The Binomial Pricing Model

The method has three moves. First, build a map of every price the underlying could take between now and expiry, assuming that in each small slice of time it can only go up by a set factor or down by a set factor. Second, at every endpoint of that map, write down what the option would be worth — which at expiry is trivial, because it is either intrinsic value or nothing. Third, walk backwards through the map, replacing each node with the discounted average of the two nodes it leads to, until you arrive back at today.

Nothing in that sequence requires advanced mathematics. It requires bookkeeping. A three-step tree can be done on paper in five minutes, which is why it is the standard way of teaching what an option premium actually is. A production system runs the same procedure with several hundred steps, and as the steps get finer the answer converges on the Black-Scholes number for a plain European contract.

The convergence matters because it tells you the two models are not rivals with different opinions. They are two routes to the same destination for the simple case. The tree is slower but stops at every intermediate point, and those stops are where it earns its keep. Everything distinctive about the binomial model comes from being able to inspect the position before expiry rather than only at it.

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 being modelled. A higher volatility input pushes the branches further apart and the tree fans out wider. A shorter step brings them closer together. The down factor is set as the reciprocal of the up factor, which is the small piece of housekeeping that makes the whole structure recombine.

Take a worked example. Suppose SBIN is at ₹800 and you are valuing an ₹800 call across three steps. With an up factor of 1.05, the down factor is 1 divided by 1.05, or 0.9524. Step one gives ₹840 or ₹761.90. Step two gives ₹882, ₹800 or ₹725.62. Step three gives ₹926.10, ₹840, ₹761.90 or ₹691.07. Four possible destinations from three steps, not eight, because the middle paths land on top of each other.

The choice of how many steps to use is a trade-off between precision and computing time. Three steps produces a crude picture. Fifty steps produces a usable one. Several hundred steps is what a desk actually runs, and at that point the individual up and down moves are tiny — a fraction of a percent each — and the tree is a dense cloud of possible paths rather than the tidy diagram you can draw by hand.

A recombining tree is one where an up move followed by a down move lands on exactly the same node as a down move followed by an up move. In the SBIN example, ₹800 up to ₹840 then down comes back to ₹800. Down to ₹761.90 then up also arrives at ₹800. The two paths are different histories but they share a destination, so the model stores one node rather than two.

The arithmetic reason is that the down factor is the reciprocal of the up factor, so multiplying by both in either order returns you to where you started. The practical reason it is done this way is that without it the model becomes unusable. Paths double at every step. Ten steps would be 1,024 endpoints, thirty steps over a billion, and a realistic several-hundred-step tree would exceed anything that can be computed. Recombining turns that explosion into simple addition: a tree of n steps has n plus one endpoints.

There is a modelling assumption hiding in this convenience, and it is worth naming. Recombination says the option’s value depends only on where the underlying is now, not on how it got there. For a plain call or put that is true. For path-dependent contracts, where the payoff depends on the route taken, a standard recombining tree is the wrong tool and the pricing has to be done another way.

When folding a node backwards you need to weight the two branches. The weight the model uses is not anybody’s forecast of an up move. It is the specific number that makes the underlying itself grow at the risk-free rate inside the tree. That number is the risk-neutral probability, usually written p, and it is a mathematical device rather than a prediction.

The logic is the same as in the previous module. An option can be replicated by holding a calculated quantity of the underlying financed by borrowing. If the replication is exact, then whatever anyone privately believes about direction has to drop out of the price, otherwise there would be a risk-free profit sitting on the table. The risk-neutral weights are what enforce that. In the SBIN example, with the risk-free rate set to zero for arithmetic clarity, p works out to about 0.49 and the down branch takes about 0.51.

Reading p as "there is a 49% chance SBIN rises" is the standard beginner error. It is not a forecast, it is a bookkeeping weight, and the same misreading resurfaces later when N(d₂) or an option’s delta is treated as a real-world probability. Module 9 handles that distinction carefully for delta. What is true is that the resulting premium is the one that removes any arbitrage from the structure — no more, no less.

Start where you know the answer. On expiry day an ₹800 call is worth the stock price minus ₹800 if that is positive, and nothing otherwise. Write that value on every endpoint. In the SBIN tree that gives ₹126.10 at ₹926.10, ₹40 at ₹840, and zero at both lower endpoints.

Now step back one column. Each node in that column leads to exactly two nodes you have already valued. Take the weighted average using the risk-neutral probability, discount it by one step at the risk-free rate, and write the result on the node. Repeat, column by column, until you reach the single node at the left. That final number is the theoretical premium today — in this example, about ₹29.27.

This is the whole method, and it exposes something a formula hides. The premium is not one number arrived at by magic. It is a chain of conditional values, each one saying "if the stock is here at this point in time, the remaining option is worth this much". That chain is what lets the model insert an extra test at each node, which is the subject of the next section.

Because it visits every intermediate node, the tree can add one extra test at each of them: is the option worth more held or exercised right now? It already knows the held value from the fold. The exercise value is simply intrinsic value at that node. Whichever is larger becomes the node’s value, and that choice then propagates backwards into every earlier node. A closed-form formula that only evaluates expiry day has no place to run that comparison.

The same node-by-node structure is what lets a specific dividend be handled properly. A dividend knocks the underlying price down on the ex-date by roughly the amount paid. In a tree you subtract it at the column that corresponds to that date and continue. Black-Scholes has no concept of a date between now and expiry, so implementations approximate by reducing the spot price at the start by the present value of expected dividends — usable, but an approximation to a discrete event.

For an American-style call on a dividend-paying share, these two features work together. Exercising early to capture a dividend can be worth more than holding, and only a model that inspects the node just before the ex-date can detect it. That combination — early exercise plus a dated cash flow — is the single clearest case where a tree gives a materially different and better answer than a formula.

Here is the part most Indian material gets wrong. Options listed on the NSE — both index contracts such as NIFTY and BANKNIFTY and single-stock contracts such as RELIANCE or SBIN — are European style. The holder cannot exercise before expiry. So the early-exercise machinery that makes the binomial model famous is not something an NSE trader has to act on. It matters for American-style contracts on overseas exchanges, and it matters for understanding why the two model families exist at all.

What does carry over is the dividend handling. Single-stock options in India sit on companies that declare dividends, and a dividend inside the life of a contract shifts the underlying on its ex-date. Any pricing that ignores that date will read calls as slightly rich and puts as slightly cheap into the event. A tree places the adjustment where it belongs.

The second thing that carries over is intuition. Once you have folded a tree by hand you stop seeing the premium as a black-box output. You see it as the present value of a set of conditional futures, each weighted and discounted. That is the single most useful mental model for what happens at expiry, when those conditional futures collapse into one — which is where module 38 on settlement picks up the thread.