Options pricing, Greeks and hedging strategies

In short

Option pricing finds the fair premium of a call or put from the spot price, strike, time to expiry, volatility, interest rate and dividends, using the no-arbitrage logic of the binomial model and the Black-Scholes-Merton formula. The Greeks - delta, gamma, theta, vega, rho - measure how that premium changes when each input moves, and so tell a trader or treasurer what to buy or sell to neutralise a risk. Hedging strategies (protective put, covered call, collar, spreads, straddles) combine options to shape a payoff for a stated view and budget. Examined in MBA/PGDM derivatives and risk papers, NISM Series-VIII (Equity Derivatives), CFA and FRM, and used daily by Indian equity desks, broker risk teams, treasury and quant roles at NSE/BSE members, banks, mutual funds and prop firms.

Explained

Inputs, moneyness and the two components of premium

• Six inputs: spot S, strike K, time to expiry T (years), volatility σ (annualised, decimal), risk-free rate r (continuously compounded), dividend yield q (or discrete dividends).
• Premium = intrinsic value + time value. Intrinsic = max(S−K, 0) for a call, max(K−S, 0) for a put; the rest is time value, which decays to zero at expiry.
• Moneyness: ITM (intrinsic > 0), ATM (S ≈ K, maximum time value), OTM (intrinsic = 0, premium is pure time value).
• Direction of each input: call value rises with S, σ, T, r and falls with K and q; put value rises with K, σ, T, q and falls with S and r.
• Bounds (no model needed): c ≤ S; c ≥ max(S e^{−qT} − K e^{−rT}, 0); p ≤ K e^{−rT}; p ≥ max(K e^{−rT} − S e^{−qT}, 0).
• European vs American: European options are exercisable only at expiry; American any time. All NSE/BSE index options and (since 2011) single-stock options are European-style, cash-settled for indices and physically settled for stocks.
• Why American calls on a non-dividend stock are never exercised early: selling the option fetches intrinsic + time value, so early exercise throws away time value. Deep ITM American puts and calls just before a large dividend can be worth exercising early - hence American options need a lattice, not Black-Scholes.

Put-call parity and synthetic positions

• Parity for European options on the same S, K, T: c + K e^{−rT} = p + S e^{−qT}. Derived by showing that a long call + cash K and a long put + share both pay max(S_T, K) at expiry, so by no-arbitrage they cost the same today.
• Any one instrument can be synthesised from the other three: synthetic long stock = long call + short put + bonds; synthetic call = long put + long stock; synthetic put = long call + short stock.
• Conversion / reversal arbitrage: if the left side is dearer, sell the call, buy the put and the stock, and borrow the net outlay - a locked spread. In practice Indian traders check parity against the futures price (F = S e^{(r−q)T}), because futures are more liquid than the cash leg.

Binomial (Cox-Ross-Rubinstein) model

1. Over one step of length Δt the price goes to Su or Sd (u > 1 > d; CRR sets u = e^{σ√Δt}, d = 1/u).
2. Build a replicating portfolio of Δ shares and B borrowing that pays the option's payoff in both states: Δ = (c_u − c_d)/(S u − S d) - the hedge ratio, and the origin of delta.
3. Option value = cost of the replicating portfolio = e^{−rΔt}[p·c_u + (1−p)·c_d] with risk-neutral probability p = (e^{(r−q)Δt} − d)/(u − d).
4. Risk-neutral valuation: p is not a real-world probability and the actual expected return of the stock never appears - only r does, because the hedge removes the risk. This is the single most asked conceptual point.
5. Multi-step: build the price lattice forward, compute payoffs at the terminal nodes, then roll back node by node. For American options take max(continuation value, immediate exercise) at every node.
6. As the number of steps → ∞, the binomial price converges to Black-Scholes. The lattice handles early exercise, discrete dividends and exotic features that the closed form cannot.

Black-Scholes-Merton: assumptions, formula, variants, limits

• Assumptions: lognormal price (GBM) with constant σ and r, no transaction costs or taxes, continuous trading and continuous delta hedging, no arbitrage, divisible assets, European exercise, known constant dividend yield.
• Derivation outline (as examined): form a portfolio long one option and short Δ = ∂c/∂S shares; by Itô's lemma its value change has no dW term, so it is riskless and must earn r. This gives the Black-Scholes PDE ∂c/∂t + rS ∂c/∂S + ½σ²S² ∂²c/∂S² = rc, solved with the payoff boundary condition to give the formula.
• Variants: Merton (continuous dividend yield q) for indices; Black's model on the futures price for options on futures and interest-rate options; Garman-Kohlhagen for currency options (q = foreign interest rate) - used for USD/INR options on NSE.
• Limitations: real returns have fat tails and jumps; volatility is neither constant nor known; the model under-prices far OTM options, which is exactly why a volatility smile/skew exists. Discrete hedging, bid-ask spreads, STT and margin costs make the perfect hedge impossible.

Volatility: historical, implied and the smile

• Historical (realised) volatility: annualised standard deviation of log returns, σ_ann = σ_daily × √252.
• Implied volatility (IV): the σ that makes the model price equal the market price - found by iteration (Newton-Raphson using vega), because the formula cannot be inverted. IV is the market's forward-looking volatility quote; traders quote options in vol, not rupees.
• India VIX: NSE's volatility index, computed from near-month and next-month Nifty option order books on the CBOE VIX methodology; it is the expected annualised Nifty volatility over the next 30 days. It spikes on budget days, election results and global shocks.
• Smile and skew: plotting IV against strike gives a skew for equity indices (OTM puts carry higher IV - crash insurance demand) and a smile for currencies. Term structure of IV = IV against expiry.
• IV crush: after a scheduled event (results, Union Budget, RBI policy) IV collapses, so a long straddle can lose even if the underlying moves - a standard interview trap.

The Greeks

• Delta (Δ = ∂V/∂S): rupee change in premium per ₹1 move. Call 0 to +1, put −1 to 0; ATM ≈ ±0.5; also read as the hedge ratio and (roughly) the risk-neutral probability of finishing ITM. Futures delta ≈ 1, stock delta = 1.
• Gamma (Γ = ∂²V/∂S²): rate of change of delta. Always positive for a long option, largest ATM and near expiry. High gamma means the delta hedge must be rebalanced often; a short-gamma book bleeds on big moves.
• Theta (Θ = ∂V/∂t): time decay, usually quoted per day and negative for long options, largest for ATM options close to expiry and non-linear (decay accelerates). Option sellers are long theta, short gamma.
• Vega (ν = ∂V/∂σ): change per 1 percentage-point change in IV; positive for long calls and puts, highest for ATM and long-dated options; zero at expiry.
• Rho (ρ = ∂V/∂r): positive for calls, negative for puts; small for short-dated equity options, material for long-dated and FX options.
• Second-order: vanna (∂Δ/∂σ), charm (∂Δ/∂t), volga/vomma (∂ν/∂σ) - used by vol desks.
• Greeks are additive across a book after multiplying by position size and lot size; risk systems report net delta, gamma, theta, vega per underlying.
• Taylor approximation: ΔV ≈ Δ·δS + ½Γ·(δS)² + Θ·δt + ν·δσ. Delta alone is a first-order hedge; gamma is the convexity correction.

Hedging and strategy construction

• Delta hedging: hold −Δ of the underlying (shares or futures) against the option position to make the book delta-neutral; rebalance as delta drifts (dynamic hedging). Hedging error rises with gamma and with the rebalancing interval.
• Gamma and vega hedging: a delta-neutral book is still exposed to large moves and to IV; neutralise gamma/vega by trading a second option (different strike or expiry), then re-delta-hedge. Gamma scalping monetises a long-gamma book by buying low/selling high while re-hedging.
• Protective put (portfolio insurance): long asset + long put = floor at K, cost = premium; breakeven = S + p. Used by Indian mutual funds and promoters to protect a holding without selling.
• Covered call: long asset + short OTM call; premium income caps upside at K. Suits a range-bound view.
• Collar: long asset + long put + short call, often zero-cost; locks a band. Common for promoter stakes and ESOP holders.
• Vertical spreads: bull call spread (buy lower K, sell higher K) and bear put spread - limited risk, limited reward, lower cost and lower vega than a naked option.
• Straddle / strangle: long = buy call and put (same strike / different strikes) for a big move either way; short = sell them for a range-bound view with unlimited risk. Strangle is cheaper with wider breakevens.
• Butterfly / iron condor / calendar: butterfly (buy 1 low, sell 2 middle, buy 1 high) profits if the price pins the middle strike; iron condor is a wider range trade; calendar spread trades the term structure of IV.
• Ratio and risk reversal: risk reversal (long OTM call, short OTM put) trades the skew; ratio spreads take uneven quantities.
• Hedging a cash portfolio with index options: number of contracts ≈ (β × portfolio value) / (index level × lot size), using Nifty or Bank Nifty puts.
• Indian market practice: SEBI/exchange rules govern margins (SPAN + exposure, intraday peak margin, upfront collection); option sellers post margin while buyers pay premium only. Position limits, strike intervals and lot sizes are set in the contract specification and revised periodically - Nifty's lot size was raised to 75 in December 2024. Following SEBI's 2024-25 derivatives measures, each exchange offers one weekly index expiry (NSE Tuesday, BSE Thursday) and index derivatives carry higher contract values.
• STT (since 1 October 2024): 0.1% of premium on the sale of an option and 0.125% of the intrinsic value on exercise; 0.02% on the sale of a futures contract. Exchange-traded derivative trading is excluded from the definition of a speculative transaction, so F&O profits are taxed as non-speculative business income (Income-tax Act, 2025).

Key points

• Option premium - intrinsic value plus time value; time value is zero at expiry.
• Put-call parity - c + Ke^{−rT} = p + Se^{−qT}; the basis of synthetics and parity arbitrage.
• Risk-neutral pricing - discount expected payoff at r using p, not real probabilities; the stock's expected return never enters.
• Binomial hedge ratio - Δ = (c_u − c_d)/(S_u − S_d); the lattice is the only practical route for American options.
• Black-Scholes - closed form for European options under lognormal prices and constant σ; d1, d2 are standardised distances to the strike.
• N(d1) - the call's delta; N(d2) - risk-neutral probability of finishing ITM.
• Implied volatility - the only input backed out of the market price; options are traded in vol terms.
• Volatility skew - OTM index puts trade at higher IV than OTM calls, contradicting constant-σ Black-Scholes.
• Delta - hedge ratio and sensitivity to spot; ATM ≈ 0.5, deep ITM → 1.
• Gamma - convexity; peaks ATM near expiry; long options are long gamma, sellers short gamma.
• Theta - decay per day; sellers earn it, buyers pay it; accelerates in the last week.
• Vega - ₹ per 1 vol point; largest for long-dated ATM options, zero at expiry.
• Rho - rate sensitivity; negligible for weekly equity options, important for FX and long-dated options.
• Delta-neutral ≠ risk-free - gamma, vega and jump risk remain.
• Protective put / collar - buy insurance / buy insurance funded by capping upside.
• Covered call - income strategy; the real cost is forgone upside above the strike.
• Spreads - limited risk and reward, cheaper and less vega-sensitive than naked options.
• Short straddle - highest theta, unlimited loss; margin-heavy and the classic blow-up trade.

Formulas and examples

Example 1 - Put-call parity. Nifty-like index at 1,000; 6-month 1,000-strike call = ₹80; r = 7%, q = 0. p = c − S + K·e^{−rT} = 80 − 1,000 + 1,000 × e^{−0.035} = 80 − 1,000 + 965.61 = ₹45.61. A put quoted at ₹55 is dear: sell the put, buy the call, short the index, lend ₹965.61 - a riskless ₹9.39 per unit before costs.

Example 2 - One-step binomial. S = ₹1,000, K = ₹1,000, T = 1 year, r = 6%, u = 1.25 (S_u = 1,250), d = 0.80 (S_d = 800).

1. e^{rT} = e^{0.06} = 1.06184; p = (1.06184 − 0.80)/(1.25 − 0.80) = 0.26184/0.45 = 0.5819.
2. Payoffs: c_u = 250, c_d = 0.
3. c = e^{−0.06} × (0.5819 × 250) = 0.94176 × 145.46 = ₹137.0.
4. Check by replication: Δ = 250/450 = 0.5556; borrowing B = e^{−rT}(Δ·S_d − c_d) = 0.94176 × 444.44 = 418.56; cost = 0.5556 × 1,000 − 418.56 = ₹137.0. Same answer - the hedge, not any probability guess, sets the price.

Example 3 - Black-Scholes and all five Greeks. S = ₹1,000, K = ₹1,000, T = 0.5, r = 7%, σ = 25%, q = 0.

1. σ√T = 0.25 × 0.7071 = 0.17678; d1 = [0 + (0.07 + 0.03125)(0.5)]/0.17678 = 0.050625/0.17678 = 0.2864; d2 = 0.2864 − 0.17678 = 0.1096.
2. N(d1) = 0.6127, N(d2) = 0.5436; K·e^{−rT} = 965.61.
3. Call = 1,000(0.6127) − 965.61(0.5436) = 612.7 − 524.9 = ₹87.8.
4. Put = 965.61(0.4564) − 1,000(0.3873) = 440.7 − 387.3 = ₹53.4 (parity confirms: 87.8 − 1,000 + 965.61 = 53.4).
5. φ(d1) = 0.39894 × e^{−0.0410} = 0.3829.
   • Δ_call = 0.613; Δ_put = −0.387.
   • Γ = 0.3829/(1,000 × 0.17678) = 0.002166 per ₹ (a ₹10 move changes delta by 0.022); same for call and put.
   • ν = 1,000 × 0.7071 × 0.3829 = 270.8 → ₹2.71 per vol point, so IV 25% → 26% lifts the call to about ₹90.5.
   • Θ_call = −(1,000 × 0.3829 × 0.25)/(2 × 0.7071) − 0.07 × 965.61 × 0.5436 = −67.69 − 36.74 = −104.43 per year = −₹0.29 per day.
   • ρ_call = 1,000 × 0.5 × 0.96561 × 0.5436/100 = ₹2.62 per 1% rate rise.

Example 4 - Delta hedge and why gamma still hurts. A desk is short 10 lots of a 25,000-strike Nifty call (lot 75, so 750 units) with Δ = 0.55 and Γ = 0.0003 per point.

1. Position delta = −750 × 0.55 = −412.5; buy 412.5 index units ≈ 5.5 futures lots (round to 6) to go delta-neutral.
2. Nifty rises 100 points. First order: option loses 750 × 0.55 × 100 = ₹41,250; futures gain 412.5 × 100 = ₹41,250 - offset.
3. Gamma term: ½ × 750 × 0.0003 × 100² = ₹1,125 loss, because the short option's delta grew to 0.58 during the move. Short gamma always loses on re-hedging; the compensation is theta of about 750 × 4 = ₹3,000 a day.

Example 5 - Strategy payoffs on a ₹1,000 stock (using the Example 3 premia; call 1,000 = 87.8, put 1,000 = 53.4, assume call 1,100 = ₹45).

Protective put check at S_T = 800: stock loses 200, put pays 200, so value = 1,000 − 53.4 = ₹946.6 - the floor. Bull spread at S_T = 1,150: 150 − 50 = 100 gross − 42.8 = ₹57.2, the cap.

Where it is used

• Broking and prop trading: option market making and jobbing on NSE Nifty/Bank Nifty and BSE Sensex contracts; risk desks at Zerodha, Angel One, ICICI Securities and prop firms monitor net delta-gamma-vega limits intraday.
• Treasury and corporate hedging: USD/INR options and option collars for importers and exporters; interest-rate options priced with Black's model; commodity options on MCX for gold, crude and base metals.
• Mutual funds, insurers and PMS: protective puts and covered calls within SEBI's derivative exposure limits; structured debentures whose payoff is a bond plus a Nifty call, priced with these models.
• Banks and NBFCs: embedded optionality in prepayable loans and callable bonds; Basel market-risk capital and the Greeks-based sensitivities approach under FRTB.
• Job roles: derivatives analyst, options trader, quant/risk analyst (Nuvama, Edelweiss, Goldman Sachs and Morgan Stanley India centres), model validation, product control; NISM Series-VIII certification is a hiring filter for dealing roles.
• ESOP and promoter stake management: collars and forwards to monetise or protect concentrated holdings without an outright sale.

Interview questions

• Q: Why does the expected return of the stock not appear in the Black-Scholes formula? A: Because the option is priced by replication - a delta hedge removes the stock's risk, so the portfolio must earn the risk-free rate, and only r survives in the risk-neutral measure.
• Q: What does N(d2) represent? A: The risk-neutral probability that the option expires in the money; N(d1) is the call's delta and the discounted expected spot conditional on exercise, scaled.
• Q: An ATM option has delta 0.5 - does that make it a coin toss? A: Roughly, but not exactly; because prices are lognormal, N(d2) for an ATM call is slightly below 0.5 for low rates, and delta is a hedge ratio first and a probability only approximately.
• Q: Why is a volatility smile evidence against Black-Scholes? A: The model assumes one constant volatility for all strikes; the market charges higher IV for OTM puts because real returns have fat tails and jumps, so one σ cannot fit every strike.
• Q: Your book is delta-neutral - what can still lose money? A: Gamma (large or fast moves), vega (an IV shift), theta drift, dividend and rate changes, plus rebalancing costs and gap risk.
• Q: Who wins from time decay, and what is the catch? A: The seller earns theta, but is short gamma - one big move can wipe out weeks of decay, and the margin requirement is far heavier than the buyer's premium.
• Q: When is early exercise of an American option rational? A: For a deep ITM put when the interest earned on the strike exceeds the remaining time value, and for a call just before a large dividend.
• Q: How would you hedge a ₹10 crore equity portfolio with beta 1.2 using Nifty options? A: Buy roughly (1.2 × 10,00,00,000)/(Nifty level × lot size) OTM put contracts of the chosen expiry - a floor for a known premium; a collar funds it by selling calls.
• Q: Why do we iterate to find implied volatility? A: The Black-Scholes price is not invertible in σ analytically, so Newton-Raphson (using vega as the derivative) or bisection converges to the σ that matches the quoted premium.
• Q: What happens to gamma and vega as expiry approaches? A: ATM gamma spikes towards infinity while vega collapses to zero - so a near-expiry ATM option is a pure spot bet, not a volatility bet.
• Q: You expect a large move around results but IV is already 70% - what do you do? A: Avoid a naked long straddle; IV crush after the event can lose money even on a correct direction. A spread, ratio or calendar that is short the inflated IV is usually better.
• Q: Put-call parity is violated in the market - would you arbitrage it? A: Only after costs: STT, brokerage, impact cost, dividend and rate assumptions, and margin funding often exceed the apparent gap, and for American or stock options the parity is an inequality.

Common mistakes

• Using the stock's expected return instead of r, or treating the risk-neutral probability p as a real probability.
• Mixing time units - putting T in days, σ in daily terms, or r in percent rather than decimal - and getting d1 wrong by an order of magnitude.
• Applying Black-Scholes to American options with dividends, or forgetting the e^{−qT} adjustment for index options.
• Quoting vega or theta without a unit: vega is per 1 percentage point of IV and theta per day only after dividing the annual figure.
• Forgetting lot size and position sign when aggregating Greeks, so a hedge is 75 times too small.
• Believing delta-neutral means risk-free, and ignoring gamma and vega on a short-option book.
• Computing strategy breakevens on intrinsic value alone - the premium paid or received must be included, as must the second breakeven of a straddle.

Related topics

Derivatives: futures and options basics · Financial risk management: market, credit and operational risk · Security analysis and portfolio management · Financial markets and instruments · International finance and foreign exchange risk management · Financial and risk analytics