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Show how the binomial model is modified to price options on futures, currencies, stock indices and stocks that pay dividends.

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

The idea

The tree is adjusted only through the growth factor a, which needs d < a < u. The up probability, between 0 and 1, is p = (a − d) ÷ (u − d). Discounting is always at the risk-free rate r. For a stock paying no dividends, a = e^(r × Δt). With a continuous dividend yield q, the growth factor for a stock or index becomes a = e^((r − q) × Δt). A currency behaves in the same way, because the foreign interest rate rf acts as the yield. For a currency option, with r the domestic rate and rf the foreign rate, the growth factor is a = e^((r − rf) × Δt). A futures contract costs nothing to enter and has no expected growth in a risk-neutral world. So for an option on a futures price, the growth factor is 1. The probability then becomes p = (1 − d) ÷ (u − d). Frequent errors are discounting at r − q or at the foreign rate, and using the plain stock formula for futures or currencies.

Formulas

Risk-neutral up probability with a general growth factor

p = (a − d) ÷ (u − d)

a
the growth factor for one step: e^(r × Δt) for a stock with no dividends, e^((r − q) × Δt) with a yield, e^((r − rf) × Δt) for a currency, 1 for a futures price
u
the up factor
d
the down factor

The growth factor must lie between d and u. Whatever a is, the expected payoff is discounted at the risk-free rate r.

Binomial growth factor for an underlying with a continuous yield

a = e^((r − q) × Δt)

r
the continuously compounded risk-free rate
q
the continuous dividend yield, or the foreign rate rf for a currency
Δt
the step length in years

Used for indices, stocks with a dividend yield and currencies. Discounting is still at r, not r − q.

A practice question

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

In a binomial tree for an option on a futures price, what is the risk-neutral probability of an up move?

  1. p = (e^(r x dt) - d) / (u - d), as for a non-dividend stockWhy not: Applies the spot-stock growth factor, but a futures price does not grow at the risk-free rate.
  2. p = (1 - d) / (u - d), because the futures price has no expected growth in a risk-neutral world - correct
  3. p = (e^(-r x dt) - d) / (u - d), using the discount factor as the growth factorWhy not: Uses the discount factor where a growth factor belongs.
  4. p = 0.5, because futures prices are unbiased predictorsWhy not: Confuses zero drift with equal probabilities; p still depends on u and d.

A futures contract costs nothing to enter, so its price has zero expected growth under the risk-neutral measure: the growth factor is a = 1 and p = (1 - d) / (u - d). Discounting is still at the risk-free rate.

A harder one

A futures price is 60 and will be either 69 or 54 in one year. The risk-free rate is 8% continuously compounded. What is the value of a one-year European call on the futures with strike 58, using a one-step tree?

  1. 4.06 - correct
  2. 4.40Why not: Computes the correct risk-neutral expected payoff but does not discount it.
  3. 7.44Why not: Uses e^(r x dt) in the probability, as for a stock, so the up probability is too high.
  4. 6.09Why not: Applies the probability of a down move to the up-state payoff.

For futures a = 1, so p = (1 - 0.90) / (1.15 - 0.90) = 0.40. Only the up node pays 69 - 58 = 11. Call = e^-0.08 x 0.40 x 11 = 4.06.

The hardest tier

A stock is priced at 100 and pays a continuous dividend yield of 8%. In each of two six-month steps it rises 20% or falls 15%. The risk-free rate is 3% continuously compounded. What is the value of a one-year American call with strike 95?

  1. 9.22Why not: Gives the European value, missing that early exercise at the up node is optimal when the yield is high.
  2. 10.38 - correct
  3. 13.97Why not: Ignores the dividend yield in the growth factor, which overstates p and the option value.
  4. 10.87Why not: Builds p correctly but discounts at r - q instead of at the risk-free rate.

a = e^(-0.025) so p = 0.3580. At the up node (120) holding is worth 21.71 but exercising gives 25.00, so the node is 25.00. The root is e^-0.015 x (0.3580 x 25 + 0.6420 x 3.00) = 10.38, against 9.22 for the European call.

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