Risk Latte - Estimating Implied Volatility using Newton-Raphson method

Estimating Implied Volatility using Newton-Raphson method

Team Latte
September 27, 2009

One of the most efficient algorithms to estimate the implied volatility from the market observed price and the theoretical Black-Scholes formula is the Newton-Rahpson method. Newton Raphson method is used to find the zeros of a real valued function. This means if there is a function then the root of the function would be such that .

If then let say we have some current approximation that produces the zero of the function and we want to better that approximation to which gives . Then using simple differential calculus we can show:

From the above, we get

...................................... (1)

The above algorithm is an iterative algorithm, i.e. the proces works as an iterative method. Our objective is to find where is the desired tolerance limit. We start with some initial value (using guess work) and run the iteration till the value converges.

Now we can apply the above to the implied volatility problem. For more see C++ Programs in Finance at the home page of Professor Bernt Arne Odegaard at the University of Stavanger, Stavanger, Norway

The function - the volatility function - is the difference between the market observed option price and the Black-Scholes theoretical option price. The root of the function is that value of the volatility the implied volatility that makes the value of the function zero. Our objective is therefore to estimate that makes .

...................................... (2)

In the above equation for the function, is the market observed price of the option which is a constant and is the Black-Scholes theoretical (model) price.

Let us say that we have some approximate value of the implied volatility, to start with and we want to better that estimate with the next best value which would be the zero of the function. Using the above Newton-Raphson estimation algorithm (1) we get:

...................................... (3)

Differentiating (2) we get

...................................... (4)

Therefore, combing (3) and (4) we get the iterative algorithm for the implied volatility

...................................... (5)

The above formula gives an iterative algorithm for estimating the implied volatility from the market observed option price and the theoretical Black-Scholes price. The more the number of iterations the better would be the convergence to the desired level of accuracy.


Reference: C++ Programs for Finance on the Home Page of Bernt Arne Odegaard at the University of Stavanger, Stavanger, Norway

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