FINM6041 · Applied Derivatives
Forward & Futures Pricing and Hedging
Forward & Futures Pricing and Hedging (Lecture 2) turns the mechanics of Chapter 1 into numbers. The engine is the cost-of-carry (no-arbitrage) model, and its skeleton is a single formula, F₀ = S₀e^{rT}, adapted to whatever the underlying happens to carry. Known cash income I subtracts as F₀ = (S₀ − I)e^{rT}; a continuous yield d gives F₀ = S₀e^{(r−d)T}; a currency uses the foreign rate, F₀ = S₀e^{(r−r_f)T}; storage costs add on. The course works in continuous compounding, so every quoted rate is first converted with r_c = n·ln(1 + r/n). If the observed forward price departs from the model, the arbitrage direction rule says which side to take: short the contract when it is dear, long it when it is cheap. The second half is hedging: basis = spot − futures, why basis risk makes real hedges imperfect, and the minimum-variance cross hedge h* = ρ·σ_S/σ_F with the optimal contract count N* = h*·(Q_A/Q_F).
What this chapter covers
- 01Continuous compounding and rate conversion: r_c = n·ln(1 + r/n) before any e^{rT}
- 02The cost-of-carry skeleton F₀ = S₀e^{rT} and its no-arbitrage justification
- 03The six variants: known income I, yield d, currency r_f, dollar storage U, percentage storage q, convenience yield y
- 04The arbitrage direction rule: F₀ too high → short the forward and buy the asset; too low → long the forward and short the asset
- 05Contract multipliers and applying the model to index and currency forwards
- 06Basis = spot − futures; strengthening vs weakening basis and the three sources of basis risk
- 07Long hedge vs short hedge, and choosing a delivery month just after the horizon
- 08Cross hedging: the minimum-variance hedge ratio h* = ρ·σ_S/σ_F and optimal contract number N* = h*·(Q_A/Q_F)
Pricing a currency forward and setting up the arbitrage
- 1 mark(a) A currency is an asset paying a continuous 'yield' equal to the foreign interest rate, so use F₀ = S₀·e^{(r − r_f)T} with r = 0.04, r_f = 0.03, T = 0.5.
- 1 mark(a) Exponent: (r − r_f)T = (0.04 − 0.03)×0.5 = 0.005.
- 2 marks(a) F₀ = 1.65·e^{0.005} = 1.65 × 1.0050125 = AUD 1.65827 per euro. Because the domestic rate exceeds the foreign rate, the forward exceeds spot — the euro trades at a forward premium.
- 1 mark(b) The dealer's AUD 1.68 is above the model value AUD 1.65827, so the forward is overpriced ⇒ sell the dear thing (short the forward) and buy the cheap thing (the euro today).
- 2 marks(b) Concretely: borrow AUD now, convert to euros at spot 1.65, invest the euros at the foreign rate 3.0% for 6 months, and short the 6-month forward at 1.68. At maturity deliver the euros into the forward, receive AUD 1.68 per euro, and repay the AUD loan.
Key terms
- Cost of carry
- The net cost of holding the underlying to delivery — financing at r, plus storage, minus any income or yield. The forward price is spot compounded forward at the net carry: F₀ = S₀e^{rT} in the simplest case.
- Continuous compounding
- The compounding convention the course uses throughout: FV = PV·e^{rT}. A rate quoted m times per year converts via r_c = m·ln(1 + r/m); doing this before any e^{rT} step is a common exam requirement.
- Arbitrage direction rule
- If the traded forward is above its cost-of-carry value, short the forward and borrow to buy-and-carry the asset; if it is below, long the forward and short-sell the asset. This tells you which side of the trade earns the riskless profit.
- Basis
- Basis = spot price of the asset hedged − futures price of the contract used. A hedge closed out before delivery locks in F₁ + b₂ (initial futures plus final basis), so uncertainty in the closing basis is the residual risk.
- Basis risk
- The risk that the basis moves unpredictably, making a hedge imperfect. It arises when the hedged asset differs from the futures underlying, when the transaction date is uncertain, or when the position must be closed before the delivery month.
- Minimum-variance hedge ratio
- The hedge ratio that minimises the variance of the hedged position: h* = ρ·(σ_S/σ_F), the slope of a regression of spot changes on futures changes. The optimal number of contracts is N* = h*·(Q_A/Q_F).
Forward & Futures Pricing and Hedging FAQ
Does FINM6041 use continuous compounding for forward pricing?
Yes — continuous compounding is the default throughout the course, so forward and futures prices are written as F₀ = S₀e^{rT} and its variants. Any rate quoted with periodic compounding must first be converted with r_c = m·ln(1 + r/m) before it goes into an e^{rT} formula. Exam questions deliberately quote a rate 'compounded quarterly' to check that you do this conversion.
What is the cost-of-carry model?
It is the no-arbitrage rule that the forward price equals the spot price carried forward at the net cost of holding the asset — financing plus storage, minus any income or yield. One skeleton, F₀ = S₀e^{rT}, adapts to six cases: known cash income subtracts as (S₀ − I)e^{rT}, a yield gives e^{(r−d)T}, a currency uses the foreign rate e^{(r−r_f)T}, and storage adds to carry. Recognising which case applies is the whole skill.
What's the difference between a forward's price and its value?
The forward price F₀ is the delivery price agreed today that makes the contract worth zero at inception — it is what pricing questions ask for. The value is what an existing forward is worth as time passes and the market moves away from the original delivery price; it starts at zero and drifts thereafter. Confusing the two is a flagged exam trap: pricing F₀ and valuing an existing contract are different tasks.
How do I know whether to long or short the forward in an arbitrage?
Compare the traded forward with its cost-of-carry value. If the forward is too high it is overpriced, so you sell it (go short the forward) and buy-and-carry the underlying by borrowing; if it is too low you buy the forward (go long) and short-sell the underlying, investing the proceeds. The rule is always sell the dear leg and buy the cheap leg.
What is basis risk and why does it stop a hedge being perfect?
Basis is spot minus futures. A hedge closed out before delivery locks in F₁ + b₂ — the initial futures price plus the final basis — so if the closing basis is uncertain, the hedged outcome is uncertain too. Basis risk arises when the hedged asset isn't exactly the futures underlying (a cross hedge), when the transaction date is unknown, or when you must close before the delivery month. It's why a hedge ratio of 1.0 is rarely optimal.
Studying with AI? Sia — free AI financial modeling tutor works through FINM6041 step by step.
Exam move
The exam move for Lecture 2 is 'one skeleton, six underlyings.' Memorise F₀ = S₀e^{rT} and drill the substitutions until they are reflex — subtract PV of income (S₀ − I)e^{rT}, use a yield e^{(r−d)T}, swap in the foreign rate for a currency e^{(r−r_f)T}, add storage. Always convert rates to continuous compounding first. Then rehearse the two hedging skills that recur every year: reading off the locked price F₁ + b₂ and computing a cross hedge with h* = ρ·σ_S/σ_F and N* = h*·(Q_A/Q_F). When a question says a forward is mispriced, state the arbitrage as a time-0 / time-T cash-flow table — the marks are for showing the riskless profit, not just naming the direction.
