The Black-Scholes-Merton Model

The model asks one narrow question: what is a fair price today for the right to buy or sell something at a fixed level on a fixed date? Fischer Black, Myron Scholes and Robert Merton answered it in 1973. Their idea was to avoid guessing direction altogether. Suppose you sold an option and at the same time held the right amount of the underlying against it. As the price moves, your gain on one cancels your loss on the other, so the combined position is close to riskless. A riskless position should earn only a plain interest rate. Work backwards from that rule and one price is left.

Look at the picture below. For a Nifty 24,500 call with 30 days left, the copy of the option is: own about half a unit of Nifty (0.507 × 24,500 = ₹12,418) and owe a loan of about ₹12,082. The difference, ₹336, is the fair price of the option per unit. These are model values with a 12% volatility assumption and interest ignored, for illustration.

This is why the formula has no forecast of direction. Nowhere does it ask whether Nifty will go up or down. It prices the cost of building and keeping the copy, and that cost depends mainly on how far the index is likely to travel. So the model works like a translator, not a crystal ball. It turns an assumption about movement into a rupee price. Run backwards, it turns a price into an assumption about movement. When your terminal shows an implied volatility next to a strike, it is running the model backwards on the last traded price.

Five numbers go in and one price comes out. Spot is the live price of the underlying, for example Nifty at 24,500. Strike is the level written in the contract, for example 24,700. Time to expiry is written as a fraction of a year, so 14 days becomes 14 ÷ 365 = 0.038. The interest rate is the return on money with no credit risk, usually taken from short-dated government securities. Volatility is the yearly size of the swings the model assumes from now to expiry.

The formula has two parts. In simple words: the first part is the value of the index you would have to own to copy the option. The second part is the loan you would have to take to pay for it. The option price is the first minus the second, as in the picture above.

N(d₁) and N(d₂) are numbers between 0 and 1. N(d₁) is the amount of the index you hold to copy the option, and it is the option’s delta. N(d₂) is the model’s own probability that the option ends in the money, in a special make-believe world called risk-neutral. It is not a forecast of the real world.

Four of the five inputs are facts. Spot is on your screen. Strike is printed in the contract. Time is on the calendar. The interest rate is published, and over a two-week option it changes the price very little. Two traders with the same terminal will not disagree on these four.

Volatility is different. Nobody can see how much Nifty will move between now and expiry, because it has not happened. You can measure how much it moved in the past, which is called historical or realised volatility, but the model needs the future. So this input is always someone’s estimate. The price you see quoted is the market’s combined estimate, written in rupees.

This is why most real option work is really work on volatility. Two traders can agree on spot, strike, time and rate, and still disagree on the price, because one expects a quiet fortnight and the other has read the calendar of events. India VIX, published by NSE, is the market’s combined estimate of swings for near-dated Nifty options.

Keep the limits in mind. A high volatility reading does not mean an option is too dear, and a low reading does not mean it is cheap. The market may be right that a big move is coming, or right that things will stay calm. Volatility only becomes useful when you compare it with something else, such as the index’s own past.

There is no simple formula that takes a price and gives a volatility. The equation cannot be rearranged that way. So platforms do the next best thing: they guess, check and correct until the model’s price equals the traded price. The volatility that makes them match is called implied volatility, because it is implied by the price and not measured from anything.

The search is quick, because the price rises steadily with volatility and there is only one answer. A handful of tries find it. That is why your terminal can show a figure for every strike in a Nifty chain without delay. It repeats this loop for each row on each tick.

The result needs care. Implied volatility is not an independent measurement against which you can check a price for a bargain. It is the same price written in different units. To say “this option is dear because IV is 22” and “IV is 22 because this option is dear” is to say the same thing. It becomes useful only when compared with a reference, such as the same index’s own history, or the volatility that actually happens.

The first assumption is that the option can be used only on expiry day. This matches Nifty index options on the NSE, which are European style according to the exchange’s contract specification. Check the specification of any other contract before assuming. It matters for American-style options on foreign exchanges, and it is the gap that the binomial model in the next article was built to fill.

The second assumption is that volatility is one fixed number until expiry. It is not. Volatility bunches up, jumps before policy announcements and results, and falls the next morning. If a company reports results nine days from now, the coming fortnight is a quiet stretch, a sudden jump and another quiet stretch. One number cannot describe this, so the market quotes different implied volatilities for different strikes and expiries. That pattern is called the volatility skew.

The third assumption is that price changes follow a smooth bell-shaped pattern, which treats a crash as nearly impossible. Real market history has days when prices gapped down by amounts the model would call once-in-many-lifetimes events. Because of this, buyers pay more than the model price for far out-of-the-money puts. That gap is not an error to be harvested. It is the market charging for a risk the formula cannot see.

A worked example makes this real. Suppose Nifty is at 24,500 and you look at the 24,500 call with 30 days left. Keep spot, strike, time and interest fixed, and change only the volatility. The values below come from the Black-Scholes call formula with interest ignored, so a live screen will show slightly different figures. They show the shape correctly.

Moving from 12% to 21% raises the price by 75%, from about ₹336 to about ₹588 per unit. Nothing about the index changed and nobody made a call on direction. The market only changed its view of how far Nifty may travel in a month. On one lot of 65 units, that is the difference between about ₹21,840 and about ₹38,220 for the same contract.

This is behind many “I was right and still lost” stories. A trader buys a call before an event when expected swings are high. The event passes, the swings expected fall, and the price drops even though the index rose. The opposite is equally true for sellers: they can lose when a calm market suddenly becomes nervous. The vega article in this series puts a number on how much price each volatility point is worth.

The Greeks are not a separate system attached to the model. Each one answers the question: if I nudge one input, how much does the price move? Nudge spot and you get delta. Nudge delta itself and you get gamma. Nudge time and you get theta. Nudge volatility and you get vega. Nudge the interest rate and you get rho. They all come from the same equation, so they must be read together.

This also means every Greek inherits the model’s assumptions. A theta figure assumes volatility stays put from today to tomorrow. A delta figure assumes smooth price moves. When the market gaps, these numbers do not just become inaccurate. They describe the wrong kind of world, because the move was one the model does not allow.

In practice, the order of use is simple. Delta tells you how directional you are now. Gamma tells you how quickly that changes. Theta tells you what waiting costs. Vega tells you what a change of mood costs. The next articles take them one at a time.