Learning Module 4 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

A non-dividend-paying stock trades at spot price S0 = 120. The annual risk-free rate (discrete) is 4% and a one-year forward on the stock trades at 124.80. Is there an arbitrage opportunity, and if so, what arbitrage trade and profit per share would realize? Assume no transactions costs.

A commodity spot price S0 = 50 has a known storage cost of 2 payable at T = 0.5 years. Risk-free rate r = 3% annual (discrete). Compute the no-arbitrage forward price F0(0.5).

You observe EUR/JPY spot S0 = 130.50 (JPY per EUR). The one-year EUR risk-free rate = -0.2% and the one-year JPY rate = 0.5% (continuous compounding). What is the one-year forward EUR/JPY no-arbitrage rate (continuous)?

A two-period zero rate z1 = 2% (one year), z2 = 3% (two years). Compute the implied one-year forward rate one year from now (IFR1,1) using discrete compounding.

An asset has S0 = 200. The continuous risk-free rate r = 1% p.a. There is a continuous dividend yield i = 3% p.a. for T = 0.5 years. What is the continuous-compounding forward price F0(T)?

You can borrow in USD at 2% and in EUR at 0.5% (both continuous). Spot EUR/USD = 1.10 (USD per EUR). Using continuous rates, what is the six-month EUR/USD forward rate?

Assume a commodity S0 = 300, storage cost payable at time T of 6, and convenience yield negligible. Risk-free rate r = 3% (annual discrete). For T = 1 year, compute F0(T).

Investor A can borrow at repo rate 1.5% (annual) and lend at 1.4% unsecured. S0 = 100, risk-free (repo) r = 1.5%. One-year forward F_obs = 102.00. Is arbitrage possible ignoring other frictions? If so, what trade? Use discrete compounding.

A 6-month equity index forward has price F0(T) = 10200. Spot S0 = 10000. The continuous risk-free rate r = 2% and continuous dividend yield of index i = 0.5%. Check if forward price satisfies continuous no-arbitrage. Compute theoretical F0(T).

A zero-coupon bond paying 100 in two years trades at 94.0. What is the two-year zero rate z2 (annual discrete)?

A trader sees a one-year forward on gold at 1,800 with spot 1,760. The one-year risk-free rate is 2% and storage cost PV is 5. Is there arbitrage? If yes, which trade yields profit?

A bond has a face value 100, annual coupon 4, price 98. If one-year zero z1 = 1.5%, solve for two-year zero z2 given price and coupons (discrete).

Spot S0 = 80, forward F0(T) for T=0.5 years is 82. You can borrow at 4% continuously. Is there an arbitrage? If so, what trade? Use continuous compounding.

An investor sees a three-month forward on an equity index with S0 = 5000, dividend yield annual i = 2%, and continuous risk-free rate r = 3%. Using continuous compounding, compute the 3-month forward.

An asset with no income has S0 = 250 and implied one-year forward F0 = 260. What is the implied discrete annual risk-free rate r used by the market (solve for r)?

A six-month forward on a dividend-paying stock has F0 = 52, S0 = 50, and a PV of expected dividend over six months equals 0.5. Risk-free rate (semi-annual discrete) is r = 1%. Does forward satisfy no-arbitrage? Compute theoretical F0.

You observe two identical zero-coupon bonds from same issuer: Bond A maturing at T pays 100 and trades at 97.5; Bond B with same maturity trades at 97.0. What arbitrage exists?

A forward commitment to buy 100 shares at F0(T) is replicated by borrowing S0 and buying 100 shares. If S0 = 120, r = 3% p.a., T=1 year, show the replicating cash flows and confirm equality at T if F0(T) = S0(1 + r).

A forward on currency AUD/USD spot = 0.7500. AUD rate = 1% p.a., USD rate = 2% p.a. (both discrete). For a one-year forward, what is F (USD per AUD)?

A trader notices a one-year forward on a commodity in contango (F > S). Which combination of carry elements could explain this (select the best explanation)?

If the convenience yield on oil rises sharply due to low inventories, what is the likely direction of the oil forward curve (short-term) assuming other variables constant?

A one-year forward on a non-dividend paying equity is priced at 110, while S0 = 100 and risk-free r = 8% (discrete). Ignoring frictions, does arbitrage exist? If yes, indicate profit per share and trade.

Which of the following best describes why forward and futures prices could differ for the same underlying and maturity?

Given a spot S0 = 400 and continuous r = 3% with no income, what must F0(T) be for a six-month forward to preclude arbitrage under continuous compounding?

A trader wants to create a synthetic long forward on a non-dividend stock using spot and bonds. Which pair of trades at t=0 replicates a long forward with delivery at T?

A spot gold price is $1,900/oz, no storage, r = 1% (continuous). A 3-month futures contract trades at 1,905. Is this consistent with no-arbitrage? Compute theoretical continuous forward and conclude.

A three-month forward on GBP/USD is 1.3000 with spot 1.2800. If the USD 3-month rate is 0.5% and GBP 3-month rate is 0.1% (both discrete annualized), check if forward matches interest differential approximation. Compute implied forward via discrete formula.

A forward on a bond with known coupon payments must reflect which additional items when pricing the forward compared with a non-coupon asset? Choose best answer.

Which of the following is the correct discrete formula for forward price when the underlying pays known cash income amounts I_k at times t_k before T and has known costs C_j at times u_j before T?

A trader sees a mispriced forward: S0 = 50, PV(dividend to be paid at 0.25) = 1, r = 4% discrete, T = 0.5 year. The forward traded at 52.20. Compute theoretical forward and indicate whether selling the forward and buying spot financed is profitable.

A forward on USD/JPY is quoted as JPY per USD. Spot S0 = 110, USD rate = 0.5% discrete, JPY rate = -0.1% discrete, T = 1 year. Compute F0.

When storage costs increase materially for a commodity, what immediate effect should you expect on the forward term structure, all else equal?

A one-year forward on EUR/GBP is trading at 0.85 (GBP per EUR). Spot S0 = 0.84. Market-implied forward from rates is 0.842. What arbitrage exists ignoring frictions?

Assume S0 = 500, r = 3% (annual discrete), no carry. A two-year forward observed price is 530. Is this consistent? If not, what arbitrage trade yields profit? (Compute theoretical and indicate trade.)

Which of the following best describes the cost-of-carry expression in continuous form for an underlying with continuous income yield i and continuous storage cost c?

A forward contract has price F0(T) equal to PV(ST) discounted by risk-free rate. Which of the following statements is true for an asset with no additional cash flows?

A dealer hedges a client forward sale by entering an equal and opposite futures position on a liquid exchange. What practical effect does daily margining have on dealer exposure relative to the forward?

A forward contract on a currency has price F0 = 1.50 (price currency per base). If the market’s spot S0 = 1.46 and continuous interest differentials imply forward should be 1.455, what theoretical arbitrage trade would you execute ignoring frictions?

A forward contract on a physical commodity is priced noticeably below the spot (F < S). Which of the following is a plausible explanation?

A one-year forward on a non-dividend stock is priced at 104 while spot is 100 and risk-free rate is 4% (discrete). A market participant borrows at 4%, buys the stock, and sells forward. Which of the following describes the position's profit at maturity if ST = 110?

An investor compares a one-year forward priced at F0 = 150 on an index that pays no income, with an 1-year collateralized repo rate (used as r) of 3%. Spot S0 = X. Solve S0 if F0 = S0(1 + r).

A dealer who centrally clears OTC forwards must post margin to CCP. How does this development change theoretical pricing differences between forwards and futures?

If interest rates are constant over the life of a contract, what is the expected relationship between forward and futures prices for identical underlying and maturity?

Which funding rate is most appropriate to use when pricing a collateralized forward contract executed via repo financing?

A stock pays a discrete dividend of $2 in 3 months. S0 = 60, r = 2% per annum (discrete), T = 6 months. Compute the forward price F0(T).

Which statement correctly summarizes how the cost of carry affects the spot-forward relationship for equities, FX, bonds, and commodities?

If an investor expects ST to be substantially higher than F0(T), which forward strategy is optimal to profit from this view while keeping no initial cash outlay?

A stock index futures price converges to spot as contract approaches expiration. Which principle explains this behavior?

You observe a one-year forward on an index F0 = 2000, spot S0 = 1900, and PV(dividends in year) = 30. What implied discrete annual r is the market using (compute r)?

Which of the following best summarizes the primary reason a forward contract's value to either party can become positive or negative between inception and maturity?