CFA Level I · CFA Level I Exam

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives: formula sheet

Full chapter guide

Key formulas

Law of one price: Same payoffs in every state ⇒ same price today; If two portfolios have identical cash flows, any price difference is an arbitrage opportunity.
Replication (derivative price): Price of derivative = Price of replicating portfolio; The replicating portfolio uses the underlying plus risk-free borrowing or lending.
Type 1 arbitrage (general finance usage): Net cash flow today > 0 and net cash flow later = 0 in all states; Riskless profit received up front, with no future liability. A general label, not an official CFA term.
Type 2 arbitrage (general finance usage): Net cash flow today = 0 and later cash flow ≥ 0 in all states, > 0 in some; No cost, no chance of loss, some chance of gain. A general label, not an official CFA term.
Risk-free rate and no-arbitrage: A riskless position with zero investment must earn 0 return; If it earns more, that is arbitrage. A hedged position earns the risk-free rate.
Forward price (no income or carry cost): F0(T) = S0 × (1 + r)^T; Comes from replicating a long forward with the asset bought using borrowed money.
Replicated long forward: Long forward = Long underlying + Borrow PV of F0(T); A short forward is the reverse: short the underlying and lend.
Up and down factors: u = S1(up) ÷ S0 ; d = S1(down) ÷ S0; Used in the one-period binomial tree.
Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d); The down probability is 1 − π. Here r is the per-period risk-free rate and there is no income on the underlying.
Hedge ratio (delta): h = (Vu − Vd) ÷ (Su − Sd); Units of underlying held to replicate the option. Vu and Vd are option values in the up and down states.
Option value by replication: V0 = h × S0 + B, where B = (Vu − h × Su) ÷ (1 + r); B is the bond position. A negative B means borrowing.
Option value by risk-neutral pricing: V0 = [π × Vu + (1 − π) × Vd] ÷ (1 + r); Gives the same value as replication.
Forward price, general (PV form): F₀ = (S₀ + PVC − PVB) × (1 + r)^T; PVC = present value of carry costs (e.g. storage); PVB = present value of carry benefits (e.g. dividends, convenience yield). Use the same r and T throughout.
Forward price, future value form: F₀ = S₀ × (1 + r)^T + FVC − FVB; FVC and FVB are values at the contract expiry date. Use this when costs or benefits are already given at expiry.
Forward price, no carry: F₀ = S₀ × (1 + r)^T; Applies when the underlying pays no income and has no holding costs.
Continuous compounding with yield: F₀ = S₀ × e^((r − γ) × T); This is the standard curriculum form. r is the continuously compounded risk-free rate and γ is the continuously compounded benefit yield (dividend yield or convenience yield). If a storage cost rate c is also given as a continuous rate, treat it as a negative benefit, so the net yield is γ − c and the exponent becomes (r − γ + c) × T. Use when the question states continuous rates.
Value of carry sign rule: Costs ↑ → F₀ ↑; Benefits ↑ → F₀ ↓; Use to check direction and to eliminate wrong options.
Forward price, no cash flows (discrete compounding): F0 = S0 × (1 + r)^T; r is the annual risk-free rate, T is in years. Reasoning: the cost of buying the asset and financing it to T.
Forward price with income and carry costs: F0 = (S0 − PV(benefits) + PV(costs)) × (1 + r)^T; Discount each benefit or cost from its payment date back to today at the risk-free rate before adjusting spot.
Forward price, continuous compounding: F0 = S0 × e^(rT); With a continuous yield q on the asset: F0 = S0 × e^((r − q)T). With a continuous carry cost θ, use r + θ − q.
Value of a long forward at time t (no cash flows): Vt = St − F0 ÷ (1 + r)^(T − t); Value of the short is −Vt. Continuous version: Vt = St − F0 × e^(−r(T − t)).
Value of a long forward at time t (with income or costs): Vt = (St − PV(benefits) + PV(costs)) − F0 ÷ (1 + r)^(T − t); PV here covers only the benefits and costs still to come between t and T.
Value at expiration: VT = ST − F0; For the long. The short's value is F0 − ST.
Value at initiation: V0 = 0; This is why the forward price is set as it is.

Quick revision

Arbitrage: riskless profit with no net investment.
Law of one price: identical payoffs must have the same price.
Arbitrage keeps prices in line, so derivative prices are set by no-arbitrage, not by expectations.
Replication: underlying plus risk-free borrowing or lending reproduces the derivative payoff.
Risk-neutral probabilities are used for pricing only; they are not real-world forecasts.
Forward price (no carry): F0 = S0 × (1 + r)^T.
Benefits of holding the underlying (dividends, interest, convenience yield) lower the forward price.
Costs of holding (storage, insurance) raise the forward price.
With carry: F0 = (S0 − PV of benefits + PV of costs) × (1 + r)^T.
Continuous form: F0 = S0 × e^((r − q) × T) when the benefit is a continuous yield q.
A forward has zero value at initiation, because the forward price is set so that neither side pays.
Value of a long forward at time t: Vt = [Ft(T) − F0(T)] ÷ (1 + r)^(T − t), the difference in forward prices discounted over the remaining time to expiry; it can be positive or negative.

Common mistakes

Calling any profitable trade an arbitrage. Fix: Check for risk. If the outcome depends on a price move, it is speculation, not arbitrage.
Treating the type 1 and type 2 labels as exam terms. Fix: Learn the core definition: a riskless profit with no net investment. Use the labels only as background, because they are not CFA curriculum terms and you should not expect them in questions.
Using real-world probabilities to price the derivative. Fix: Real-world probabilities are a distractor for pricing. Use π from (1 + r − d) ÷ (u − d).
Forgetting to discount the expected payoff at the risk-free rate. Fix: Always divide by (1 + r) per period. The discount rate is the risk-free rate, not the required return on the underlying.
Adding dividends instead of subtracting them Fix: The forward buyer misses the dividends, so the forward price is lower. Always subtract benefits and add costs.
Mixing PV and FV amounts Fix: Put every item on one date. Discount dividends to today before subtracting inside the bracket, or compound them to expiry (FV) before subtracting after compounding the spot.
Treating the forward price and the forward value as the same thing. Fix: The forward price F0 is the fixed delivery price. The forward value Vt is the contract's current worth and is zero at initiation. Ask what the question wants before you calculate.
Using Vt = St − F0 before expiration. Fix: Before expiry, discount F0 over T − t at the risk-free rate. Only at T does the discounting vanish.

Exam tips

Questions often give a price and ask what trade captures the arbitrage. Decide buy low, sell high first, then match the option.
Know the arbitrage definition: a riskless profit with no net investment. The type 1 and type 2 labels are not CFA terms, so do not expect them in questions.
Remember that arbitrage trades push prices back to the no-arbitrage level, which links to market efficiency questions.
With no penalty for wrong answers, never leave a question blank. Eliminate options that buy the expensive asset to narrow to one.
If a question gives a real-world up probability and asks for a derivative price, ignore it for pricing. It is usually a trap.
Know both routes. If a question asks for the hedge ratio or the borrowing amount, use replication. If it asks only for value, use risk-neutral pricing.
For forwards, remember the replication in words: long forward equals long asset plus borrowing the present value of the forward price.
Sanity-check every answer. A call cannot be worth more than the underlying, and a value must not be negative.