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Show how true probabilities differ from risk-neutral ones and use that gap to handle the drift in interest rates.

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

The idea

True probabilities describe how rates are expected to move in the real world. Risk-neutral probabilities are the ones that price bonds as if every bond earned the short rate in expectation. They differ because investors demand a premium for holding rate risk. The risk-neutral probability of the bad state (rates up, bond prices down) is larger than the true one. Equivalently, the drift of rates in the risk-neutral tree is higher than the true drift in the real world. The gap between the two drifts measures the market price of risk, roughly as follows. Lambda = (Bond return − Short rate) ÷ Volatility. Here the bond return is the expected return and the volatility is that of the bond's return. For pricing use the risk-neutral tree. For forecasting or risk reporting use the true drift. Do not mix them: a real-world expected payoff discounted at the riskless rate misprices.

Formulas

Market price of interest rate risk

λ = (μ − r) ÷ σ

μ
the bond's expected return under the true probabilities
r
the riskless short rate for the period
σ
the volatility of the bond's return

An approximation of the gap between the true and risk-neutral drifts. Use the excess return over the short rate, and the standard deviation rather than the variance.

A practice question

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

In a risk-neutral interest rate tree, the expected return over the next period on any default-free zero-coupon bond equals:

  1. its own yield to maturity, which is the return promised if the bond is held to maturity.Why not: Confuses a promised yield with an expected one-period return; in the risk-neutral world all bonds earn the short rate.
  2. zero, because risk-neutral probabilities remove all expected drift from bond prices.Why not: Mixes up removing the risk premium with removing return; bonds still earn the riskless rate.
  3. the real-world expected return, which is the short rate plus a term premium proportional to maturity.Why not: Describes the true measure; the risk-neutral measure strips out the premium by construction.
  4. the short-term (six-month) rate, the same for every bond regardless of maturity. - correct

Under risk-neutral probabilities every bond's expected one-period return is the riskless short rate, which is why discounting at the short rate gives arbitrage-free prices.

A harder one

The six-month rate is 5.00% and will be 7.00% or 3.00% in six months (semiannual compounding). The risk-neutral probability of 7.00% is 0.60, and the real-world probability is 0.50. The one-year zero (face 100) is priced with the risk-neutral probabilities. Its expected excess return over the six-month riskless return (2.50% per six months) under the real-world probabilities is closest to:

  1. 0.00%Why not: Computes the expected return with risk-neutral probabilities, where the excess return is zero by construction.
  2. 0.20% - correct
  3. 0.40%Why not: Annualizes the excess return; the question asks for the six-month figure.
  4. 2.70%Why not: Reports the total expected six-month return rather than the excess over the 2.50% riskless return.

Six-month prices: 96.62 and 98.52. Today's price: (0.6x96.62 + 0.4x98.52)/1.025 = 95.00. Real-world expected price 97.57; return 2.70%; excess over 2.50% = 0.20%.

The hardest tier

The six-month rate is 5.00% and will be 7.00% or 3.00% in six months (semiannual compounding). Risk-neutral probability of 7.00% is 0.60, real-world probability 0.50. The one-year zero (face 100) is priced with risk-neutral probabilities. The market price of risk per six months, defined as the real-world expected excess return of the bond over the riskless six-month return divided by the standard deviation of the bond's six-month return under the real-world probabilities, is closest to:

  1. 0.10Why not: Divides by the full price range (up minus down) instead of the standard deviation, which is half the range when the probabilities are 0.5.
  2. 2.70Why not: Uses the total expected return rather than the excess return over the riskless rate.
  3. 20.0Why not: Divides by the variance of the return instead of its standard deviation.
  4. 0.20 - correct

Expected excess return 0.200%. Return standard deviation = (98.52 - 96.62)/2/95.00 = 1.002%. Ratio = 0.20.

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