Implied volatility of an option
This sub‑topic explains Implied Volatility (IV) – the market‑derived estimate of future price fluctuations embedded in option prices. Understanding IV is crucial for valuing options, managing risk, and answering several NISM exam questions. It links the theoretical Black‑Scholes model with real‑world market data and fits within the broader module on Equity Derivatives.
Learning Objectives
- 1Define Implied Volatility and differentiate it from Historical Volatility.
- 2Explain how IV is derived using the Black‑Scholes model.
- 3Identify factors that cause IV to change for Indian equity options.
- 4Apply a simple numerical method to approximate IV and interpret its exam relevance.
What is Implied Volatility?
Implied Volatility (IV) is the volatility input that, when placed into an option‑pricing model (most commonly Black‑Scholes), reproduces the observed market price of the option.
Unlike Historical Volatility, which is calculated from past price movements, IV reflects the market’s collective expectation of future volatility and therefore changes continuously as market sentiment evolves.
For the NISM exam, IV is frequently asked in the context of option pricing, volatility smile, and risk‑management scenarios. Remember: IV is a *model‑derived* number, not a directly observable market figure.
- IV helps traders gauge whether options are relatively cheap or expensive.
- Higher IV leads to higher option premiums, all else equal.
Many candidates mistakenly treat Implied Volatility as the same as Historical Volatility. In the exam, always check the wording: if the question asks for the volatility “implied by the option price,” use IV, not the past‑price‑based figure.
How Implied Volatility is Derived
IV is obtained by solving the Black‑Scholes pricing equation for the volatility variable (σ) such that the theoretical price equals the actual market price of the option.
Because the Black‑Scholes formula is non‑linear in σ, a closed‑form solution does not exist. Practitioners use numerical techniques – most commonly the Newton‑Raphson iterative method or a simple trial‑and‑error approach – to converge on the σ that matches the market price.
In the NISM syllabus, you are expected to understand the concept of “solving for σ” and to recognise that the resulting σ is the Implied Volatility. You are not required to perform a full Newton‑Raphson calculation, but you should be able to interpret a given IV or approximate it using a quick method.
Where:
C= Market price of the European call option (in rupees)S= Current spot price of the underlying equity (in rupees)K= Strike price of the option (in rupees)r= Risk‑free interest rate (annual, expressed as decimal)T= Time to maturity in years (e.g., 0.5 for six months)σ= Volatility of the underlying (annual, expressed as decimal)N(d_{1})= Cumulative standard normal distribution at d1N(d_{2})= Cumulative standard normal distribution at d2Worked Example
Given S = 100, K = 100, r = 0.05, T = 0.5 years, σ = 0.20: Step 1: d_{1} = [ln(100/100) + (0.05 + 0.20^{2}/2)\times0.5] / (0.20\times\sqrt{0.5}) = 0.2121 Step 2: d_{2} = d_{1} - σ\sqrt{T} = 0.2121 - 0.20\times\sqrt{0.5} = 0.0707 Step 3: N(d_{1}) ≈ 0.584, N(d_{2}) ≈ 0.528 (using standard normal tables) Step 4: C = 100\times0.584 - 100\times e^{-0.05\times0.5}\times0.528 ≈ 58.4 - 100\times0.9753\times0.528 ≈ 58.4 - 51.5 = 6.9 Verification: The Black‑Scholes formula with σ = 20% yields a call price of approximately 6.9 rupees.
To extract Implied Volatility, set the market price (C_{mkt}) equal to the Black‑Scholes price and solve for σ. Because σ appears inside the normal‑distribution terms, the equation must be solved iteratively.
A common exam shortcut is the Brenner‑Subrahmanyam approximation: \(\sigma \approx \sqrt{\frac{2\pi}{T}} \times \frac{C_{mkt}}{S}\). This provides a quick estimate when the option is at‑the‑money and T is short.
Remember that the approximation works best for Indian equity index options (e.g., NIFTY) with low dividend yields. In practice, trading platforms display the exact IV computed by sophisticated algorithms, but the concept remains the same.
If a question gives you the option price, spot price, and time to expiry, you can quickly estimate IV with the Brenner‑Subrahmanyam formula. Ensure the option is near‑the‑money; otherwise the estimate may be off.
Factors Influencing Implied Volatility
IV is not static; it reacts to several market forces. The primary drivers are:
- Underlying price movement – Sudden jumps raise IV as market participants price in higher risk.
- Time to expiry – Short‑dated options often show higher IV because of uncertainty compression.
- Strike price (moneyness) – Out‑of‑the‑money (OTM) options typically have higher IV, creating the "volatility smile".
- Market sentiment and news – Earnings releases, macro‑economic data, or geopolitical events can cause IV spikes.
For the NISM exam, you may be asked to identify which factor would cause IV to increase or decrease, or to interpret a volatility smile chart.
Effect of Moneyness on Implied Volatility for Indian Equity Options
| Moneyness | Typical IV Trend | Exam Note |
|---|---|---|
| In‑the‑Money (ITM) | IV tends to be lower than ATM | IV rises as option moves OTM |
| At‑the‑Money (ATM) | IV usually highest for short‑dated contracts | Most common reference point in questions |
| Out‑of‑the‑Money (OTM) | IV often higher, forming the smile | Watch for steep smile in volatile markets |
Implied Volatility in the Indian Market
SEBI does not prescribe a specific method for calculating IV, but Indian brokers and exchanges (NSE, BSE) publish IV for index options such as NIFTY and BANKNIFTY. The values are usually expressed in annualised percentage terms.
Typical IV ranges for NIFTY options are 12‑25% in calm markets and can surge above 40% during periods of heightened uncertainty (e.g., general elections or major policy announcements).
Exam questions may present a table of IVs for different strikes and ask you to identify the strike with the highest IV or to comment on the shape of the volatility curve.
Sample Implied Volatility Smile for NIFTY (30‑day expiry)
Practical Example – Approximating Implied Volatility
Scenario
An investor sees a NIFTY call option (strike 10,200) trading at ₹120. The current NIFTY spot is ₹10,150, time to expiry is 0.25 years, and the risk‑free rate is 6% p.a. The option is near‑the‑money. Estimate the Implied Volatility using the Brenner‑Subrahmanyam approximation.
Solution
Step 1: Use the approximation \(\sigma \approx \sqrt{\frac{2\pi}{T}} \times \frac{C_{mkt}}{S}\).\nStep 2: Compute the square‑root factor: \(\sqrt{\frac{2\pi}{0.25}} = \sqrt{\frac{6.2832}{0.25}} = \sqrt{25.1328} \approx 5.013\).\nStep 3: Ratio of option price to spot: \(\frac{120}{10,150} \approx 0.01183\).\nStep 4: Multiply: \(5.013 \times 0.01183 \approx 0.0593\) or 5.93% annualised.\nStep 5: Because the option is slightly OTM, the true IV will be a bit higher; rounding to the nearest whole number gives an exam‑friendly answer of 6% IV.
Conclusion
The quick approximation yields an IV of about 6%, illustrating how market participants can gauge volatility without full iterative calculations. Remember to adjust upward for OTM options in the exam.
Common Mistakes to Avoid
1. Treating IV as a constant – IV varies with strike and expiry; assuming a single value for all options leads to pricing errors.
2. Confusing the input σ with the output IV – In the Black‑Scholes formula, σ is the volatility input. When the market price is known, solving for σ gives the IV.
3. Ignoring the effect of dividends – For equity options that pay dividends, the dividend yield must be incorporated; otherwise the IV estimate will be biased.
4. Misreading the time unit – Always express T in years. Using months or days without conversion causes large IV miscalculations.
If the question gives time to expiry in days, convert to years (e.g., 30 days = 30/365 ≈ 0.082 years) before applying any formula.
Exam Relevance and Quick Recall
Key points that frequently appear in NISM multiple‑choice questions:
- IV is the volatility that makes the Black‑Scholes price equal to the market price.
- Higher IV → higher option premium; lower IV → cheaper premium.
- IV varies with strike (volatility smile) and expiry (term structure).
- Use the Brenner‑Subrahmanyam formula for a fast IV estimate when the option is ATM and T is short.
Memory aid: "IV = Implied = Input Volatility that fits the market price." This helps you quickly decide which variable to solve for.
⭐Exam Takeaways
- Implied Volatility is the model‑derived volatility that matches the observed option price.
- IV differs from Historical Volatility; it reflects market expectations, not past price movements.
- The Black‑Scholes formula is used to back‑solve for IV; the equation is solved iteratively.
- Brenner‑Subrahmanyam approximation provides a quick IV estimate: σ ≈ √(2π/T) × (C/S).
- IV rises for out‑of‑the‑money options, creating the volatility smile commonly shown in charts.
- In the Indian market, IV for NIFTY options typically ranges between 12% and 25% in stable periods.
- Always convert time to years and include dividend yields when relevant.
- Common exam traps: confusing IV with Historical Volatility and ignoring units of time.
Practice Questions
8 questions on Implied volatility of an option
What is Implied Volatility (IV) as defined in the study material?
How does Implied Volatility differ from Historical Volatility?
Which market factor is most likely to cause Implied Volatility to increase?
When an at‑the‑money Indian equity index option has a short time to expiry, which shortcut does the material recommend for a quick IV estimate?
In the example where S=₹100, K=₹100, r=5%, T=0.5 years and the Black‑Scholes call price is ₹6.9, what volatility σ was used?
Using the Brenner‑Subrahmanyam approximation, estimate the IV for a NIFTY call with C=₹120, S=₹10,150 and T=0.25 years. Which value is closest?
Based on the sample volatility smile for NIFTY, which strike exhibits the highest implied volatility?
What is the typical range of Implied Volatility for NIFTY options in calm market conditions?