Part IV · Derivatives · M07 · week 4 · 16 h

Forwards and futures

The no-arbitrage forward price, futures versus forwards, basis and convergence, Indian index futures and margins, open interest, minimum-variance hedging, and a preview of commodity carry.

Start reading · Learning outcomes

Builds on M01 Market structureM02 Statistics of returns

Prerequisites M01 (order books, clearing, transaction costs), M02 (returns, covariance, fat tails). Maths: continuous compounding, conditional expectation, least squares, Gaussian processes (one optional calculation with a Vasicek short rate uses the Itô isometry, introduced properly in M08). Time: ~16 hours (notes 8 h, problems 5 h, code 3 h). Primary texts: Hull, Options, Futures, and Other Derivatives (11th ed., 2021), chapters on futures mechanics, hedging and forward/futures prices; Cox, Ingersoll and Ross (1981); NISM Series VIII (Equity Derivatives) for Indian F&O plumbing.

Learning outcomes

Derive the forward price $F_0 = S_0 e^{(r-q)T}$ by constructing both the cash-and-carry and the reverse cash-and-carry arbitrage, and the discrete-dividend version $F_0=(S_0-I)e^{rT}$ .
Value an existing forward contract and state the no-arbitrage band for a forward price when borrowing and lending rates differ and trading costs are present.
Compute the daily mark-to-market cash flows and margin calls of a futures position, and the return on margin for a given price move.
Prove that futures and forward prices coincide when interest rates are deterministic, and sign and size the futures–forward gap when rates are stochastic.
Decompose a hedger's effective price into the futures price and the basis at close-out, and explain basis risk.
Derive the minimum-variance hedge ratio $h^*=\rho\sigma_S/\sigma_F$ , its hedge effectiveness $\rho^2$ , and the beta-hedge contract count $N^*=\beta V_P/V_F$ ; estimate them from data and size a Nifty futures hedge in lots.
Read open interest correctly, including the accounting identity behind it and the reasons the popular price/OI labels carry little predictive information.
Write the cost-of-carry relation with storage cost and convenience yield, $F_0=S_0e^{(r+u-y)T}$ , and explain why it is an inequality for consumption commodities.

Sections ≈ 2 h 50 min of study

Foundation 5 sections · ≈ 55 min
Core 7 sections · ≈ 1 h 45 min
Advanced 1 section · ≈ 10 min

Learning outcomes Foundation 5′
7.0 Conventions and notation Foundation 5′
7.1 Forward contracts and the cost-of-carry price Core 30′
7.2 Futures: daily settlement and the futures–forward relation Core 20′
Figure Daily settlement: same P&L, different cash
7.3 Basis and convergence Core 10′
Figure Cost of carry and the convergence of the basis
7.4 Index futures in India Foundation 20′
7.5 Open interest Foundation 10′
7.6 Hedging with futures Core 15′
Figure The minimum-variance hedge ratio
Lab Minimum-variance hedge of Bank Nifty with the Nifty
7.7 Commodity futures preview: storage and convenience yield Advanced 10′
Figure Storage, convenience yield and the futures curve
7.8 Case studies Core 20′
Summary of key results Foundation 15′
Readings and resources Core 5′
Facts that change Core 5′