FINM6041 · Applied Derivatives
Options on Indices, Currencies & Futures
This chapter merges Lecture 07 (options on indices and currencies) and Lecture 08 (options on futures) into one idea worth mastering: almost every option-pricing result you have met carries over to a new underlying by a single substitution. For any asset paying a continuous yield q, value a European option by replacing S₀ with S₀e−qT and then treating it as an ordinary non-dividend problem. The genius is that q wears three hats — for a stock index it is the index dividend yield, for a currency it is the foreign risk-free rate r_f, and for a futures contract it is q = r (a futures price has zero risk-neutral drift). That last case collapses Black-Scholes into Black's model, c = e−rT[F₀N(d₁) − XN(d₂)], the standard tool for European options on futures. Alongside the pricing you cover the put-call parity variants, the binomial risk-neutral probabilities for each underlying, portfolio insurance with index puts, and the multiplier trap the exam loves — the ASX 200 index option (XJO) is A$10 per point while the SPI 200 futures is A$25 per point.
What this chapter covers
- 01The master rule: value any option on a yield-paying asset by replacing S₀ with S₀e^{−qT}, then treat it as non-dividend
- 02The three faces of q — index dividend yield, currency foreign rate r_f, and futures q = r
- 03Put-call parity variants: index c + Xe^{−rT} = p + S₀e^{−qT}; currency uses r_f; futures uses F₀e^{−rT}
- 04Black's model for European options on futures: c = e^{−rT}[F₀N(d₁) − XN(d₂)], d₁ = [ln(F₀/X) + σ²T/2]/(σ√T)
- 05Binomial risk-neutral probabilities: p = (e^{(r−q)Δt} − d)/(u − d) for a yield; p = (1 − d)/(u − d) for a futures
- 06Contract multipliers you must not swap: ASX 200 index option (XJO) A$10/point vs SPI 200 futures A$25/point
- 07Portfolio insurance with index puts: number of puts = β · (portfolio value)/(index level × multiplier), a protective-put floor
- 08Why a futures call and a spot call diverge before maturity (contango vs backwardation)
Pricing a European call on a futures with Black's model
- 1 markRecognise the underlying is a futures price, so set q = r and use Black's model: c = e^{−rT}[F₀N(d₁) − XN(d₂)] with d₁ = [ln(F₀/X) + σ²T/2]/(σ√T), d₂ = d₁ − σ√T.
- 2 marksCompute d₁: ln(75/72) = 0.040822; σ²T/2 = 0.0784 × 0.5/2 = 0.0196; numerator = 0.060422. Denominator σ√T = 0.28 × 0.70711 = 0.197990. d₁ = 0.060422/0.197990 = 0.3052.
- 1 markCompute d₂ = d₁ − σ√T = 0.3052 − 0.197990 = 0.1072.
- 2 marksRead the cumulative normal values: N(d₁) = N(0.3052) ≈ 0.6199 and N(d₂) = N(0.1072) ≈ 0.5427.
- 3 marksAssemble the bracket: F₀N(d₁) − XN(d₂) = 75 × 0.6199 − 72 × 0.5427 = 46.4925 − 39.0744 = 7.4181. Discount the whole bracket: c = e^{−0.025} × 7.4181 = 0.97531 × 7.4181.
- 1 markFinal value: c = $7.24. Sanity check — the call is in the money (F₀ > X) and the price exceeds the discounted intrinsic value e^{−rT}(F₀ − X) = 0.97531 × 3 = $2.93, as it must once time value is added.
Key terms
- q-substitution (master rule)
- The unifying trick of this chapter: to value a European option on any asset paying a continuous yield q, replace S₀ with S₀e^{−qT} everywhere and use d₁ = [ln(S₀/X) + (r − q + σ²/2)T]/(σ√T). One skeleton then covers indices, currencies and futures.
- Black's model
- The pricing formula for European options on futures, obtained by setting q = r in the yield-adjusted Black-Scholes: c = e^{−rT}[F₀N(d₁) − XN(d₂)], p = e^{−rT}[XN(−d₂) − F₀N(−d₁)], with d₁ = [ln(F₀/X) + σ²T/2]/(σ√T). A futures price has zero risk-neutral drift, which is why q = r.
- Currency option / foreign rate r_f
- An option on a foreign currency is priced with the master rule where q = r_f, the foreign risk-free rate: parity is c + Xe^{−rT} = p + S₀e^{−r_f·T}. The foreign currency behaves like an asset yielding its own interest rate.
- Index option multiplier
- The dollar value of one index point for a cash-settled index option. The ASX 200 index option (XJO) uses A$10 per point — distinct from, and frequently confused with, the SPI 200 futures multiplier.
- SPI 200 futures multiplier
- The ASX SPI 200 index futures contract is worth A$25 per index point (the Mini SPI is A$5). The exam deliberately sets a distractor that swaps this A$25 futures figure with the A$10 index-option figure.
- Portfolio insurance (protective put)
- Buying index puts to put a floor under a portfolio's value without selling the assets. The number of puts = β · (portfolio value)/(index level × multiplier); as beta rises, both the number of puts and the required strike rise, so insurance costs more.
Options on Indices, Currencies & Futures FAQ
Does FINM6041 use Black's model?
Yes. Black's model is the Lecture 08 tool for pricing European options on futures: c = e^{−rT}[F₀N(d₁) − XN(d₂)], with d₁ = [ln(F₀/X) + σ²T/2]/(σ√T). It appears directly in the 2022 sample paper as a short-answer calculation, and it is really just Black-Scholes with the substitution q = r. Learn to derive it from the master rule rather than memorising it in isolation.
What is the difference between the XJO and SPI 200 multipliers in FINM6041?
They belong to different products. The ASX 200 index option (XJO) is worth A$10 per index point; the ASX SPI 200 index futures is worth A$25 per index point. The concept-MCQ block plants a distractor that swaps these two, so tie each multiplier to its product in your notes: A$10 for the option, A$25 for the futures.
How do you price an option on a stock index?
Use the master rule: replace S₀ with S₀e^{−qT}, where q is the index dividend yield, and price it like a non-dividend Black-Scholes option. The dividend yield lowers the effective spot, so an index call is worth a little less and an index put a little more than the same option on a non-dividend asset.
Is portfolio insurance examinable in FINM6041?
Yes. Portfolio insurance is the protective-put drill from Lecture 07 (and extended to beta ≠ 1 in Workshop 07): buy long index puts, with the number of puts = β · (portfolio value)/(index level × multiplier). Be ready to state the position, count the contracts, find the strike that sets the floor, and show the put payoff offsets the portfolio loss.
How is a foreign-currency option priced differently?
Exactly the same master rule, but q becomes the foreign risk-free rate r_f. Parity reads c + Xe^{−rT} = p + S₀e^{−r_f·T}, and Black-Scholes uses d₁ = [ln(S₀/X) + (r − r_f + σ²/2)T]/(σ√T). A foreign currency is treated as an asset that pays a continuous yield equal to its own interest rate.
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Exam move
Do not learn five separate formulas — learn one substitution and its three disguises. Write the master rule at the top of a page (S₀ → S₀e^{−qT}, and in d₁ use r − q + σ²/2), then underneath it list what q means for each underlying: index dividend yield, currency r_f, and futures q = r. From that single line you can regenerate the index/currency/futures parity variants, the binomial probabilities, and Black's model on the spot, which matters because the FINM6041 formula sheet gives you the pieces but not the mapping. Drill the futures case until c = e^{−rT}[F₀N(d₁) − XN(d₂)] is automatic, discounting the whole bracket. Finally, memorise the two multipliers as a matched pair — A$10 index option, A$25 SPI futures — because that one-line distinction is a near-guaranteed concept mark that many students throw away.
