Financial Derivatives – Principles of Pricing Forwards and Futures

32 question-and-answer flashcards covering key definitions, formulas, and arbitrage principles for pricing forwards, futures, interest-rate parity, valuation, cost of carry, and related concepts from the lecture notes.

What are Treasury rates?

The returns earned on Treasury bills and bonds issued by a government in its own currency; regarded as default-risk-free.

Why are Treasury securities considered ‘default-risk’ free?

Investors are virtually certain that both interest and principal will be paid by the issuing government.

What is LIBOR?

The London Interbank Offered Rate—the rate at which a bank is willing to make a large wholesale deposit with other AA-rated banks.

Why do derivatives traders often use LIBOR rather than Treasury rates as the risk-free rate?

Because LIBOR reflects the short-term opportunity cost of capital for AA-rated banks and is considered a better proxy for borrowing/lending in derivatives markets.

What are repo rates?

Rates that apply to repurchase agreements, where securities are sold with an agreement to buy them back; effectively a secured short-term borrowing rate.

What is continuous compounding?

The limiting case of compounding interest infinitely often; an amount S grows to Se^{rT} after time T at rate r.

How does $100 grow under continuous compounding at rate r for time T?

To $100·e^{rT}.

Give the PV formula with continuous compounding.

Present Value = Future Value · e^{-rT}.

Distinguish between investment and consumption assets.

Investment assets are held primarily for investment (e.g., shares, bonds, gold); consumption assets are held primarily for use/consumption (e.g., copper, oil, grains).

Why is the investment/consumption distinction important for forwards pricing?

Arbitrage pricing arguments reliably determine forward prices for investment assets but not always for consumption assets due to potential convenience yields.

List the three key conditions for an arbitrage opportunity.

1) Actual price differs from theoretical price, 2) Ability to enter simultaneous offsetting trades, 3) Resulting position is risk-free and yields a profit.

Define the notation S0, F0, T, and r used in forward pricing.

S0 = spot price today; F0 = forward/futures price today; T = time to delivery; r = risk-free interest rate for maturity T.

What is the forward price formula for an investment asset with no income or storage costs (continuous compounding)?

F0 = S0 · e^{rT}.

What arbitrage strategy is used when F0 > S0·e^{rT}?

Buy the asset in the spot market and short the forward to lock in a risk-free profit.

What arbitrage strategy is used when F0 < S0·e^{rT}?

Short-sell the asset and go long the forward contract to lock in a risk-free profit.

Give the forward price formula when the asset provides a known cash income stream I.

F0 = (S0 – I) · e^{rT}, where I is the present value of the income.

Give the forward price formula when the asset provides a known dividend yield q.

F0 = S0 · e^{(r – q)T}.

What is the cost-of-carry (c) in futures pricing?

c = r – q + u – y, where r is risk-free rate, q dividend yield, u storage-cost percentage, and y convenience yield.

State the pricing formula for a foreign currency forward (domestic currency quoted per unit of foreign).

F0 = S0 · e^{(r – rf)T}, where r is domestic risk-free rate and rf is foreign risk-free rate.

What does Interest Rate Parity (IRP) imply?

The forward/spot rate differential equals the interest-rate differential; otherwise arbitrage will occur to restore parity.

Under IRP, what happens when the foreign rate rf > domestic rate r?

The forward price of the foreign currency will be lower than the spot price (forward trades at a discount).

Under IRP, what happens when rf < r?

The forward price of the foreign currency will be higher than the spot price (forward trades at a premium).

At initiation, what is the value of a forward contract?

Zero—no money changes hands and the delivery price equals the theoretical forward price.

Give the value formula for a long forward contract entered earlier (using F0 and K).

f = (F0 – K) · e^{-rT}.

Give the alternative value formula for a long forward (using spot price).

f = S0 – K·e^{-rT}.

How is the value of a short forward position calculated?

f = K·e^{-rT} – S0 (or f = (K – F0)·e^{-rT}).

What is the basic storage-cost adjusted forward pricing formula using dollar costs U?

F0 = (S0 + U) · e^{rT}, where U is the PV of storage costs minus any income.

What is the alternative storage-cost formula using a percentage cost u?

F0 = S0 · e^{(r + u)T}, where u is storage cost as a percentage of asset value.

Define ‘convenience yield’.

An implicit benefit of physically holding a commodity, lowering the effective cost of carry and thus the forward price.

When do index arbitrageurs sell futures and buy stocks?

When F0 > S0 · e^{(r – q)T} (futures overpriced relative to the index).

When do index arbitrageurs buy futures and short stocks?

When F0 < S0 · e^{(r – q)T} (futures underpriced).

Why is LIBOR considered the short-term opportunity cost for AA-rated banks?

Because it represents the rate at which these banks can lend or borrow large sums in the interbank market, reflecting their marginal funding cost.

What is meant by ‘futures are marked to market’ and how does that contrast with forwards?

Futures are re-priced daily with gains/losses settled each day, whereas forwards are not marked to market and gains/losses accrue until maturity.