Option Pricing — Calls, Puts & Payoff Diagrams

An option is a contract giving the buyer the right but not the obligation to buy (call) or sell (put) an asset at a fixed strike price $K$ before or on an expiry date.

• Call payoff at expiry: $\max(S_T - K,\ 0)$ — profitable only if the stock price $S_T$ rises above $K$.
• Put payoff at expiry: $\max(K - S_T,\ 0)$ — profitable only if $S_T$ falls below $K$.

The buyer pays an upfront premium. The Black-Scholes model prices the option as the discounted expected payoff under a risk-neutral world. Intuitively: higher volatility $\sigma$ or longer time $T$ increases the chance that the option finishes in the money, so both increase the premium. The four key Greeks — Delta ($\partial C/\partial S$), Gamma, Theta (time decay), and Vega ($\partial C/\partial \sigma$) — measure how the price changes with each input.

Black-Scholes call price:

$C = S \cdot N(d_1) - K e^{-rT} \cdot N(d_2)$

$d_1 = \frac{\ln(S/K) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$

where $N(\cdot)$ is the standard normal CDF. Key takeaway: $C$ rises with $S$, $\sigma$, $T$ and falls as $K$ increases.

A call option: $S = ₹100$, $K = ₹105$, premium = ₹4, expiry in 3 months.

• At expiry $S_T = ₹115$: payoff $= 115 - 105 = ₹10$; net profit $= 10 - 4 = ₹6$.
• At expiry $S_T = ₹102$: payoff $= 0$ (out of the money); net profit $= -₹4$ (full premium lost).

Break-even price at expiry $= K + \text{premium} = 105 + 4 = ₹109$.

Airlines buy call options on jet fuel to cap fuel costs when prices spike. Exporters buy put options on foreign currency to protect rupee revenues when the dollar weakens. Mispriced options on mortgage-backed securities contributed to the 2008 financial crisis — demonstrating how critically option valuation matters in practice.

1. A put option has $K = ₹200$ and premium ₹8. At expiry $S_T = ₹185$. What is the net profit or loss for the put buyer?

2. All else equal, does a call option become more or less expensive if volatility $\sigma$ increases? Give a one-sentence explanation using the concept of payoff probability.