Basics of Option Pricing and Option Greeks
This sub‑topic covers the fundamentals of option pricing and the five major Option Greeks. Understanding how an option premium is built and how Greeks measure sensitivity is essential for NISM Series VIII questions. The content links pricing theory to practical exam scenarios involving Indian equity derivatives.
Learning Objectives
- 1Identify the components of an option premium
- 2Apply the Black‑Scholes formula for a European call
- 3Define and interpret Delta, Gamma, Theta, Vega and Rho
- 4Calculate Greeks using standard syllabus formulas
Option Premium – Components
Option premium is the price an investor pays to acquire an option contract. It consists of two parts – intrinsic value and time value. Intrinsic value reflects the immediate profit if the option were exercised today, while time value compensates the writer for the possibility of future favourable moves.
For a call option, intrinsic value is the excess of the underlying price (S) over the strike price (K). For a put, it is the excess of K over S. If the option is out‑of‑the‑money, intrinsic value is zero, and the entire premium is time value. The time value declines as expiry approaches, a phenomenon known as time decay.
Exam questions often ask you to break down a quoted premium into its intrinsic and time components, or to identify why a deep‑in‑the‑money option has a small time value. Remember that the premium can never be less than the intrinsic value – a common trap for candidates.
- Intrinsic value = max(0, S‑K) for calls, max(0, K‑S) for puts
- Time value = Premium – Intrinsic value
Where:
S= Spot price of the underlying asset in rupeesK= Strike price of the option in rupeesWorked Example
Given S = 1,200 ₹ and K = 1,150 ₹: Step 1: Compute S - K = 1,200 - 1,150 = 50 Step 2: Intrinsic Value = max(0, 50) = 50 ₹ Verification: max(0, 1,200 - 1,150) = 50.
Where:
Option Premium= Quoted price of the option in rupeesIntrinsic Value= Value calculated from the intrinsic value formulaWorked Example
If the option premium is 80 ₹ and the intrinsic value (from previous example) is 50 ₹: Step 1: Time Value = 80 - 50 = 30 ₹ Verification: 80 - 50 = 30.
Students sometimes treat the quoted premium as the intrinsic value. Always verify that the premium is at least the intrinsic amount; the excess is the time value.
Black‑Scholes Option Pricing Model
The Black‑Scholes model provides a closed‑form solution for pricing European‑style equity options. It assumes a frictionless market, constant risk‑free rate, and log‑normal distribution of the underlying price. SEBI’s guidelines reference this model for valuation of listed index options such as NIFTY.
Key inputs are the spot price (S), strike price (K), time to expiry (T in years), risk‑free rate (r), and volatility (σ). The model yields separate formulas for call and put prices, but the call formula is most frequently tested.
In the exam, you may be given the inputs and asked to compute the theoretical price, or you may need to identify which input influences the price most. Remember that higher volatility or longer time increases the premium, while higher interest rates raise call prices but lower put prices.
Where:
C= Theoretical price of the European call option in rupeesS= Spot price of the underlying asset in rupeesK= Strike price in rupeesr= Continuously compounded risk‑free rate (decimal)T= Time to expiry in yearsN(·)= Cumulative standard normal distribution functiond_1= Intermediate variable = [ln(S/K) + (r + σ^2/2)T] / (σ\sqrt{T})d_2= Intermediate variable = d_1 - σ\sqrt{T}Worked Example
Given S = 1,200 ₹, K = 1,150 ₹, r = 5% p.a. (0.05), σ = 20% p.a. (0.20), T = 0.5 yr: Step 1: Compute d1 = [ln(1200/1150) + (0.05 + 0.20^2/2)×0.5] / (0.20×√0.5) = 0.50 (approx) Step 2: Compute d2 = d1 - 0.20×√0.5 = 0.30 (approx) Step 3: N(d1) ≈ 0.6915, N(d2) ≈ 0.6179 (standard normal table) Step 4: C = 1,200×0.6915 - 1,150×e^{-0.05×0.5}×0.6179 = 829.8 - 1,150×0.9753×0.6179 = 829.8 - 57.3 ≈ 11.9 ₹ Verification: 1,200×0.6915 - 1,150×e^{-0.025}×0.6179 = 11.9.
The exam provides N(d) values when required. Focus on understanding the steps to compute d1 and d2 rather than trying to remember tables.
Option Greeks – Quick Reference
Key Option Greeks and Their Effect on Option Price
| Greek | Definition | Effect on Option Price |
|---|---|---|
| Delta | Rate of change of option price with respect to underlying price | Positive for calls, negative for puts |
| Gamma | Rate of change of Delta with respect to underlying price | Measures curvature; highest near‑the‑money |
| Theta | Rate of change of option price with respect to time (per day) | Usually negative for long positions |
| Vega | Rate of change of option price with respect to volatility | Positive for both calls and puts |
| Rho | Rate of change of option price with respect to risk‑free rate | Positive for calls, negative for puts |
Delta – Sensitivity to Underlying Price
Delta (Δ) quantifies how much the option price moves for a one‑rupee change in the underlying asset. For a European call, Δ lies between 0 and 1; for a put, it lies between –1 and 0. A Delta of 0.60 means the option price will increase by 0.60 ₹ for every 1 ₹ rise in the underlying.
Delta is also used as a proxy for the probability that the option will finish in‑the‑money at expiry, a fact often tested in NISM questions. Higher Delta indicates a deep‑in‑the‑money option, while a Delta close to zero signals an out‑of‑the‑money contract.
In practice, traders hedge their option positions by taking an opposite position in the underlying equal to the Delta. The exam may ask you to compute the hedge ratio or to choose the correct Delta value from a given set of N(d1) numbers.
Where:
\Delta= Delta of the call optionN(d_1)= Cumulative standard normal value at d1d_1= Same d1 as in Black‑Scholes formulaWorked Example
Using the previous example where d1 = 0.50 and N(d1) ≈ 0.6915: Step 1: \Delta = 0.6915 Verification: N(0.50) = 0.6915, so \Delta = 0.6915.
Gamma – Curvature of the Price Curve
Gamma (Γ) measures the rate of change of Delta with respect to the underlying price. It is always positive for both calls and puts, indicating that Delta becomes more responsive as the option moves closer to the money. High Gamma values are observed for near‑the‑money options with short time to expiry.
Gamma is crucial for risk management because a portfolio with large Gamma can experience rapid changes in Delta, leading to potential over‑hedging or under‑hedging. NISM exams often test the relationship between Gamma, time to expiry, and volatility.
When Gamma is high, a small move in the underlying can cause a large swing in the option’s Delta, which in turn amplifies the profit or loss of a Delta‑hedged position.
Where:
\Gamma= Gamma of the optionN'(d_1)= Standard normal probability density at d1 = \frac{1}{\sqrt{2\pi}}e^{-d_1^2/2}S= Spot price of the underlying in rupees\sigma= Annual volatility (decimal)T= Time to expiry in yearsWorked Example
Using S = 1,200 ₹, σ = 0.20, T = 0.5 yr, and d1 = 0.50: Step 1: N'(d1) = (1/√(2π))·e^{-(0.5)^2/2} ≈ 0.3521 Step 2: Denominator = 1,200 × 0.20 × √0.5 = 1,200 × 0.20 × 0.7071 ≈ 169.7 Step 3: \Gamma = 0.3521 / 169.7 ≈ 0.00207 ≈ 0.0021 per rupee Verification: 0.3521 ÷ (1,200×0.20×0.7071) = 0.0021.
Theta – Time Decay
Theta (Θ) represents the loss in option value for each passing day, assuming all other factors remain constant. For long option holders, Theta is typically negative, reflecting the erosion of time value as expiry approaches. For writers (short positions), Theta is positive, indicating they earn premium each day.
The magnitude of Theta increases as the option gets closer to expiry and is higher for at‑the‑money options. In the Indian market, SEBI’s daily settlement mechanism makes Theta a practical concept for daily P&L calculations.
Exam questions may ask you to identify which Greek is responsible for the daily decline in a deep‑in‑the‑money call, or to compare Theta values across different maturities.
Vega – Volatility Sensitivity
Vega (ν) measures how much the option price changes for a 1 % (0.01) change in implied volatility. Both calls and puts have positive Vega, meaning higher volatility raises the premium. Vega is highest for at‑the‑money options with moderate time to expiry.
In the Indian derivatives market, volatility spikes during earnings announcements or macro‑economic events, causing noticeable Vega‑driven price movements. NISM exams often test the direction of Vega’s impact rather than exact numerical values.
Remember that Vega diminishes as expiry approaches because there is less time for volatility to affect the payoff.
Greek Sensitivities vs. Underlying Price (Near‑the‑Money Call)
Scenario
Rohit, an Indian retail investor, buys a NIFTY call option with strike 15,000 ₹, expiry in 30 days. The current NIFTY spot is 15,200 ₹. The option premium quoted on NSE is 120 ₹. The risk‑free rate is 6% p.a., and the implied volatility is 18% p.a. Rohit wants to know the intrinsic value, time value, and Delta of the option.
Solution
Step 1: Intrinsic Value = max(0, 15,200 ₹ – 15,000 ₹) = 200 ₹. Step 2: Time Value = Premium – Intrinsic = 120 ₹ – 200 ₹ = –80 ₹. Since the premium cannot be less than intrinsic, the quoted premium must be a typo; assume the premium is 320 ₹. Then Time Value = 320 ₹ – 200 ₹ = 120 ₹. Step 3: Compute d1 = [ln(15,200/15,000) + (0.06 + 0.18^2/2)×(30/365)] / (0.18×√(30/365)) ≈ 0.45. Using standard normal tables, N(d1) ≈ 0.673. Therefore Delta ≈ 0.673. Rohit’s option will gain about 0.673 ₹ for each 1 ₹ rise in NIFTY.
Conclusion
The example shows how to decompose a premium, correct a common data error, and compute Delta using the Black‑Scholes inputs – a typical NISM exam requirement.
⭐Exam Takeaways
- Option premium = Intrinsic value + Time value; intrinsic is max(0, S‑K) for calls.
- Black‑Scholes call price: C = S N(d1) – K e^{-rT} N(d2); know how to compute d1 and d2.
- Delta = N(d1) for calls; it indicates price sensitivity and approximate ITM probability.
- Gamma = N'(d1)/(S σ √T); high near‑the‑money, short‑dated options have larger Gamma.
- Theta is usually negative for long positions and measures daily time decay.
- Vega is positive for all options; higher volatility raises the premium.
- Rho (not shown) affects price with interest‑rate changes – positive for calls, negative for puts.
- Use the Greeks to assess risk, hedge ratios, and the impact of market moves in exam scenarios.
Practice Questions
8 questions on Basics of Option Pricing and Option Greeks
What is the formula for the intrinsic value of a call option?
Which Option Greek is positive for both call and put options?
A call option has a spot price of 1,200 ₹ and a strike price of 1,150 ₹. What is its intrinsic value?
According to the study material, how does an increase in volatility affect the premium of a European call option?
Using S = 1,200 ₹, σ = 0.20, T = 0.5 yr, and d1 = 0.50, what is the approximate Gamma of the call option?
If the Delta of a European call option is 0.6915, how many shares of the underlying must be bought to delta‑hedge one option contract?
Which Greek measures the rate of change of Delta with respect to the underlying price?
An option premium is quoted at 40 ₹ for a call where S = 1,200 ₹ and K = 1,150 ₹. According to the study material, why is this premium likely erroneous?