The Binomial Pricing Model

The method has three moves. First, draw a map of the prices the share could take between now and expiry, assuming that in each short slice of time it can only go up by a fixed factor or down by a fixed factor. Second, at every end point of the map, write down what the option would be worth. At expiry this is easy: it is the in-the-money amount, or nothing. Third, walk backwards, replacing each point with a weighted average of the two points it leads to, until you are back at today.

None of this needs advanced mathematics. It needs careful bookkeeping. A three-step tree can be done on paper in five minutes, which is why it is the usual way to teach what an option price really is. A working system runs the same steps with hundreds of hops. As the steps get finer, the answer moves closer to the Black-Scholes price for a plain option that can be used only at expiry.

This closeness matters. It shows that the two models are not rivals with different opinions. They are two routes to the same place for the simple case. The tree is slower, but it stops at every point on the way, and those stops are where it is useful.

The up and down factors are not chosen by taste. They come from the volatility assumption and the length of each step, so that the spread of prices at expiry matches the volatility in the model. Higher volatility means the branches spread wider. A shorter step means they sit closer. The down factor is set equal to 1 divided by the up factor.

Take an example. SBIN is at ₹800 and you value an ₹800 call over three steps, using an up factor of 1.05. The down factor is 1 ÷ 1.05 = 0.9524. After one step the price is ₹840 or ₹761.90. After two steps it is ₹882, ₹800 or ₹725.62. After three steps it is ₹926.10, ₹840, ₹761.90 or ₹691.07. Three steps give four end prices, not eight, because paths that mix ups and downs land on the same price. If three steps cover three months, this 5% step means a yearly volatility of about 17%.

How many steps? Three gives a rough picture, enough to learn from. Fifty gives a usable one. A trading desk may use several hundred, and then each hop is a fraction of a percent and the tree looks like a dense cloud, not the neat diagram you can draw by hand. More steps cost more computing time. The share prices and hop sizes here are made up for illustration.

A recombining tree is one in which an up move followed by a down move ends at exactly the same price as a down move followed by an up move. In the SBIN example, ₹800 up to ₹840 and then down comes back to ₹800. ₹800 down to ₹761.90 and then up also reaches ₹800. The two paths have different histories but the same destination, so the model keeps one price, not two.

The reason is arithmetic. Because the down factor is 1 divided by the up factor, multiplying by both, in either order, returns you to where you began. The practical reason is that without this the model cannot be used. The number of paths doubles at every step: ten steps give 1,024 end points and thirty steps give more than a billion. Recombining turns that explosion into simple counting: a tree of n steps has n + 1 end prices.

There is an assumption hidden in this convenience. Recombining says the option’s value depends only on where the price is now, not on how it got there. For a plain call or put that is true. For contracts whose payoff depends on the path taken, a standard recombining tree is the wrong tool.

When you fold the tree backwards you must give each of the two branches a weight. The weight the model uses is not anyone’s forecast of an up move. It is the specific number that makes the share itself grow at the risk-free rate inside the tree. It is called the risk-neutral probability, written p, and it is a bookkeeping device, not a prediction.

The logic is like the previous article. An option can be copied by holding the right amount of the underlying financed by a loan. If the copy is exact, then what anyone believes about direction cannot matter to the price. Otherwise there would be a free profit lying around. The risk-neutral weights enforce this. In the SBIN example, with the interest rate set to zero to keep the arithmetic clear, p = (1 − 0.9524) ÷ (1.05 − 0.9524) = 0.4878. The down branch takes 0.5122.

Reading p as “there is a 48.78% chance SBIN rises” is the standard beginner mistake. It is not a forecast. The same mistake appears later when an option’s delta is read as a real-world probability. The price that results is the one that leaves no room for a free profit. It says nothing about which way the share will actually go.

Start where you already know the answer. On expiry day an ₹800 call is worth the share price minus ₹800 if that is positive, and nothing otherwise. Write that on every end point. In the SBIN tree, this gives ₹126.10 at ₹926.10, ₹40.00 at ₹840, and zero at the two lower end points.

Now step back one column. Each price in that column leads to two prices you have already valued. Take the weighted average using p = 0.4878 and 0.5122, and write the result on the price. With interest set to zero, no discounting is needed. Repeat, column by column, until you reach the single price on the left. That final number is the theoretical price today. Here it is about ₹29.27.

This is the whole method, and it shows something a formula hides. The premium is not a number pulled from nowhere. It is a chain of “if the share is here at this time, the remaining option is worth this much”. That chain is what allows an extra test at each point, which is the subject of the next section. The limits: the answer depends on the assumed hop size and interest rate, and a real share can jump by more than one hop between steps.

Because it visits every price on the way, the tree can add one more test at each: is the option worth more if I keep it, or if I use it right now? It already knows the value of keeping it from the fold. The value of using it now is the in-the-money amount at that point. The larger of the two becomes the value there, and this choice passes backwards into every earlier point. A formula that looks only at expiry day has no place to make that comparison.

The same step-by-step structure allows a dividend to be placed properly. A dividend lowers the share price on the ex-date by about the amount paid. In a tree you subtract it at the column for that date and continue. Black-Scholes has no idea of a date between now and expiry, so implementations reduce today’s spot by the present value of expected dividends. That is usable but only an approximation of a single event.

For an American-style call on a share that pays a dividend, both features work together. Using the option early to capture the dividend can be worth more than waiting, and only a model that looks at the point just before the ex-date can see it. This is the clearest case where a tree gives a better answer than a formula.

The tree also has weaknesses. It still assumes one volatility for the whole life of the contract and small, smooth hops. It is slower than a formula, and for a plain option that can be used only at expiry it gives no better answer than Black-Scholes.

Nifty index options on NSE are European style according to the exchange’s contract specification: they can be used only at expiry. So the early-exercise feature that makes the binomial model well known is not something a Nifty option holder acts on. It matters for American-style options on foreign exchanges, and it explains why two model families exist. Check the contract specification of any other option before assuming.

What carries over is the handling of dividends. Single-stock options are written on companies that pay dividends, and a dividend inside the life of a contract moves the share price on its ex-date. A price that ignores this date will make calls look a little rich and puts a little cheap before the event. A tree puts the adjustment at the right place.

The second thing that carries over is intuition. After you fold a tree by hand once, you stop seeing the option price as the output of a black box. You see it as today’s value of a set of possible futures, each weighted and brought back to the present. That is a helpful picture of what happens at expiry, when all those futures collapse into one, which the settlement article in this series continues.