Futures pricing
This sub‑topic covers the pricing of equity futures, focusing on the cost‑of‑carry model, its discrete approximation for Indian markets, and the impact of interest rates, dividends, storage costs and convenience yield. Understanding futures pricing is essential for NISM Series VIII as it forms the basis of valuation questions and helps you calculate fair values during the exam. The content links directly to the module on Introduction to Forwards and Futures and prepares you to answer both conceptual and numerical items.
Learning Objectives
- 1Explain the cost‑of‑carry theory and its relevance to futures pricing.
- 2Derive and use the continuous and discrete futures pricing formulas.
- 3Identify how each component (interest, dividend, storage, convenience yield) influences the futures price.
- 4Apply the pricing steps to solve typical NISM exam scenarios.
Cost‑of‑Carry Model
The cost‑of‑carry model states that the fair price of a futures contract equals the spot price of the underlying asset plus all costs incurred to carry (hold) the asset until the contract’s maturity, minus any benefits earned from holding it. In equity markets, the primary costs are the financing cost (risk‑free rate) and any storage or insurance charges, while the main benefit is the dividend yield that the holder would receive.
Mathematically, the model is expressed using continuous compounding because the Indian securities market (regulated by SEBI) assumes a continuously compounded risk‑free rate for valuation. Continuous compounding simplifies the algebra and matches the way interest rates are quoted in the RBI’s policy rates, which are the benchmark for futures pricing.
For the NISM exam, you must remember that the formula integrates four variables: spot price (S₀), risk‑free rate (r), storage/insurance cost (u), and dividend yield (d). Forgetting any one of these leads to an incorrect fair price and a common trap in multiple‑choice questions.
- Spot price – current market price of the underlying equity or index.
- Risk‑free rate – annualised rate of a government security (e.g., 10‑year RBI bond) expressed in decimal form.
Many candidates use only the risk‑free rate and spot price, overlooking the dividend yield. In equity futures, dividends reduce the fair price because the holder of the futures does not receive them. Always subtract the dividend component (d) in the cost‑of‑carry formula.
Where:
F= Fair futures price in rupeesS_0= Current spot price of the underlying in rupeesr= Annual risk‑free rate (decimal)u= Annual storage/insurance cost (decimal)d= Annual dividend yield (decimal)T= Time to maturity in yearsWorked Example
Given S_0 = 45,000, r = 0.06, u = 0.005, d = 0.02, T = 0.5 years: Step 1: Compute (r + u - d) = 0.06 + 0.005 - 0.02 = 0.045 Step 2: Multiply by T: 0.045 \times 0.5 = 0.0225 Step 3: Exponentiate: e^{0.0225} \approx 1.0228 Step 4: Futures price F = 45,000 \times 1.0228 \approx 46,026 Verification: 45,000 \times e^{0.0225} = 46,026 (rounded).
Discrete Approximation for Indian Markets
While continuous compounding is theoretically clean, many NISM practice questions present rates on an annual simple basis. In such cases, a discrete approximation is accepted: the futures price is the spot price multiplied by (1 + net cost) raised to the power of the time fraction. This method aligns with the way brokers quote financing charges on the NSE.
The discrete formula uses the same components as the continuous model, but they are added linearly inside the parentheses. It is especially useful when the time to maturity is expressed in whole months or when the exam explicitly states “simple interest” for the financing cost.
Remember to convert the time to years (e.g., 3 months = 0.25 years) before applying the exponent. Failure to adjust the time unit is a frequent source of calculation errors in the exam.
Where:
F= Fair futures price in rupeesS_0= Spot price in rupeesr= Annual risk‑free rate (decimal)u= Annual storage/insurance cost (decimal)d= Annual dividend yield (decimal)T= Time to maturity in years (can be fractional)Worked Example
Using the same inputs as the continuous example: Step 1: Net factor = 1 + 0.06 + 0.005 - 0.02 = 1.045 Step 2: Raise to T = 0.5: 1.045^{0.5} = \sqrt{1.045} \approx 1.0223 Step 3: Futures price F = 45,000 \times 1.0223 \approx 46,004 Verification: 45,000 \times 1.045^{0.5} = 46,004 (rounded).
Impact of Individual Components
Each component in the cost‑of‑carry model moves the futures price in a predictable direction. An increase in the risk‑free rate (r) raises the financing cost, pushing the futures price higher. Conversely, a higher dividend yield (d) lowers the price because the holder of the futures forgoes dividend income.
Storage or insurance costs (u) are usually small for equities but become significant for commodity‑linked equity futures (e.g., gold ETFs). The convenience yield (often denoted as y) represents the non‑monetary benefit of holding the physical asset; a higher convenience yield reduces the futures price, similar to a dividend.
For the NISM exam, you may be asked to predict the direction of price change when a single variable moves, without performing full calculations. Memorise the sign convention: +r, +u increase price; +d, +y decrease price.
Effect of Cost‑of‑Carry Components on Futures Price
| Component | Definition | Effect on Futures Price |
|---|---|---|
| Risk‑free rate (r) | Annual financing cost based on government securities | Higher r → Futures price ↑ |
| Dividend yield (d) | Annual cash dividend expressed as a fraction of spot | Higher d → Futures price ↓ |
| Storage/Insurance (u) | Cost to hold the underlying asset | Higher u → Futures price ↑ |
| Convenience yield (y) | Benefit of physically holding the asset | Higher y → Futures price ↓ |
Practical Example – Index Futures Pricing
Scenario
An Indian investor wants to price a 6‑month Nifty 50 futures contract. The current Nifty spot is 18,200 points. The annual risk‑free rate is 6.5%, the expected dividend yield on the index is 1.8%, and there are no storage costs. Compute the fair futures price using the continuous cost‑of‑carry model.
Solution
Step 1: Convert rates to decimals – r = 0.065, d = 0.018, u = 0.0. Step 2: Net cost = r + u – d = 0.065 – 0.018 = 0.047. Step 3: Time to maturity T = 6 months = 0.5 years. Step 4: Exponent = (0.047) \times 0.5 = 0.0235. Step 5: e^{0.0235} \approx 1.0238. Step 6: Futures price = 18,200 \times 1.0238 \approx 18,634 points. The calculation shows the futures price is roughly 434 points above the spot, reflecting the cost of financing after accounting for dividends.
Conclusion
The example illustrates how a modest dividend yield offsets most of the financing cost, a pattern frequently tested in NISM questions.
Time to Maturity and Price Curve
Futures Price vs. Time to Maturity (Same Underlying)
When T is given in months, always divide by 12 before using it in the formula. Using months directly will inflate the exponent and give a price that is too high.
Key Differences Between Forward and Futures Pricing
Both forwards and futures are derived from the same cost‑of‑carry principle, but the mechanics differ. A forward contract is settled only at maturity, so its price reflects the entire cost‑of‑carry up to that date. A futures contract, however, is marked‑to‑market daily, meaning gains and losses are realized each day.
Because of daily settlement, the futures price incorporates the expected path of interest rates and may deviate slightly from the theoretical forward price, especially in volatile markets. In the Indian context, SEBI mandates daily margining for all listed futures, reinforcing this distinction.
Exam questions often ask you to pick the statement that correctly describes the impact of daily marking‑to‑market on pricing. Remember: daily settlement reduces credit risk but does not change the theoretical fair price derived from cost‑of‑carry.
Forward vs. Futures – Core Differences
| Feature | Forward Contract | Futures Contract |
|---|---|---|
| Settlement | Only at maturity | Daily mark‑to‑market |
| Credit Risk | Higher (counter‑party risk) | Lower (exchange clearing house) |
| Pricing Basis | Cost‑of‑carry without daily cash flows | Cost‑of‑carry with daily cash flows |
| Regulation (India) | OTC, less regulated | Listed on NSE/BSE, SEBI regulated |
Step‑by‑Step Pricing Procedure for Exams
1. Identify the spot price (S₀) of the underlying equity or index.
2. Gather the annual risk‑free rate (r) from the latest RBI policy or government bond yield.
3. Determine the dividend yield (d) expected over the contract’s life; for indices, use the published dividend forecast.
4. If applicable, add any storage or insurance cost (u) and subtract convenience yield (y).
5. Convert the time to maturity (T) into years (e.g., 90 days = 90/365).
6. Choose the appropriate formula – continuous for most NISM questions, discrete if the problem explicitly states simple interest.
7. Plug the numbers, compute, and round as per the question’s requirement (usually to the nearest rupee or index point).
8. Review the direction of change: higher r or u raises the price, higher d or y lowers it – a quick sanity check can prevent calculation slip‑ups.
⭐Exam Takeaways
- Futures price = Spot × e^{(r + u - d)T} (continuous) or Spot × (1 + r + u - d)^{T} (discrete).
- Risk‑free rate and storage cost increase futures price; dividend yield and convenience yield decrease it.
- Always convert time to years; for months divide by 12, for days divide by 365.
- Daily mark‑to‑market distinguishes futures from forwards but does not alter the theoretical cost‑of‑carry price.
- When a question provides simple‑interest rates, use the discrete formula; otherwise default to continuous compounding.
Practice Questions
8 questions on Futures pricing
What is the continuous‑compounding futures pricing formula for equity futures?
In the cost‑of‑carry model, an increase in which component will lower the futures price?
Using the continuous formula, calculate the fair futures price when S₀ = 45,000, r = 0.06, u = 0.005, d = 0.02 and T = 0.5 years.
Using the discrete approximation, what is the futures price for the same inputs: S₀ = 45,000, r = 0.06, u = 0.005, d = 0.02, T = 0.5 years?
A common exam trap is to ignore which component when pricing equity futures?
How does daily mark‑to‑market settlement affect the theoretical fair price of a futures contract?
Which statement correctly describes a core difference between forward and futures pricing bases?
For S₀ = 45,000, r = 0.06, u = 0.005, d = 0.02 and T = 0.5 years, what is the net cost‑of‑carry (r+u‑d) and the resulting relationship of futures price to spot price?