Chapter 3: Futures Contracts, Mechanism and Pricing (Part 2)
1. Fundamental Principles of Futures Pricing
The Cost-of-Carry Model
The pricing of futures contracts is governed by the cost-of-carry model, which establishes a theoretical parity relationship between the spot price of an underlying asset and its futures price. The cost of carry measures the net storage and financing expenses associated with holding the asset until contract maturity, minus any income generated by the asset during the holding period.
| Component | Meaning |
|---|---|
| Spot Price | Current market price of the underlying asset. |
| Cost of Carry | Net cost of holding the asset until the futures expiry, including financing costs minus any income such as dividends. |
| Fair Futures Price | The theoretical futures price derived from the spot price and net cost of carry. |
Fair Futures Price = Spot Price + Cost of Carry
In an efficient derivatives market, any temporary deviation of the observed market futures price from its theoretical fair value creates a riskless arbitrage opportunity. Arbitrageurs execute offsetting spot and futures transactions to capture this price discrepancy, bringing the market futures price back into equilibrium with its fair value.
Theoretical Fair Value Formula
When no intermediate income or dividends are paid by the underlying asset, the fair value of a futures contract under continuous compounding is expressed as:
Futures Pricing Formula (Simple Line Format): F = S * e^(r * T)
Where:
- F: Theoretical fair futures price
- S: Current spot price of the underlying asset
- r: Cost of financing expressed as a continuously compounded annual interest rate
- T: Time remaining until contract expiration expressed in years (holding period in days / 365)
- e: Base of the natural logarithm, approximately equal to 2.71828
Basic Numerical Illustration
Consider Security XYZ Ltd trading in the cash market at a spot price of Rs. 1150. Assuming a continuously compounded risk-free borrowing rate of 11% per annum (r = 0.11), the fair value of a one-month futures contract (T = 1 / 12 years) is calculated as follows:
Calculation (Simple Line Format): F = 1150 * e^(0.11 * (1 / 12)) = Rs. 1160
2. Pricing Equity Index Futures
Characteristics of Stock Index Futures
A stock index futures contract gives its holder the legal right and obligation to buy or sell the portfolio of equities represented by the index. Index futures differ fundamentally from physical commodity futures in two key operational aspects:
- Zero Storage Costs: Holding an equity portfolio does not incur physical warehousing or inventory storage expenses.
- Dividend Income Stream: Equity shares yield dividends, which act as a negative carrying cost for long position holders and a positive cost for short position holders.
Net Cost of Carry for Equities (Simple Line Format): Net Cost of Carry = Financing Cost - Dividend Income
Consequently, accurate dividend forecasting is essential for calculating the precise fair value of equity index futures contracts. All stock index futures on Indian stock exchanges are cash settled at expiration without physical delivery of individual constituent shares.
Pricing Index Futures Given Expected Dividend Amount
When specific constituent companies in the index are expected to declare lump-sum cash dividend amounts during the life of the futures contract, the carrying cost must be reduced by the present value of those expected dividends.
Step-by-Step Numerical Example (Nifty Index Futures)
Suppose a new 2-month Nifty Index Futures contract (T = 60 / 365 years) is introduced on the exchange when the spot Nifty stands at 4000. The risk-free borrowing rate is 10% per annum (r = 0.10), and Nifty trades with a standard contract multiplier of 100.
Assume that constituent stock ABC Ltd, which holds a 7% weight in the Nifty index, is expected to declare a dividend of Rs. 20 per share after 15 days of purchasing the contract:
- Total Contract Value: 4000 * 100 = Rs. 4,00,000.
- ABC Ltd Weight Value in Contract: Rs. 4,00,000 * 0.07 = Rs. 28,000.
- Number of ABC Ltd Shares per Unit Lot: Assuming ABC Ltd trades at a spot market price of Rs. 140, the lot represents 28,000 / 140 = 200 shares.
- Total Gross Dividend Received: 200 shares * Rs. 20 = Rs. 4000.
- Dividend Compounding Adjustment: Because the dividend is received after 15 days, it earns interest for the remaining 45 days of the 60-day holding period. The dividend adjustment per index unit is obtained by dividing the compounded dividend by the multiplier of 100.
Index Futures Pricing Formula with Dividend Amount (Simple Line Format): F = S * e^(r * T) - (Dividend Amount * e^(r * (T - t_d))) / Multiplier
Substituting the example parameters:
Calculation Steps (Simple Line Format): F = 4000 * e^(0.10 * (60 / 365)) - (200 * 20 * e^(0.10 * (45 / 365))) / 100 F = Rs. 4025.80
Thus, the fair value of the two-month Nifty futures contract is Rs. 4025.80.
Pricing Index Futures Given Expected Dividend Yield
When constituent companies pay dividends fairly uniformly throughout the year without seasonal clustering, it is more practical to express dividend income as an annualized continuous dividend yield (q).
Index Futures Formula with Dividend Yield (Simple Line Format): F = S * e^((r - q) * T)
Where:
- F: Theoretical fair futures price
- S: Spot index value
- r: Cost of financing (annualized continuously compounded risk-free rate)
- q: Expected annualized dividend yield of the index
- T: Time to maturity in years
Numerical Example (Continuous Dividend Yield)
A 2-month Nifty futures contract trades on the NSE (T = 60 / 365 years). The spot value of Nifty is 4000, the financing cost is 10% per annum (r = 0.10), and the expected annualized dividend yield on the index portfolio is 2% (q = 0.02).
Calculation (Simple Line Format): F = 4000 * e^((0.10 - 0.02) * (60 / 365)) F = 4000 * e^(0.08 * 0.16438) F = Rs. 4052.95
The theoretical fair value of the 2-month Nifty futures contract is Rs. 4052.95.
Basis Dynamics and Convergence at Expiration
The structural relationship between spot and futures prices manifests through the basis, defined as the futures price minus the spot price (Basis = Futures Price - Spot Price).
| Stage | Basis Condition | Explanation |
|---|---|---|
| Contract Inception | Positive Basis | The futures price is above the spot price, resulting in a positive basis. |
| During Contract Life | Basis Converges | As the contract approaches expiry, the difference between the futures price and spot price generally narrows. |
| Expiry Date | Basis = 0 | At expiry, the futures price converges with the spot price, so the basis becomes zero. |
Key Dynamics of Basis:
- Time Decay Effect: As time progresses toward the expiration date (T -> 0), the cost of carry diminishes, causing the futures price to converge toward the spot price.
- Zero Basis at Expiration: On the exact day of contract expiration, the time to maturity is zero (T = 0), forcing the carrying cost to zero and making the futures settlement price equal to the spot closing price (Basis = 0).
- Arbitrage Parity Enforcement: If the basis fails to hit zero at contract expiration, or if futures mispricing creates an abnormal spread during the contract lifecycle, market arbitrageurs trade spot against futures to force price convergence.
3. Pricing Single Stock Futures (SSF)
Mechanics of Single Stock Futures
Single Stock Futures give the position holder the legal obligation to buy or sell a specific quantity of an individual equity share at a fixed price on a future date. Like index futures, stock futures contracts traded on the NSE are cash settled without physical delivery of individual shares.
Stock futures pricing incorporates the same net cost-of-carry principles: financing interest represents a positive carrying cost, while expected corporate dividend payouts represent a negative carrying cost.
Stock Futures Cost Breakdown: Carrying Cost = Borrowing Interest - Present Value of Dividends
Pricing Stock Futures When No Dividend is Expected
If an individual company is not expected to declare any cash dividends during the life of the futures contract, the fair value is determined solely by compounding the spot price at the risk-free financing rate.
No-Dividend Stock Futures Formula (Simple Line Format): F = S * e^(r * T)
Numerical Example (Zero Dividend Expected)
A 2-month futures contract (T = 60 / 365 years) is introduced on XYZ Ltd. The prevailing spot price of XYZ Ltd is Rs. 228, and money can be borrowed at an annual interest rate of 10% (r = 0.10).
Calculation (Simple Line Format): F = 228 * e^(0.10 * (60 / 365)) F = 228 * e^(0.016438) F = Rs. 231.90
The theoretical fair value for a unit of the 2-month XYZ Ltd futures contract is Rs. 231.90.
Pricing Stock Futures When Dividends Are Expected
When an individual stock is expected to pay a cash dividend prior to contract expiration, the fair futures price is calculated by deducting the future compounded value of the dividend from the gross compounded spot price.
Stock Futures Formula with Dividends (Simple Line Format): F = S * e^(r * T) - D * e^(r * (T - t_d))
Where:
- S: Spot price of the individual stock
- r: Continuously compounded borrowing interest rate
- T: Total tenure of the futures contract in years
- D: Absolute cash dividend amount per share
- t_d: Time when dividend is received, expressed in years
Numerical Example (Expected Corporate Dividend)
Suppose XYZ Ltd futures trade on the exchange as a 2-month contract (T = 60 / 365 years) when the spot share price is Rs. 140. Money can be borrowed at 10% per annum (r = 0.10). The company is expected to declare a dividend of Rs. 10 per share after 15 days (t_d = 15 / 365 years).
The dividend of Rs. 10 is received after 15 days and is compounded for the remaining 45 days (T - t_d = 45 / 365 years):
Calculation Steps (Simple Line Format): F = 140 * e^(0.10 * (60 / 365)) - 10 * e^(0.10 * (45 / 365)) F = 140 * e^(0.016438) - 10 * e^(0.012329) F = 142.32 - 10.12 F = Rs. 132.20
The theoretical fair value for a unit of the 2-month XYZ Ltd stock futures contract drops to Rs. 132.20 due to the dividend deduction.
Structural Comparison: Index Futures vs. Stock Futures Pricing
The table below summarizes the key pricing inputs and mathematical adjustments across derivative types:
| Pricing Parameter | Index Futures (Dividend Amount) | Index Futures (Dividend Yield) | Stock Futures (No Dividend) | Stock Futures (Dividend Expected) |
|---|---|---|---|---|
| Primary Formula | F = S*e^(r*T) - PV(D_index) | F = S*e^((r-q)*T) | F = S*e^(r*T) | F = S*e^(r*T) - D*e^(r*(T-t_d)) |
| Financing Expense (r) | Added via compounding | Added via compounding | Added via compounding | Added via compounding |
| Dividend Adjustment | PV of individual dividend weights | Deducted from rate (r - q) | None (D = 0) | Deducted as compounded value |
| Settlement Method | Cash settled on exchange | Cash settled on exchange | Cash settled on exchange | Cash settled on exchange |
Key Takeaways and Important Terms
Core Conceptual Summary
- Cost-of-Carry Parity: The theoretical fair value of a futures contract equals the spot price plus financing interest costs minus the present value of income/dividends generated by the underlying asset.
- Arbitrage Efficiency: Any market price deviation from theoretical fair value triggers cash-and-carry or reverse cash-and-carry arbitrage, restoring equilibrium.
- Dividend Impact: Dividend payments reduce the net carrying cost, thereby lowering the fair futures price relative to a non-dividend-paying asset.
- Basis Convergence: The difference between futures and spot prices narrows over time and reaches zero on contract expiration date (Basis = 0 at T = 0).
- Cash Settlement Standard: Both equity index futures and single stock futures contracts in Indian derivatives markets are cash settled without physical delivery of underlying shares.
Glossary of Essential Terms
- Cost of Carry: The net cost of holding an underlying asset until maturity, incorporating interest and storage costs minus dividend payouts.
- Fair Value: The theoretical equilibrium price of a futures contract calculated using the cost-of-carry pricing model.
- Arbitrage: The simultaneous purchase and sale of identical or equivalent financial instruments in different markets to capture riskless profit from price discrepancies.
- Dividend Yield (q): Annualized dividend income generated by an index or portfolio expressed as a percentage of its spot value.
- Basis Convergence: The natural narrowing of the gap between futures and spot prices over the contract lifetime until they converge to zero on the expiration date.
- Continuous Compounding: Interest compounding calculated over infinitely small time intervals, represented mathematically using exponential functions (e^(r*T)).