Skip to content
All library documents

Pricing Bond Options When Exercise Dates Fall Between Tree Nodes

Article Quant Q&A · Author: PLE
Read the original

Summary

The document asks how to price a bond with an embedded option in a Black–Derman–Toy short-rate tree when the exercise date does not coincide with a time step. In the aligned case, the tree is built with a fixed step and bond values are rolled backward; at the option date, values are compared with the strike and adjusted to reflect exercise. The difficulty is that an arbitrary expiry can fall between the tree’s existing dates, leaving no node at which to make that comparison directly.

The suggested response is to use interpolation. It proposes valuing the bond as though the call date were at each of the two neighboring tree dates, then interpolating between those values for the actual date. The source gives no specific interpolation formula, error analysis, or discussion of how to interpolate the tree or exercise boundary. The recommendation is therefore a high-level approximation, and the accuracy depends on the model, step size, and interpolation method.

Key ideas

  • Backward induction applies the exercise decision at nodes matching the option expiry.
  • An expiry between tree dates has no direct node for comparing continuation value with the strike.
  • The proposed workaround is to price at the neighboring tree dates and interpolate.
  • The document does not specify an interpolation formula or quantify the resulting approximation error.

Tags

Fixed incomeOptionsDerivatives pricingBacktesting
Full text
# Price of options when exercise date doesn't match nodes of BDT Tree


# Price of options when exercise date doesn't match nodes of BDT Tree












I think I'm missing something obvious here, but here I go. I'm studying pricing of bonds with embedded options using Black Derman Toy. I understand tree construction and its application for simple cases, by which I mean something like this:

> Zero Coupon Bond maturing in 5 years with embedded call option expiring in 3 years, of strike price $100

Here one would construct a binomial tree of step length, say, 0.25 years (3 months). Then from the end of the tree, apply backwards discounting using short rates calibrated on the tree. If the prices of bonds at option expiry is higher than 100 (strike) then 'fix' the price at that node to be 100;

This method works because the option expiry and bond maturity are both multiples of 0.25, so that the nodes coincide with respective dates.

> But what if this wasn't the case? For example, what if bond matured in 5 years but option expires in some random number, say 3.12 years?

If we decied on the same strategy and generate a tree of step length 0.25 years, the option maturity will fall between the nodes placed at time 3 years and 3.25 years.

In this scenario, how do I determine if option is exercised or not? Of course, situation can be resolved by constructing a tree of step length 0.01 years, but is there simpler way?

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/79743

You need to perform some sort of interpolation. For example value the bond using 3.0 as the call date , then value it using 3.25, then interpolate.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.