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Show how the arbitrage pricing principles for fixed-income derivatives carry over to several periods.

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

The idea

The one-period arbitrage argument extends to many periods by repeating it node by node. At the last date the derivative's payoff is known at each node. Step back one date. At each node, replicate the next-period payoffs with the underlying bond and cash. Equivalently, compute the value at the node with its own rate and the risk-neutral p. Value = [p × Vup + (1 − p) × Vdown] ÷ (1 + r/2). Continue back to the root. The replicating portfolio is therefore dynamic: it is rebalanced at every node as the tree unfolds, which is why the price is exact only if rebalancing is costless. Path-by-path discounting at today's rate is wrong. Early exercise is handled by comparing exercise value with continuation value at each node.

Formulas

Rolling a derivative's value back one node on a multi-period tree

V = [p × Vup + (1 − p) × Vdown] ÷ (1 + r ÷ 2)

p
the risk-neutral probability of the up move
Vup
the value at the next date's up node
Vdown
the value at the next date's down node
r
the node's own six-month rate, annualised and compounded semiannually

Apply it at every node from expiry to the root, with each node's own rate. For early exercise, take the larger of this continuation value and the exercise value.

A practice question

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

On a multi-period tree, an option's values at the two successors of a node are 6.00 and 2.00. The risk-neutral probabilities are 0.5 each. The node's six-month rate is 12.00% and today's (root) six-month rate is 4.00% (semiannual compounding). The option's value at the node is closest to:

  1. 3.92Why not: Discounts at the root rate of 4.00% instead of the node's own 12.00% rate.
  2. 3.77 - correct
  3. 4.00Why not: Takes the expected value of the successor values and forgets to discount one period at the node rate.
  4. 3.56Why not: Discounts twice at the node rate, covering a period that has already been handled.

Value = (0.5 x 6.00 + 0.5 x 2.00)/(1 + 0.12/2) = 4.00/1.06 = 3.77.

A harder one

The six-month rate tree starts at 5.00%; after six months it is 6.00% or 4.00%; after one year it is 7.00% (up-up), 5.00% (up-down or down-up) or 3.00% (down-down). Probabilities are 0.5 on each branch and rates are semiannually compounded. A call expires at one year on a 1.5-year zero-coupon bond (face 100) with strike 97.50; at expiry the bond is a six-month zero worth 100/(1 + r/2). The call's price is closest to:

  1. 0.34Why not: Gives each of the three terminal nodes weight one third, ignoring that the middle node is reached by two of the four equally likely paths.
  2. 0.21Why not: Prices a put (payoffs reversed) instead of a call.
  3. 0.27 - correct
  4. 0.53Why not: Reports the option value at the down node after six months instead of rolling back to the root.

Bond values at one year: 96.62, 97.56, 98.52; payoffs 0, 0.06, 1.02. Six-month nodes: up 0.030, down 0.531. Root: 0.5x(0.030+0.531)/1.025 = 0.27.

The hardest tier

The tree has a six-month rate of 5.00% today, 6.00% or 4.00% after six months, and 7.00%, 5.00% or 3.00% after one year (0.5 probability on each branch, semiannual compounding). An American put on a 1.5-year zero-coupon bond (face 100) has strike 97.00 and can be exercised at six months or one year. The zero is worth 94.26 at the six-month up node, 96.12 at the six-month down node, and 100/(1 + r/2) at one year. The American put's price is closest to:

  1. 1.77 - correct
  2. 0.09Why not: Prices the put as European, ignoring the value of exercising early at the six-month nodes.
  3. 1.34Why not: Exercises early at the up node but ignores early exercise at the down node, where exercise value 0.88 exceeds the continuation value of 0.
  4. 2.74Why not: Reports the up-node exercise value as the price, skipping the down node, the probability weights and the discounting.

Exercise values at six months: up 2.74, down 0.88. Continuation values: up 0.185, down 0.000. Optimal values: 2.74 and 0.88. Root: 0.5 x (2.74+0.88)/1.025 = 1.77 (European: 0.09).

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