Black-Scholes-Merton Implied Volatility Calculators:
Newton-Raphson & Bisection Models (10x Banks)
Calculate Implied Volatility using the Black-Scholes-Merton (with Dividends) Newton-Raphson and Bisection Models. This professional Implied Volatility Calculator calculates Implied Volatility percentage results given the typical BSM inputs and Option Price.
The spreadsheet includes 10 individual BSM I.V. Calculators (in Apple Numbers) and uses two robust models to derive the Implied Volatility percentage: The Newton-Raphson and Bisection models. The calculator sheet is unlocked and fully editable.
Overview of BSM Implied Volatility Calculator (x10 Banks). A higher resolution image follows below:
1) What Is Implied Volatility?
Implied volatility is the projected annual price movement of an Underlying Asset like, e.g. the Dow Jones index, or the £/$. It is presented on a one Standard Deviation (SD) basis, (a measure of variance). This figure is derived from the Options price - in other words, the Black-Scholes-Merton model's inputs solve for I.V., not the other way around.
Implied Volatility (I.V.) for Options represents the market's forecast of a potential price change in the Underlying Asset, expressed as a percentage. It indicates expected volatility but not direction and directly influences option premiums: High I.V. makes Options more expensive, while low I.V. makes them cheaper.
2) Implied Volatility is Inputted into the B.S.M. Model
So now we have an understanding of what I.V. -- whilst also observing that generally when Underlying Asset prices are rising Call Option prices also rise and Put Option prices decrease in value (and visa versa), we can see that the Black-Scholes-Merton model doesn't actually calculate Implied Volatility and that you enter a given I.V. percentage value obtained from your broker, into the model.
So how do we go about solving what the I.V. is for an Option if we know it's given Option price? This is where we need to know how the value of an Option price is derived so that we can reverse calculate the I.V. value:
3) How the Black-Scholes Formula Calculates Option Prices:
The Black-Scholes Model calculates Option Prices given these inputs:
• Current Underlying Asset Price (S)
• Strike Price (K)
• Time to Expiration (T)
• Risk-Free Interest Rate (r)
• Volatility (σ) ← this is the key
Call Price Formula:
Call Price = S·N(d₁) - K·e^(-rT)·N(d₂)
Where:
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
N() = cumulative standard normal distribution
Put Price Formula:
Put Price = K·e^(-rT)·N(-d₂) - S·N(-d₁)
Where:
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
N() = cumulative standard normal distribution
Note: The d₁ and d₂ calculations are identical to the Call formula - only the structure of the pricing equation changes, using N(-d₁) and N(-d₂) instead of N(d₁) and N(d₂).
4) How Implied Volatility is Actually Calculated:
I.V. is calculated by working backwards:
• You observe the actual market price of an Option.
• You know S, K, T, and r (all observable).
• You solve for σ (volatility) that makes the Black-Scholes-Merton price equal the market price.
The problem: There's no closed-form solution to reverse the B.S.M. formula for Volatility. You can't just rearrange the equation algebraically.
The solution: Use numerical methods:
Newton-Raphson Method (most common):
This iterative approach:
• Makes an initial guess for σ.
• Calculates what the Option price would be with that σ.
• Compares it to the actual market price.
• Adjusts σ based on how the price changes with volatility (using "Vega" - the option's sensitivity to volatility).
• Repeats until convergence (usually 3-5 iterations).
The iteration formula: σ_new = σ_old - (BS_price - Market_price) / Vega.
Bisection Method - slower but more robust and provided in my 10 calculators as a back up solution:
• Sets a lower bound (e.g., σ = 0.01) and upper bound (e.g., σ = 5.0) for volatility.
• Calculates the midpoint σ_mid between the bounds.
• Calculates what the Option price would be with σ_mid.
• Compares it to the actual market price.
• Narrows the range: if B.S.M. price is too high, the upper bound becomes σ_mid; if too low, the lower bound becomes σ_mid.
• Repeats, cutting the search space in half each time, until the range is tiny (convergence).
The iteration formula: σ_mid = (σ_lower + σ_upper) / 2. Then it updates bounds based on whether BSM_price(σ_mid) > or < Market_price.
Why is Implied Volatility Important in Options Trading?
1. I.V. Tells You How Expensive Options Are:
• High I.V. = Expensive premiums (Options cost more).
• Low I.V. = Cheap premiums (Options cost less).
• It's the market's expectation of future Volatility baked into the Option price.
2. Critical for Option Sellers:
If you're selling Options (collecting premium), you want:
• High I.V. - You collect more premium, which is ideal if you believe Volatility will decrease.
• Sell when I.V. is elevated and buy back when it drops, i.e. profit from "I.V. crush."
3. Critical for Option Buyers:
If you're buying Options, you want:
• Low I.V. - Pay less for the Option.
• Risk: If you buy high I.V. Options and I.V. drops (even if you're right on direction), you can still lose money due to "I.V. crush."
4. I.V. Helps You Find Mis-priced Options:
The B.S.M. Implied Volatility Calculator helps you:
• Compare the market price vs. the B.S.M. theoretical price.
• Spot when brokers are overpricing or underpricing options.
• See if the I.V. being used makes sense.
5. I.V. Drives Expected Moves:
As my Calculator shows, I.V. tells you the expected price range:
• 68.2% probability (1 SD)
• 95.4% probability (2 SD)
• 99.7% probability (3 SD)
This helps you choose Strike (K) prices intelligently.
Example: If Oil is trading at £66 with an I.V. of 24.95%:
• The Expected Move = ±£16.46 over a yr, (66 x 0.2495). +£16.46 = £82.46 and -£16.46 = £49.54.
• If Selling a Put at a Strike (K) of £63, you know there's a 68.2% chance oil stays above £49.54:
• Why? Because 68.2% of the time, market movement is "normal" and won't exceed the lower boundary, but 31.8% of the time (100% - 68.2%), oil could drop below £49.54.
How to Use the Implied Volatility Calculators:
- • Enter the Black_Scholes-Merton Option data inputs in the I.V. Calculator in the blue fields marked with a ">>" (not incl. C10). Enter the Asset name (in cell C6).
- • Enter the Underlying Price (in C7).
- • Select the Dropdown in C8 and choose "Call" or "Put."
- • Enter the Broker "Market Offered Option Price" (in C9). Use the Mid Price between the Bid/Ask Spread.
- • Enter the Strike Price (K) (in C11).
- • Input the Interest Rate (C12) and Dividend Yield (if applicable in C13).
- • Input the Start and End Dates (C15 and C15).
- • Using the broker Option chain or historic data enter an initial guesstimate of the Implied Volatility percentage (in C16).
- • The Implied Volatility (I.V.) result outputs are presented in the light green fields, (C18 and C19).
- • In the event that the Newton-Raphson model fails to derive the Implied Volatility, the calculator gives a warning "Can't Compute Newton I.V." and uses a second backup calculation method - the Bisection model - to find the given Implied Volatility.
Key Features Recap:
- • Input: The Option Price and other typical BSM Inputs like Spot (S), Strike (K) and Start and Expiry Dates.
- • Output: Implied Volatility (I.V.).
- • Purpose / Use: Comparing Option prices, find Option mis-pricing and Volatility surface analysis.
- • Multi-Bank System: The I.V. calculator includes 10 calculators (1 rows of 10).
- • Expected Market Move: Estimate how much an asset is likely to move over any given timeframe, from minutes to months, based on probabilities of 68.2%, 95.4%, and 99.7%.
- • Live @Now Pricing: Set option start dates to the current moment to monitor real-time price changes.
- • Delta Warning: Get a red alert when OTM Delta falls below 60%, helping short sellers stay on top of risk.
- • Accurate Implied Volatility Pricing to 2 decimal places, using two mathematical solutions: The Newton-Raphson and Bisection models.
- • 0DTE Capabilities: Calculate I.V. right down to the last minutes before expiry for real-time decision-making.
- • The Spreadsheet is unlocked and can be edited.
Don't Let Market Uncertainty Catch You Off Guard: Stop Guessing. Start Knowing.
Compare Option Prices, find Option Mis-Pricing and carry out Volatility Surface Analysis.
✓ Fast Implied Volatility Pricing
✓ Uses Industry Standard Newton-Raphson and Bisection Models
✓ Understand what Implied Volatility the Market is Implying
✓ Benefits from Extra Features including 0DTE Calculations and Expected Move Projections
Trade Like a Professional — Gain Confidence with Data-Driven Decisions to Maximise Profits and Minimise Risk.
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Best of Luck in Your Options Trading,
Ian,
B.Sc. Finance (Hons), UWIST, Wales.