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Obtain the forward price from an asset's spot price, and outline the arbitrage reasoning that links the two prices.

FRM® Part I · Financial Markets and Products · Pricing Financial Forwards and Futures · FRM1.FM.10.d

The idea

For an asset with no income or carrying costs, the forward price comes from replication. With continuous compounding, F0 = S0 × e^(rT). Here S0 is the spot price, r the continuously compounded risk-free rate and T the time to delivery in years. If the market forward price is above this value, an arbitrageur borrows at r, buys the asset and sells the forward. At maturity the asset is delivered at the forward price, the loan is repaid and the difference is kept, a trade called cash and carry. If the market forward price is below it, the arbitrageur short sells the asset, lends the proceeds and buys the forward, a trade called reverse cash and carry. Income would lower the forward price, and storage or other carrying costs would raise it. Either trade needs no initial outlay and carries no risk, which is why arbitrage enforces the price. Check the compounding convention in the question. With annual compounding, F0 = S0 × (1 + r)^T. Express T in years, and compound the spot price forward rather than discounting it.

Formulas

Forward price of a non-income asset, continuous compounding

F0 = S0 × e^(r × T)

S0
the spot price today
r
the continuously compounded risk-free rate, per year
T
the time to delivery, in years

Holds by cash-and-carry and reverse cash-and-carry arbitrage when the asset pays no income and costs nothing to hold. Compound spot forward, do not discount it, and put T in years.

Forward price of a non-income asset, annual compounding

F0 = S0 × (1 + r)^(T)

S0
the spot price today
r
the risk-free rate, annually compounded
T
the time to delivery, in years

Use only when the question gives an annually compounded rate; mixing it with the exponential form gives a slightly wrong price.

A practice question

From the question bank, with the reason each wrong option is wrong.

For an asset that pays no income, with continuously compounded risk-free rate r and time T to delivery, the forward price in terms of the spot price S is:

  1. S x e^(-rT)Why not: Discounts the spot price instead of compounding it forward.
  2. The expected future spot priceWhy not: Confuses the forward price with a forecast; replication, not expectation, sets the price.
  3. S x e^(rT) - correct
  4. S x e^(rT) - SWhy not: Gives only the financing cost, not the price.

Buying the asset today and holding it costs S financed at r, so the delivery price must be S x e^(rT).

A harder one

A stock with spot price USD 50 pays no dividends. The risk-free rate is 6% continuously compounded and the one-year forward price in the market is USD 55. An arbitrageur uses cash and carry. The profit at maturity per share is closest to:

  1. USD 1.80Why not: Reports the present value of the profit, not the amount received at maturity.
  2. USD 1.91 - correct
  3. USD 5.00Why not: Takes forward minus spot and ignores the cost of financing the purchase.
  4. USD 7.91Why not: Compares the forward with spot discounted at 6% rather than compounded.

Theoretical forward = 50 x e^0.06 = 53.09. Profit at maturity = 55 - 53.09 = USD 1.91.

The hardest tier

A non-income asset has spot price USD 100. The risk-free rate is 10% per year with annual compounding. The three-year forward price is closest to:

  1. USD 75.13Why not: Discounts the spot price instead of compounding it forward.
  2. USD 130.00Why not: Uses simple interest of 10% for three years.
  3. USD 133.10 - correct
  4. USD 134.99Why not: Applies continuous compounding e^(0.10 x 3) although the rate is stated with annual compounding.

With annual compounding F = S x (1 + r)^T = 100 x 1.1³ = USD 133.10.

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