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State the idea of risk-neutral pricing and put it to work in valuing options.

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

The idea

Risk-neutral pricing says that a derivative's value equals its expected payoff, computed with special probabilities, discounted at the riskless short rate. Those probabilities are not forecasts. They are the ones that make the traded securities, such as the bonds on the tree, come out at their market prices. For a one-period bond tree, let P0 be today's bond price, Pup and Pdown the bond prices after the rate moves up or down, and r the annual short rate compounded semiannually. Up means the higher rate. The risk-neutral probability of the up state is then as follows. p = (Pdown − P0 × (1 + r/2)) ÷ (Pdown − Pup). Because the probabilities come from market prices, they embed investors' attitude to risk, so the price is the same for any investor who accepts no arbitrage. Use them for every instrument on the same tree. If the implied p lies outside zero to one, the quoted prices admit an arbitrage.

Formulas

Risk-neutral probability of the up (higher-rate) move on a one-period bond tree

p = (Pdown − P0 × (1 + r ÷ 2)) ÷ (Pdown − Pup)

Pdown
the bond price next period after a down move in rates
P0
the bond's market price today
r
today's six-month rate, annualised and compounded semiannually
Pup
the bond price next period after an up move in rates

Up means the higher rate, where the bond price is lower. Grow P0 at today's rate, not a node rate; a p outside zero to one signals an arbitrage.

A practice question

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

Which statement best captures the idea of risk-neutral pricing of an interest rate option?

  1. The price equals the expected payoff under the real-world probabilities, discounted at a rate that includes a premium for the option's own risk.Why not: Describes a real-world valuation needing a risk premium; risk-neutral pricing changes the probabilities and discounts at the riskless rate.
  2. The price assumes that investors are in fact indifferent to risk, so true probabilities can be used with riskless discounting.Why not: Treats risk neutrality as an economic assumption; it is a pricing device and the probabilities come from market prices.
  3. The price equals the expected payoff under probabilities that make the traded bonds correctly priced, discounted at the short rate, and matches the replication price whatever investors' risk preferences. - correct
  4. The price equals the discounted payoff at the bond's yield to maturity, which already contains the market's view of risk.Why not: Discounting at a yield rather than the node short rate with implied probabilities ignores how the payoff varies by state.

Risk-neutral probabilities are backed out of market prices of traded securities; with them, discounting at the riskless rate reproduces the replication price. No assumption about investor risk preferences is needed.

A harder one

The six-month rate is 4.00% and will be 6.00% or 2.00% in six months (semiannual compounding). The risk-neutral probability of the 6.00% state is 0.6. A put expires in six months on a one-year zero-coupon bond (face 100) with strike 98.50. Its price is closest to:

  1. 0.83 - correct
  2. 0.55Why not: Applies the 0.4 probability to the state where the put pays (the 6.00% state); the 6.00% state has probability 0.6.
  3. 0.20Why not: Values a call (down-rate state payoff) instead of the put.
  4. 0.69Why not: Ignores the given 0.6 and uses 0.5.

Bond prices at six months: 97.09 (6% state) and 99.01 (2% state). The put pays 1.41 only in the 6% state: 0.6 x 1.41/1.02 = 0.83.

The hardest tier

The six-month rate is 4.50% and will be 6.50% or 2.50% in six months (semiannual compounding). The one-year zero (face 100) trades at 95.55. A call expiring in six months on that bond has strike 97.50. The price of the call, using risk-neutral pricing calibrated to the bond, is closest to:

  1. 0.69Why not: Weights the payoff by the up-state probability, but the call pays only in the down-rate state.
  2. 0.62Why not: Uses a probability of 0.5 rather than the probability implied by the 95.55 bond price.
  3. 0.55 - correct
  4. 1.27Why not: Takes the undiscounted, unweighted payoff in the in-the-money state.

Prices at six months: 96.85 and 98.77. Implied p(up) = (98.77 - 95.55 x 1.0225)/(98.77 - 96.85) = 0.56. The call pays 1.27 in the down state: (1 - 0.56) x 1.27/1.0225 = 0.55.

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