11.2 Pricing Financial Forwards and Futures

Pricing Financial Forwards and Futures

FMP–10 builds the carry engine for financial assets: stocks, stock indices, currencies (as a special case), and bonds treated as investment assets. If you can borrow/lend at known rates and trade the underlying, the forward is not a free opinion—it is an arbitrage-enforced function of spot and carry.

Investment Assets Versus Consumption Assets

An investment asset is held primarily for investment by a significant number of investors (equities, gold for many holders, currencies). A consumption asset is held chiefly for use (crude oil by refiners, copper by manufacturers). Cash-and-carry arguments are tightest for investment assets because shorting and storage decisions are not blocked by consumption convenience. Commodity complications (storage, lease rates, convenience yield) are FMP–11; here we stay with financials.

Spot, Forward, and Futures: Roles

For FRM Part I pricing formulas, treat futures price ≈ forward price for financial underlyings unless a question emphasizes daily settlement and interest correlation. The economic story—carry arbitrage—is the same.

Short-Selling and Dividends

Short-selling an equity: borrow shares, sell them spot, later repurchase and return. While short, you owe the lender any dividends (you pay dividends). That cash outflow is a cost of being short—and a benefit to the long holder who receives dividends.

Implications for arbitrage:

• Cash-and-carry (to enforce an upper bound on F): buy spot (borrow cash), short/sell the forward. You receive dividends if long the stock, which reduces the net carry cost → lower fair F.
• Reverse cash-and-carry (to enforce a lower bound): short stock, lend proceeds, long the forward. You must pay dividends on the short, which reduces the advantage of shorting → again consistent with dividends lowering F.

If short-selling is restricted or dividends are uncertain, bounds widen; with known dividends and frictionless shorting, equality holds.

Forward Price and Arbitrage

For a non-dividend financial asset with continuous risk-free rate r:

F0 = S0 × e^(r × T)

Cash-and-carry: If quoted forward F_q > S0 × e^(rT), borrow S0, buy the asset, sell the forward. At T, deliver on the forward, receive F_q, repay debt S0 × e^(rT), pocket the difference.

Reverse cash-and-carry: If F_q < S0 × e^(rT), short the asset, lend proceeds, buy the forward. At T, take delivery via the forward (pay F_q), close the short, and keep the excess from lending.

Worked example: no income

S0 = $100, r = 5% continuous, T = 0.5 years.

Fair F0 = 100 × e^(0.05 × 0.5) = 100 × e^0.025 ≈ 100 × 1.025315 = $102.53

If the forward is quoted at $104, sell the forward, borrow $100, buy the stock. In six months debt ≈ $102.53; delivery proceeds $104; arbitrage profit ≈ $1.47 per share (before frictions).

Forward Price Versus Forward Value

• Forward price F0 (or Ft later): the delivery price that would make a newly initiated forward have zero value at that moment.
• Value of an existing forward with locked-in delivery price K:

For a long forward on a non-dividend asset:

Value ≈ S0 − K × e^(−rT) (or e^(−rT) × (F0 − K) with F0 = S0 e^(rT))

At initiation, set K = F0 ⇒ value = 0. Later, if the fair forward rises above K, the long forward has positive value; if it falls, the long has negative value (the short has the opposite).

Exam trap: saying “the forward price is $102 so the contract is worth $102.” The price is the strike that zeros value; the value is usually near zero when struck at market, and otherwise is the discounted gap between F and K.

Worked value example

One year ago a long forward was struck at K = $100. Now S0 = $108, r = 4% continuous, remaining maturity T = 0.25, no dividends. Current fair F0 = 108 × e^(0.04×0.25) ≈ 108 × 1.01005 ≈ $109.09. Long value ≈ e^(−0.01) × (109.09 − 100) ≈ 0.990 × 9.09 ≈ $9.00.

Income and Yield Cases

Known dollar dividends

If the asset pays known cash dividends during the life of the forward with present value I:

F0 = (S0 − I) × e^(r × T)

You only “finance” the prepaid forward value of the asset net of PV(dividends).

Continuous dividend yield q

For a stock index or equity with continuous yield q:

F0 = S0 × e^((r − q) × T)

Interpretation: holding the index earns yield q, which reduces net carry the same way a foreign risk-free rate does in FX (FX is the case q = r_foreign when S is domestic per foreign).

Known yield as discrete proportional income

If the asset provides a known yield with PV of income equal to a fraction of spot, formulas align with subtracting PV(income) or using an effective yield.

Worked dividend example

Stock at $50 pays a $1 dividend in three months. r = 6% continuous, forward maturity T = 0.5 years.

PV of dividend I = 1 × e^(−0.06×0.25) ≈ 0.985. F0 = (50 − 0.985) × e^(0.06×0.5) ≈ 49.015 × e^0.03 ≈ 49.015 × 1.03045 ≈ $50.51

Without the dividend, F0 would be 50 × e^0.03 ≈ $51.52. The known dividend cuts the forward by roughly the future value of that cash.

Worked index yield example

Index S0 = 4,000, r = 5%, q = 2%, T = 1 year (continuous). F0 = 4,000 × e^((0.05−0.02)×1) = 4,000 × e^0.03 ≈ 4,000 × 1.03045 = 4,121.8

Index Futures and Index Arbitrage

Equity-index futures settle to the index (cash-settled in many markets) with fair value from the yield formula above. Index arbitrage trades the futures against a replicating basket (or ETF):

• Futures rich (quoted F too high): sell futures, buy basket (cash-and-carry).
• Futures cheap: buy futures, short basket (reverse cash-and-carry).

Profits are clipped by transaction costs, tracking error, borrow fees, and dividend uncertainty. Program trading and ETF creations/redemptions are the plumbing that keeps index futures near fair value in liquid markets.

Basis = spot index − futures (sign conventions vary by market; know the definition in the question). As expiry approaches, futures converge to spot (for cash-settled indices, to the settlement index).

Synthesis

Price financial forwards from carry: finance the prepaid asset, credit income. Keep F (fair delivery price) separate from value (PV of F versus K). When an index future prints away from S e^((r−q)T) by more than costs, index arbitrage—not forecasting—is the first FRM answer.