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Value American and European options, both calls and puts, with binomial models of one and of two steps.

FRM® Part I · Valuation and Risk Models · Binomial Trees · FRM1.VR.14.a

The idea

A binomial tree prices an option by working backward from expiry. Each step the stock moves up by u or down by d, and no-arbitrage puts the one-step growth between d and u, so 0 < p < 1, where the risk-neutral up probability is p = (e^(r × Δt) − d) ÷ (u − d). Here r is the continuously compounded risk-free rate and Δt the step length. At expiry each node is worth its payoff: max(S − K, 0) for a call and max(K − S, 0) for a put. One step earlier, a node is the discounted risk-neutral average of the next two, Vup and Vdown: V = e^(−r × Δt) × [p × Vup + (1 − p) × Vdown]. Repeat at every step, so two steps mean two discounts. For an American option, at every node, including the first, compare continuation with exercise and keep the larger. Common errors are using 0.5 for p and skipping a discount. An American put can be worth more than a European put. An American call on a non-dividend-paying stock equals a European call when rates are not negative.

Formulas

Risk-neutral up probability in a binomial tree, stock with no dividends

p = (e^(r × Δt) − d) ÷ (u − d)

r
the continuously compounded risk-free rate
Δt
the length of one step in years
u
the up factor
d
the down factor

No-arbitrage needs d below the one-step growth and u above it, so p lies between 0 and 1. It is a pricing weight, not a forecast; 0.5 is not a substitute.

Value at a node of a binomial tree

V = e^(−r × Δt) × [p × Vup + (1 − p) × Vdown]

r
the continuously compounded risk-free rate
Δt
the length of one step in years
p
the risk-neutral up probability
Vup
the option value at the up node one step on
Vdown
the option value at the down node one step on

Applied step by step from expiry, so two steps mean two discounts. For an American option, take the larger of V and the exercise value at every node.

A practice question

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

A stock is priced at 50. In one year it will be either 60 or 40. The risk-free rate is 4% per year, continuously compounded. What is the value today of a one-year European call option with strike 52?

  1. 4.63 - correct
  2. 3.84Why not: Uses 0.5 for each branch instead of the risk-neutral probability, then discounts.
  3. 4.82Why not: Weights the payoff with the correct risk-neutral probability but forgets to discount it back one year.
  4. 3.06Why not: Applies the risk-neutral probability of a down move, 1 - p, to the up-state payoff.

p = (e^0.04 - 0.8) / (1.2 - 0.8) = 0.6020. Only the up node pays 60 - 52 = 8, so the call = e^-0.04 x 0.6020 x 8 = 4.63. The probability is risk-neutral, not the real-world chance of a rise.

A harder one

A stock is priced at 40 and moves up by 10% or down by 10% in each of two successive six-month steps. The risk-free rate is 5% continuously compounded. What is the value of a one-year European put with strike 42?

  1. 2.72Why not: Allows early exercise at the down node, which gives the American put value rather than the European one.
  2. 2.34 - correct
  3. 2.46Why not: Takes the risk-neutral expected terminal payoff but does not discount it at all.
  4. 2.40Why not: Discounts the expected terminal payoff for only one six-month step instead of two.

p = (e^0.025 - 0.9) / 0.2 = 0.6266. Terminal puts are 0 at 48.4, 42 - 39.6 = 2.4 at 39.6 and 42 - 32.4 = 9.6 at 32.4. Expected payoff = 2.4618, discounted over a full year at e^-0.05 gives 2.34.

The hardest tier

A stock is priced at 50 and moves up 20% or down 20% in each of two annual steps. The risk-free rate is 5% continuously compounded. What is the value of an American put with strike 52 and two years to expiry?

  1. 4.19Why not: Gives the European put value because it never tests the down node for early exercise.
  2. 5.35Why not: Handles the early exercise correctly but does not discount the root value back one step.
  3. 5.09 - correct
  4. 6.38Why not: Uses 0.5 for the up and down probabilities instead of the risk-neutral probability.

p = (e^0.05 - 0.8) / 0.4 = 0.6282. Down node (40): holding is worth 9.46 but exercising gives 52 - 40 = 12, so the node is 12.00. Up node: 1.41. Root = e^-0.05 x (0.6282 x 1.41 + 0.3718 x 12) = 5.09, against 4.19 for the European.

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