Advanced Financial Management · Risk Management

Hedging with Derivatives and Greeks for CA Final AFM

Updated 5 October 2026 · Fact-checked

Hedging means taking a derivative position whose gains offset losses on an existing exposure. Delta tells you how much an option moves per ₹1 move in the underlying, so shares ÷ delta gives options needed. For futures, contracts = portfolio value × beta ÷ contract value. Then check the net outcome.

Understand Hedging with Derivatives and Greeks

Hedging means taking a position in a derivative that moves opposite to an exposure you already hold. If you own shares and fear a fall, you sell index futures or buy puts. A good hedge does not make profit. It fixes the outcome and removes the risk you do not want.

The three tools work differently. Futures lock in a price and remove both gain and loss. Options give a right, not an obligation, so you pay a premium and keep the upside (a protective put is the usual example). Swaps exchange cash flows, for example fixed interest for floating interest, to convert the nature of a liability.

The Greeks measure how an option price reacts to its inputs. Delta is the change in option price for a ₹1 change in the underlying. Gamma is the change in delta for a ₹1 change in the underlying. Vega is the change in option price for a 1 percentage point change in volatility. Theta is the change in option price as time passes, usually per day, and it is normally negative for a long option. Rho is the change in option price for a 1 percentage point change in the interest rate.

Delta hedging uses delta as the hedge ratio. A call has delta between 0 and 1. A put has delta between −1 and 0. A position is delta-neutral when the total delta of everything you hold is zero. Delta changes as the price moves, and gamma tells you how fast. That is why a delta hedge must be rebalanced.

Key rules to remember

Delta: Delta = Change in option price ÷ Change in price of underlying; Call delta lies between 0 and 1. Put delta lies between −1 and 0. Under Black-Scholes, call delta = N(d1) and put delta = N(d1) − 1, for a non-dividend-paying share.
Delta-neutral hedge: Number of options = Number of shares held ÷ Delta of option; Shares held long are hedged by writing calls (or buying puts). Neutral means shares × 1 + options × delta = 0, with sign for buy or sell.
Gamma: Gamma = Change in delta ÷ Change in price of underlying; Approximate new delta = old delta + gamma × price change. Gamma is highest for at-the-money options.
Vega, Theta, Rho: Vega = ΔPrice ÷ ΔVolatility (per 1%); Theta = ΔPrice ÷ Δtime (per day); Rho = ΔPrice ÷ ΔRate (per 1%); Use the unit given in the question. Do not mix per-day and per-year theta.
Futures hedge ratio (beta method): Number of contracts = (Portfolio value ÷ Value of one futures contract) × Portfolio beta; Value of one contract = futures price × lot size. Sell contracts to hedge a long portfolio.
Minimum variance hedge ratio: h = ρ × (σS ÷ σF); Number of contracts = h × Exposure ÷ Value of one contract; ρ is the correlation between spot and futures price changes. σS and σF are the standard deviations of those changes.
Changing portfolio beta: Number of contracts = (Target beta − Current beta) × Portfolio value ÷ Value of one contract; A negative answer means sell futures. A positive answer means buy futures.

How to solve Hedging with Derivatives and Greeks questions

Use this order for any hedging or Greeks question. It keeps signs and units under control.

1Identify the exposure: are you long or short, in shares, a portfolio, a currency or a loan? Write down its size in units and in rupees.
2Decide the direction of risk. A long exposure loses when prices fall, so the hedge must gain when prices fall. This fixes whether you buy or sell.
3Pick the instrument given in the question (futures, calls, puts or swap) and note its lot size, price and premium.
4Compute the hedge ratio: shares ÷ delta for options, or value × beta ÷ contract value for futures. Round to a whole number of contracts or lots only at the end.
5If the question asks for rebalancing, find the new delta using old delta + gamma × price change, then recompute the position and state how many options to buy or sell.
6Verify with a small price move. Compute the gain or loss on the exposure and on the hedge, and show the net figure.
7State the conclusion in a line: what you bought or sold, how many, and what the net result shows.

Quickest way: Delta and beta shortcut

When to use it: Use this when the question gives delta or beta and asks only for the number of options or contracts, or asks for a check of hedge effectiveness.

For options: write options needed = shares ÷ delta. For 10,000 shares and delta 0.40, that is 25,000 options.
For futures: write contracts = (portfolio × beta) ÷ (futures price × lot size).
Fix the sign by asking: do I lose if the market falls? If yes, sell the hedge.
Test with one move. Exposure change = units × price move. Hedge change = options × delta × price move, or contracts × lot × index move.
Round contracts to the nearest whole number and mention the small residual risk.

Common mistakes in Hedging with Derivatives and Greeks

Multiplying shares by delta instead of dividing when finding the number of options.
Students remember delta as a ratio and apply it the wrong way round.
Fix: Think of one option as moving only delta shares. To cover 10,000 shares you need 10,000 ÷ delta options, which is more than 10,000 for a delta below 1.
Buying a hedge when you should sell, or the reverse.
Students do not first ask which price move hurts the exposure.
Fix: Write the direction of loss first. A long portfolio is hedged by selling futures. A short exposure is hedged by buying.
Forgetting the lot size when computing the value of one futures contract.
The index level looks like the contract value.
Fix: Contract value = futures price × lot size. Always write this line before dividing.
Using a positive delta for a put or ignoring the sign.
Students memorise the call delta only.
Fix: A put gains when the price falls, so its delta is negative. Put delta = call delta − 1 for the same strike and expiry, on a non-dividend share.
Calling a delta-neutral position fully hedged against all moves.
Delta is treated as constant.
Fix: Delta-neutral protects only against small moves. Gamma changes delta, so the position needs rebalancing. Vega and theta are still exposed.
Mixing units for theta, vega and rho.
The question gives theta per day but the period is in months, or vega per 1% but volatility changes by 3 points.
Fix: Scale by the exact change. A vega of ₹2 per 1% and a 3 point rise gives ₹6 change in option price.

Worked examples

Example 1

An investor holds 10,000 shares of a company priced at ₹500. A call option on the share has delta 0.40 and gamma 0.02. (a) How many call options must she write to be delta-neutral? (b) Verify with a ₹1 rise in the share price. (c) After the share rises by ₹5, what is the approximate new delta and how should she rebalance?

Show the solution

(a) Options needed = 10,000 ÷ 0.40 = 25,000 calls. She writes them because she is long the shares.
(b) Gain on shares for a ₹1 rise = 10,000 × ₹1 = ₹10,000. Loss on short calls = 25,000 × 0.40 × ₹1 = ₹10,000. Net change = ₹0, so the position is neutral.
(c) New delta ≈ 0.40 + 0.02 × 5 = 0.50.
Calls needed now = 10,000 ÷ 0.50 = 20,000.
She is short 25,000 calls and needs to be short only 20,000, so she buys back 5,000 calls.

Answer: (a) Write 25,000 calls. (b) Net change for a ₹1 move is ₹0. (c) New delta is about 0.50, so buy back 5,000 calls to stay delta-neutral.

Example 2

A fund manager holds an equity portfolio worth ₹5,00,00,000 with a beta of 1.2 against the Nifty. Nifty futures trade at 22,000 and one contract has a lot size of 50 units (assumed for this problem). (a) How many contracts should she sell to hedge fully? (b) If the Nifty falls 5% (futures also fall 5%), show the net result.

Show the solution

(a) Value of one contract = 22,000 × 50 = ₹11,00,000.
Contracts = (5,00,00,000 ÷ 11,00,000) × 1.2 = 6,00,00,000 ÷ 11,00,000 = 54.55, which rounds to 55 contracts to sell.
(b) The Nifty falls 5%, so the portfolio falls 5% × 1.2 = 6%. Loss = 6% × ₹5,00,00,000 = ₹30,00,000.
Futures fall 5% of 22,000 = 1,100 points. Gain on 55 short contracts = 1,100 × 50 × 55 = ₹30,25,000.
Net result = ₹30,25,000 − ₹30,00,000 = ₹25,000 gain. The small difference comes from rounding 54.55 up to 55.

Answer: Sell 55 Nifty futures contracts. For a 5% fall, the portfolio loses ₹30,00,000 and the futures gain ₹30,25,000, a net gain of ₹25,000.

Exam tips

Show the hedge ratio formula, the substitution and the verification. Numerical answers are marked step by step, so a neat check often earns marks.
State clearly whether you buy or sell and how many contracts or options. A correct number with no direction loses marks.
When delta, gamma, vega, theta or rho are given as values, apply them directly. Do not bring in Black-Scholes unless asked.
In theory questions, explain what each Greek measures in one line and add its practical use, such as gamma showing why rebalancing is needed.
If rounding changes the answer, round contracts at the last step and mention the residual risk in one line.

Practice questions from Risk Management

Hedging with Derivatives and Greeks in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Hedging with Derivatives and Greeks: frequently asked questions

What is the difference between delta and gamma?

Delta measures how much the option price changes for a ₹1 change in the underlying. Gamma measures how much delta itself changes for that same ₹1 move. Delta is the speed and gamma is the acceleration.

How do I calculate the hedge ratio for delta hedging?

Divide the number of shares you hold by the option's delta. For 10,000 shares and delta 0.50, you need 20,000 options. Write calls against long shares, or buy puts.

Why is theta usually negative for a long option?

An option has time value, and that value shrinks as expiry approaches, other things equal. So the buyer loses a little each day. The seller of the option gains from this decay.

How many futures contracts do I need to hedge a portfolio?

Divide the portfolio value by the value of one futures contract, then multiply by the portfolio beta. Value of one contract is futures price × lot size. Sell the contracts if the portfolio is long.