Solving for implied volatility

Why search at all? The volatility sits inside the normal curve in the formula, so it cannot be moved to one side. There is no formula for it. But one fact makes the search easy: a higher volatility always gives a higher price, for a call and for a put. The slope of that rise is vega. Because price only climbs, one price fits one volatility, never two.

There are two ways to move the guess.

Newton has a weak spot. Far out of the money, vega is close to zero, the divide blows up and the next guess can leap out of range. A sound solver keeps the answer fenced and falls back to bisection when Newton leaves the fence. RupeeCase's engine does exactly this.

How do I calculate IV for Nifty options? The short version, which the question page repeats:

Two choices matter more than the maths. Use the option that is out of the money at each strike: the put below the forward, the call above it. In the money options hold mostly intrinsic value, carry little vega and wide spreads, so a small price error becomes a big volatility error. And use the same inputs every time you compare, because the answer depends on all of them.

Each input pushes the answer in a known direction. A higher forward lowers the volatility a call needs to reach its price, and raises it for a put. More days to expiry lower the volatility needed, because time alone adds value. The rate matters least for an option of a few weeks, since it only moves the discount factor, but it grows with the time left. Get the rate or the days wrong and every strike shifts the same way. Get the forward wrong and the calls shift one way while the puts shift the other, which is why a smile built on a wrong forward shows a step where it switches from puts to calls.

Sometimes there is no answer. A call cannot be worth less than the discounted gap between the forward and the strike, and cannot be worth more than the discounted forward. A price outside those bounds has no volatility behind it. That signals a stale or wide quote, and a good solver returns nothing instead of a wrong number.

Now Newton, starting from 20%. The model gives Rs 427.98, which is Rs 106.33 too high. Vega at that point is Rs 21.64 a unit for each volatility point. Divide: 106.33 over 21.64 is 4.91 points. The next guess is 15.09%. Price it again and the model returns Rs 321.645 against Rs 321.65. One step brought it within a paisa. The solved value is 15.09%.

Do the same for every strike and you get the smile of the chain.

The 21,400 put solves to 19.72%, the curve bottoms near 24,000 at 14.38%, and the 25,050 call solves to 16.27%. The shape has its own lesson in the smile and skew module. Here the point is that each dot is a separate solve.

Back to the strike where the call and put disagreed. Here are both sides.

At 23,250 the call gives 15.09% and the put 14.65%, a gap of 0.44 points. With a vega of Rs 21.64 a unit, that gap is worth about Rs 9.52 of premium. Settlement prices are not trades, so a few rupees of disagreement between a call and a put is ordinary. Put call parity ties the two prices together, and the parity module shows how the leftover is measured. It is also why the smile uses one option per strike.

And the edge case. The 24,800 put settled at Rs 1,553.20 on 146 contracts. The least a put that deep in the money can be worth is the discounted gap to the forward, Rs 1,555.54. The settlement price sits Rs 2.34 below that floor, so no volatility reproduces it. The solver returns nothing, and a screen that shows a number there has done something extra.

Over a longer stretch, the same solve run daily on the nearest monthly expiries gives a constant 30 day at the money series.

Both lines come from option prices, and both rise and fall together. They differ because they read different strikes. The India VIX lesson takes the index apart.