Forward and Futures Pricing and Hedging › Summary

This chapter established the theoretical and practical foundations for pricing and hedging with forward and futures contracts. The key insights center on no-arbitrage pricing and risk transfer.

The cost-of-carry model provides the fundamental pricing relationship. The forward price equals the spot price compounded at the cost of carry: $F_0 = S_0 e^{(r - q + u - y)T}$, where $r$ is the financing rate, $q$ is income yield, $u$ is storage cost, and $y$ is convenience yield. Any deviation from this formula creates arbitrage opportunities through cash-and-carry or reverse cash-and-carry trades.

Forward contracts can be synthetically replicated. A long forward is equivalent to buying the asset and borrowing to fund the purchase; a short forward is equivalent to short-selling and investing the proceeds. This replication principle underlies all derivative pricing and enables arbitrage when market prices deviate from theoretical values.

Futures differ from forwards through daily settlement. Exchange-traded futures eliminate counterparty risk through central clearing and daily marking-to-market. For most practical purposes, futures and forward prices are identical, but stochastic interest rates introduce a convexity adjustment when asset prices and rates are correlated.

Hedging requires matching exposure characteristics. Perfect hedges use contracts identical to the underlying exposure. When mismatches exist, the minimum variance hedge ratio $h^* = \rho \sigma_S / \sigma_F$ minimizes residual risk. Basis risk from asset mismatch, timing mismatch, or specification differences reduces hedging effectiveness.

For equity portfolios, beta determines the hedge ratio. The number of index futures contracts to hedge or target a specific beta is $N = (\beta_{\text{current}} - \beta_{\text{target}}) \times V / (F \times m)$. This allows portfolio managers to adjust market exposure without trading the underlying stocks.

Commodity hedging transfers price risk from producers and consumers to speculators. Producers short futures to lock in sale prices; consumers go long to lock in purchase prices. Cross-hedging using related but not identical commodities is common when direct hedges are unavailable or illiquid.