Binomial Model

CFA Level II Fixed Income
#1

Correct me if I am wrong, but the binomial model is used to value any bond with or without embedded options.

  1. In the practical world, why do we need to obtain an arbitrage-free price for a bond?

  2. In what circumstances should I use the binomial model and when should I not?

Please forgive me if these questions sound so naive.

#2

In theory, binomial interest rate trees could be used to value any bond. However, in practice binomial trees are derived using noncallable bonds and the trees so derived are used to value bonds with embedded options for early retirement. (This discussion will not address convertible bonds as they would likely need to be valued using, in part at least, an equity option binomial mode).

  1. Arbitrage-free price in the sense implied in the curriculum would only apply where markets were complete enough for arbitrage strategies to dictate relatively narrow arbitrage pricing boundaries. The US Treasury market would be an example: it’s easy to buy and short sell securities, there is an active Treasury futures market, coupon Treasuries can be stripped and reconstituted. In that market segment the value of any coupon bond or note must trade within a narrow band circumscribed by the cumulative value of the stripped coupon and principal securities that could be used to reconstruct (reconstitute) the coupon securities’ cash flows or that the coupon securities could be divided (stripped) into. The same limitation would generally apply to the cheapest-to-deliver Treasury versus the related Treasury futures contract.

Corporate bond prices should always be “arbitrage free” (the chance to make risk-free profits will not be passed up by market participants in this world), but there would seldom be a single price (or only a narrow range of prices) that would be arbitrage free. It’s not always easy to short-sell corporate bonds, corporates can’t be stripped, there are no corporate bond futures traded on individual issuers, etc. Should the conditions that would require bonds trade within a narrow range of related securities or derivatives do not exist, arbitrage-free prices are not enforceable via arbitrage trading in most cases.

  1. Binomial interest rate trees are most commonly used to estimate the fair value of bonds with embedded options. (Going forward I will use callable bonds rather than the more precise but wordier bonds with embedded option for early retirement. Putable bonds are not that common but would be valued in a similar fashion.) Binomial trees are also used when doing an option-adjusted spread (OAS) analysis. OAS analysis is used to estimate the yield a callable bond would trade at if it were not callable.

The rates on the binomial tree are used to estimate the fair value of a callable bond. If a callable bond is trading at a market price significantly above or below the fair value so estimated, it might indicate the bond is trading rich or cheap. The bond may therefore be an attractive purchase if undervalued (i.e. trading cheap) or an attractive sale (if you own it) since it appears overvalued (i.e. trading rich).

Binomial trees are also used to determine the effective duration and convexity of callable bonds. This is done by shifting all of the interest rates on the tree up and down by the same number of basis point (e.g. 10 bps) and then recalculating the price under those circumstances. The prices obtained in that fashion are then plugged into the formulas to obtain the duration and convexity of the callable securities.