Derivatives: options, futures and forwards
A derivative takes its value from an underlying asset. Options give a right without obligation; forwards and futures bind both sides, and futures are marked to market daily.
Option payoffs at expiry (S = spot at expiry, X = strike)
| Position | Profit | Break-even | Maximum loss or gain |
|---|---|---|---|
| Long call | max(S − X, 0) − premium | X + premium | Loss limited to the premium; gain unlimited |
| Short call | premium − max(S − X, 0) | X + premium | Gain limited to the premium; loss unlimited |
| Long put | max(X − S, 0) − premium | X − premium | Loss limited to the premium; gain up to X − premium |
| Short put | premium − max(X − S, 0) | X − premium | Gain limited to the premium; loss up to X − premium |
Moneyness
| State | Call | Put |
|---|---|---|
| In the money | S > X | S < X |
| At the money | S = X | S = X |
| Out of the money | S < X | S > X |
Worked example 1: call and put payoffs
Strike ₹500. Call premium ₹20, put premium ₹15. Find the net profit for a call buyer and a put buyer if the share ends at ₹540, and if it ends at ₹470.
- At ₹540: the call is exercised; payoff 540 − 500 = 40; profit = 40 − 20 = ₹20. The put lapses; loss = the ₹15 premium.
- At ₹470: the call lapses; loss = ₹20. Put payoff 500 − 470 = 30; profit = 30 − 15 = ₹15.
- Break-even: call 500 + 20 = ₹520; put 500 − 15 = ₹485.
- The writers' results are exact mirror images: options are a zero-sum game.
At ₹540: call +₹20, put −₹15. At ₹470: call −₹20, put +₹15.
Worked example 2: put-call parity
A share trades at ₹100. A one-year European call with strike ₹100 costs ₹12. The risk-free rate is 10% a year (annual discounting). Find the fair price of the matching put.
- PV(X) = 100 ÷ 1.10 = ₹90.91.
- Parity: P = C + PV(X) − S0 = 12 + 90.91 − 100 = ₹2.91.
- If the market put price differs, arbitrage is possible: buy the cheaper side of the parity equation and sell the dearer side.
Put = ₹2.91. Books that use continuous discounting write PV(X) = X × e^(−rT); the logic is the same.
Worked example 3: one-step binomial call value
A share at ₹100 can rise to ₹120 (u = 1.2) or fall to ₹80 (d = 0.8) in one period. The risk-free rate is 10% per period. Value a call with strike ₹100.
- Payoffs: up 120 − 100 = ₹20; down ₹0.
- Risk-neutral probability p = (1 + r − d) ÷ (u − d) = (1.10 − 0.80) ÷ (1.20 − 0.80) = 0.30 ÷ 0.40 = 0.75.
- C = [0.75 × 20 + 0.25 × 0] ÷ 1.10 = 15 ÷ 1.10 = ₹13.64.
- Check by replication: hedge ratio (delta) = (20 − 0) ÷ (120 − 80) = 0.5. Buy 0.5 share (₹50) and borrow 0.5 × 80 ÷ 1.10 = ₹36.36. Cost = 50 − 36.36 = ₹13.64.
Call value ₹13.64 by both methods.
Forwards vs futures
| Feature | Forward | Future |
|---|---|---|
| Where traded | Over the counter, between two parties | On an exchange |
| Terms | Customised | Standardised lot size, quality and dates |
| Counterparty risk | High | Low: the clearing corporation guarantees settlement |
| Settlement | At maturity | Marked to market daily, with margins |
| Liquidity | Low; hard to exit early | High; can be closed out any time |
| Oversight | Contract terms between the parties | Exchange rules and the regulator (SEBI in India) |
Pricing inputs, Greeks and facts
- Black-Scholes inputs: S, X, T, r and σ. The stock's expected return is not needed. The basic model prices European options on non-dividend shares, with constant σ and r and lognormal prices.
- Call value rises with S, T, σ and r, and falls with X. Put value rises with X and σ, and falls with S and r.
- Option value = intrinsic value + time value. Time value is highest at the money and falls to zero at expiry.
- Delta: change in option price per ₹1 change in the underlying. Gamma: change in delta. Theta: time decay. Vega: sensitivity to volatility. Rho: sensitivity to the interest rate.
- American options can be exercised any time up to expiry; European options only at expiry.
- Forward price by cost of carry: F = S0 × (1 + r)^T; for example 1,000 × 1.08 = ₹1,080 for one year at 8% with no income.
- Swaps exchange streams of cash flows, such as fixed for floating interest.
- In India, the L. C. Gupta Committee (1998) recommended exchange-traded derivatives; index futures began in 2000.
NET asks payoff and break-even numericals, put-call parity, the five Black-Scholes inputs, moneyness, the Greeks and forwards vs futures in match or statement form.
Typical MCQ: “A put has a strike of ₹250 and a premium of ₹10. What is the buyer's break-even price?” ₹240 (strike − premium). Trap: ₹260 applies the call rule.
Typical MCQ: “Is the stock's expected return a Black-Scholes input?” No. The five inputs are spot price, strike, time to expiry, risk-free rate and volatility; pricing is risk-neutral.
Call = you can call the share to you (buy); put = you can put it on someone else (sell). Parity: C + PV(X) = P + S, 'calls plus cash equal puts plus stock'.