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Price a call on a zero-coupon bond through an arbitrage argument that rests on a replicating portfolio.

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

The idea

To price a call on a zero-coupon bond by arbitrage, build a portfolio of the underlying bond and cash that has the same payoff as the option in both the up and the down state of the next period. Take payoffs Cup and Cdown and bond prices Pup and Pdown. Call the number of bonds Delta. Delta = (Cup − Cdown) ÷ (Pup − Pdown). The cash position, usually a borrowing, is called Cash. Cash = (Cup − Delta × Pup) ÷ (1 + r/2). Here P0 is the market price of the bond today. The option price is called Price. Price = Delta × P0 + Cash. The result never uses the real-world chance of up or down, only the prices of the traded bond and cash. The classic mistake is to price the option with an expected payoff computed from a trader's own probabilities.

Formulas

Bonds held in the replicating portfolio

Δ = (Cup − Cdown) ÷ (Pup − Pdown)

Cup
the option payoff in the up state
Cdown
the option payoff in the down state
Pup
the bond price in the up state
Pdown
the bond price in the down state

Payoff change over price change, never the reverse. Both the call payoff and the bond price fall when rates rise, so Δ is positive.

Cash position in the replicating portfolio

Cash = (Cup − Δ × Pup) ÷ (1 + r ÷ 2)

Cup
the option payoff in the up state
Δ
the number of bonds held
Pup
the bond price in the up state
r
today's six-month rate, annualised and compounded semiannually

Usually negative, a borrowing. Discount it one period at today's rate; forgetting that discounting is the usual slip.

Arbitrage price of an option on a zero from replication

Price = Δ × P0 + Cash

Δ
the number of bonds held
P0
the bond's market price today
Cash
the cash position, negative when borrowing

Uses only traded prices and the riskless rate, never anyone's real-world probabilities.

A practice question

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

In a one-period binomial tree for six-month rates, which portfolio replicates a call option that expires in six months on a one-year zero-coupon bond?

  1. A position in the one-year zero only, sized so that its expected payoff under the real-world probabilities equals the option's expected payoff.Why not: Matches an expected payoff rather than the payoff in each state, and uses a single asset where two are needed for two states.
  2. A position in the one-year zero-coupon bond plus a position in cash lent or borrowed at the six-month rate, sized to match the option payoff in both states. - correct
  3. A position in the one-year zero and a long-maturity coupon bond, with weights chosen to match duration.Why not: Duration matching hedges small yield shifts; replication on a tree matches exact payoffs in the up and down state using cash.
  4. Cash only, invested at the six-month rate in the amount of the expected option payoff.Why not: Cash pays the same in both states, so it cannot reproduce a payoff that differs by state.

With two states you need two assets with linearly independent payoffs: the bond (payoff differs by state) and cash (the same in both). Solving two equations gives the bond units and the cash position.

A harder one

Same tree as before: the six-month rate is 5.00% today and 6.00% or 4.00% in six months with risk-neutral probability 0.5 each (semiannual compounding). A call expiring in six months on the one-year zero (face 100) has strike 97.80. Its arbitrage-free price is closest to:

  1. 0.23Why not: Discounts the down-state payoff without weighting it by its probability of 0.5.
  2. 0.06Why not: Applies the probability twice (0.5 x 0.5) to the down-state payoff.
  3. 0.25Why not: Reports the hedge ratio (option payoff change over bond price change) as if it were the option price.
  4. 0.12 - correct

Bond prices at six months: 97.09 and 98.04. Payoffs: 0 and 0.24. Price = 0.5 x 0.24 / 1.025 = 0.12; replication gives the same value.

The hardest tier

The six-month rate is 3.00% and will be 5.00% or 2.00% in six months (semiannual compounding). The one-year zero (face 100) trades at 96.40. A trader believes the real-world probability that the rate rises is 70%. A call expires in six months on the one-year zero with strike 98.50. Its arbitrage-free price per option is closest to:

  1. 0.15Why not: Weights the down-state payoff by the trader's 30% real-world probability; arbitrage pricing does not use real-world probabilities.
  2. 0.10 - correct
  3. 0.25Why not: Assumes a 0.5 probability instead of calibrating the probability to the 96.40 market price of the bond.
  4. 0.35Why not: Puts the 70% on the in-the-money down-rate state, mixing up which state the probability belongs to.

Bond prices at six months: 97.56 and 99.01. The market price implies p(up) = (99.01 - 96.40 x 1.015)/(99.01 - 97.56) = 0.80. The call pays 0.51 only in the down state: (1 - 0.80) x 0.51/1.015 = 0.10. The 70% belief is irrelevant.

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