Option pricing and the Greeks

Nobody expects a distributor to derive the Black-Scholes formula, and the exam does not ask you to. It does expect you to say which way a premium moves when volatility rises, to use put-call parity to find a missing price, and to turn a delta into a rupee number. This chapter gives you the inputs, the parity relation and a working feel for each Greek.

Six things that move a premium

An option's premium is intrinsic value plus time value, and six variables drive it. Learn the direction for both calls and puts.

• Spot price (S): higher spot raises call premiums and lowers put premiums.
• Strike price (K): a higher strike makes a call cheaper and a put dearer.
• Time to expiry (T): more time means more chance of a favourable move, so both calls and puts are worth more. Time value decays as expiry approaches.
• Volatility (σ): higher expected volatility raises both call and put premiums, because the buyer's loss is capped while the upside is not.
• Interest rate (r): a higher rate raises call premiums slightly (buying a call defers paying for the stock) and lowers put premiums slightly.
• Dividends: an expected dividend lowers the forward price, so it reduces call premiums and raises put premiums.

Volatility is the one input nobody can observe. Historical volatility is measured from past returns; implied volatility is the number that, fed into the pricing model, reproduces the market premium. India VIX is the implied volatility of near-term Nifty options, expressed as an annualised percentage.

Put-call parity

A European call and put with the same strike and expiry on the same underlying are tied together by a no-arbitrage relation. Holding a call and a bond that pays K at expiry gives the same payoff as holding the stock and a put. So the two packages must cost the same today.

If the put actually trades at ₹190, it is ₹27.7 too dear relative to the call: sell the put, sell the stock, buy the call and invest the strike's present value, and the position earns the difference whatever Nifty does at expiry. In practice the ATM call is worth more than the ATM put by roughly the interest on the strike, which is why ATM calls on Nifty are always a little dearer than ATM puts.

Black-Scholes: the inputs

The Black-Scholes model gives a fair premium for a European option from five inputs: spot price, strike, time to expiry, risk-free rate and volatility; a sixth, dividend yield, is added for index and stock options. It assumes continuous trading, constant volatility, no transaction costs and log-normally distributed prices. The exam asks for the inputs and the assumptions, not the equation. For American options or stocks with discrete dividends, the binomial model, which steps the price up or down over small intervals, is the standard alternative. Both models output a theoretical price and the Greeks below.

The Greeks

A Greek measures how much the premium changes when one input changes and the rest stay still.

• Delta (Δ): change in premium per one-point change in spot. Calls have delta between 0 and +1, puts between −1 and 0. ATM options are near ±0.5; deep ITM near ±1; deep OTM near 0. Delta is also the approximate probability of expiring ITM and the hedge ratio: a writer of 10 lots of a 0.5-delta call needs long 5 lots of futures to be delta-neutral.
• Gamma (Γ): change in delta per one-point change in spot. Highest for ATM options near expiry. Long options have positive gamma; writers have negative gamma, which is why short option positions become dangerous fast.
• Theta (Θ): change in premium per day as time passes, with everything else constant. Negative for the buyer (time decay), positive for the writer. Decay accelerates in the final week.
• Vega: change in premium per 1% change in implied volatility. Positive for long calls and puts; largest for ATM options with time left.
• Rho (ρ): change in premium per 1% change in the interest rate. Positive for calls, negative for puts, and small for short-dated options.

The gamma term is why a long option gains more than delta suggests on a big favourable move and loses less on an adverse one. For exam purposes, the first-order answer, delta times the move, is usually what is asked; add gamma only if the question gives it.