Charta is on Google Play: practise on the go. Get the app
Charta

Assess how fitting the Black-Scholes-Merton model is as a tool for pricing fixed-income derivatives.

FRM® Part II · Market Risk Measurement and Management · Arbitrage Pricing with Term Structure Models · FRM2.MR.12.j

The idea

Black-Scholes-Merton assumes the underlying has a constant volatility and a lognormal price, and that the riskless rate is constant. For fixed-income derivatives these assumptions fail. A bond's price volatility shrinks to zero as it approaches par at maturity, so constant volatility is wrong. A lognormal price lets a zero trade above par, which implies negative rates. The short rate that drives the bond is also the one used for discounting, so holding it constant is inconsistent. The model is tolerable only when the option is short-dated relative to a long-maturity bond, so the bond's volatility barely changes and the pull to par is minor. For options on rates, or long-dated options on short bonds, use a term structure model. Convert yield volatility to price volatility with duration before using BSM.

Formulas

Bond price volatility from yield volatility

σprice = D × σyield

D
the bond's modified duration
σyield
the absolute yield volatility, in percentage points a year

An approximation for converting before using BSM on a bond. 100 bp is 1.00%, not 0.1%; and the result shrinks as the bond nears maturity.

A practice question

From the question bank, with the reason each wrong option is wrong.

A practitioner applies Black-Scholes-Merton to an option on a bond, using a constant short rate for discounting. The main conceptual flaw in this use is:

  1. the short rate is used instead of the bond's yield to maturity, which would be more accurate.Why not: Using a yield instead of a short rate would not solve the inconsistency with stochastic rates.
  2. bond prices are known at expiry, so the option has no volatility to price.Why not: The bond price at option expiry is uncertain unless the bond matures at expiry.
  3. the interest rate is the very variable that moves the bond price, so holding it constant for discounting is inconsistent with the underlying's own randomness. - correct
  4. the model needs a rate for each maturity, which a constant rate cannot provide.Why not: BSM can use a term-structure of discount rates; the flaw is that rates are treated as non-random.

Interest rate randomness drives both the bond price and the discount factor; BSM treats the rate as known and so ignores that link.

A harder one

Which option on a bond is BSM (with constant price volatility) least likely to mis-price badly?

  1. A five-year option on a bond with six years to maturity.Why not: Heavy pull to par: the bond's volatility falls greatly over the option's life.
  2. A two-year option on a two-and-a-quarter-year zero.Why not: At expiry the bond has three months left, so its price volatility will be tiny compared with today's.
  3. A ten-year option on a ten-and-a-half-year zero.Why not: Very long option life and very short remaining maturity at expiry make constant volatility unrealistic.
  4. A three-month option on a bond with 20 years to maturity. - correct

For a short option on a long bond, the remaining maturity and so the bond's price volatility barely change during the option's life, so constant volatility is a tolerable approximation.

The hardest tier

A one-year zero-coupon bond (face 100) trades at 96.00. An analyst models its price at the expiry of a six-month option as lognormal with annual volatility 3.00%, a continuously compounded riskless rate of 4.00%, and a forward price at expiry of 96.00 x exp(0.04 x 0.5) = 97.94. Under this model, the risk-neutral probability that the bond's price at expiry exceeds its face value of 100 is closest to:

  1. 2.6%Why not: Uses the current price of 96.00 instead of the forward price when computing d2.
  2. 16% - correct
  3. 84%Why not: Computes the probability that the price is below 100 (N of minus d2).
  4. 0%Why not: Assumes the model respects the par cap; a lognormal price has no upper bound, which is exactly why BSM misfits zeros.

d2 = [ln(97.94/100) - 0.5 x 0.03² x 0.5]/(0.03 x sqrt(0.5)) = -0.99; N(d2) = 0.161. A zero cannot exceed par if rates are non-negative, yet BSM gives it a 16% chance.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Arbitrage Pricing with Term Structure Models