higher than the par, we substitute it with par. When the value at the node does not exceed par, we leave it as it is and continue the backward induction process.

Valuation of a callable bond is demonstrated by the following example:

According to the stated call rule, the bond is called at the four special nodes, because in these cases the estimated value at the node exceeds par. Let's demonstrate the last step in this backward induction process - calculation of the value at the root of the tree. The one-year rate at the root is 2.5%. If the higher rate (4.516) is effective at the end of year 1, the value at the next node is 98.108. The bond is not expected to be called at this node, because its value is below par. On the contrary, if rates increase by a lesser amount, to 3.3456, the estimated value of the bond at the next node would be 101.229, so the bond would be called, since it is valued above par. We substitute the value at this node with par (call price). We now discount the two values and coupons to find the value at the root:

V₀ = 1/2 x (V_{1, H}+ C) / (1 + r₀) + 1/2 x (V_{1, L} + C) / (1 + r₀) = 1/2 x (98.108 + 5) / (1+0.025) + 1/2 x (100 + 5)/(1+0.025) = 101.516.

In contrast, if the bond is putable, the decision to exercise the option is made by the bondholder, who will exercise the put option when the discounted value of the bond's future cash flows is lower than the put price.

Effect of Interest Rate Volatility

Volatility does not affect the value of a straight bond. However, higher volatility does increase the value of any embedded option, and:

• decreases the value of a callable bond.
• increases the value of a putable bond.

Changes in the level and shape of the yield curve also affect the value of a callable or putable bond.

When interest rates are high, the bond is unlikely to be called. Therefore, the callable bond will have similar price/yield relationship (positive convexity) as a comparable straight bond. As interest rates decline, the value of a straight bond will rise. The value of a callable bond will also rise, but not as much as the straight bond. This is because the value of the call option will rise too:

Value of callable bond ↑ = Value of straight bond ↑ - value of issuer call option ↑

The call price will set an upper limit on the price of the callable bond.

As interest rates rise, the value of a straight bond will decline. The value of an otherwise identical putable bond will also decline, but not as much as the straight bond. This is because the value of the put option will rise:

Value of putable bond ↓ = Value of straight bond ↓ + value of investor put option ↑

As the yield curve flattens:

• The value of the call option increases.
• The value of the put option decreases.

Valuation

Valuing a bond with embedded options assuming an interest rate volatility requires three steps:

• Generate a tree of interest rates based on the given yield curve and volatility assumptions;
• at each node of the tree, determine whether the embedded options will be exercised; and
• apply the backward induction valuation methodology to calculate the present value of the bond.

Let's see what happens to our interest rate tree based on the same yield curve, when we increase the assumed volatility from 15% to 20%:

15% Annual Volatility Assumption

20% Annual Volatility Assumption

As demonstrated by the exhibit above, when yield volatility is higher, the interest rate tree becomes more dispersed. Consequently, there is a greater probability that the values at future nodes will be higher than the call price or lower than the put price. The higher probability of options being exercised makes these options more valuable.