## Implied Volatility Calculator

The Black-Scholes model can be used to estimate implied volatility. Implied Volatility can be estimated using spot price, strike price, asset price, risk-free rate, time to maturity, and dividend yield. To achieve this, given an actual option value, you have to iterate to find the volatility solution. There are various techniques available; however we will use the Newton-Raphson bisection method for Implied Volatility Calculator.

Following below is the equation and explanations.

### Newton-Raphson Method and Implied Volatility.

Implied Volatility is distinctively different from historical volatility measures. The term implied volatility comes from the fact that the volatility is removed from the market prices of options. Using Black-Scholes option pricing model, we can calculate Implied Volatility using trial and error.

A great benefit of Newton-Raphson bisection method is that it gives fast convergences and the error approximation reduces rapidly with each additional iteration. The equation to calculate Implied Volatility of an option:

$$\sigma_{n+1} = \sigma_n - \frac {P_m-p_t(\sigma_n)}{dP_t({\sigma_n}/{d \sigma})}$$

Where,

Pm is the market price of the option which we are trying to solve a fit for,

Pt is the option price given by Black-Scholes equation,

σ is the implied volatility

Once Black-Scholes is structured, we use an iterative technique to solve for σ. This method works for options where Black-Scholes model has a closed form solution.

### Implied Volatility Calculator

An ITM option has 10 days for expiration. The strike price is 55 and the current stock price is 50. The stock has daily volatility of 0.03. The risk free interest rate is assumed to be 0.02.

$$d_1 = \frac {{ln(50/55)+(0.22/365+0.03^2/2)}10}{0.03 \sqrt 10}$$ $$d_2 = \frac {{ln(50/55)+(0.22/365-0.03^2/2)}10}{0.03 \sqrt 10}$$

Alternatively, d2 can be calculated as

$$d_2 = d_1 - \sigma \sqrt t$$

So, the European Call value can be calculated as

$$C = S * Norm(d_1)-K*e^{-rt/365}*Norm(d2)$$
 Variables C = call value, S = spot price of 50, t = expiry period which is 10 days, Norm = Normal probability distribution with mu=0 and sigma = 1, K = option strike price of 55, r = risk free rate of interest – we have assumed to be 0.02 in this example, σ = daily stock volatility of 3% or 0.03

 Stock price: Strike price: Risk free rate (%): Day to maturity: Volatility guess value (%): Dividend yield Optional (%): Option price:

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