The Binomial Pricing Model

A tree is a map of prices. Each dot, called a node, is one possible price at one moment. Each line, a branch, joins it to a price in the next interval. NIFTY 50 is measured in points, and the strike is the contract’s fixed level. A Call (C or CE) pays max(final level − strike, 0) at expiry. A Put (P or PE) pays max(strike − final level, 0). “Max” means take the larger number, so a negative difference becomes zero.

The method has two passes. First, write the payoff at every expiry node. Second, step back: each earlier node is the discounted, weighted average of the two option values it leads to. Discounting turns future money into money at the earlier time. The number you reach at the starting node is the theoretical premium.

Payoff and profit are different. A buyer’s profit also subtracts the premium paid and costs. Before expiry, an option’s model value depends on the chances still ahead, so it is not just intrinsic value. Do not write the expiry payoff on the middle nodes and call that a price.

How far should each branch reach? Volatility sets the size of the move, not its direction. CRR uses an up multiplier u = exp(σ√Δt) and a down multiplier d = 1/u. Here σ is annual volatility and Δt is one interval in years: the time left T divided by the number of steps N. Higher volatility spreads the branches. Shorter intervals bring them closer.

Because an up move followed by a down move ends where a down move followed by an up move ends. Split the same 30 days into two 15-day steps and the multipliers change to u = 1.037164 and d = 0.964168. The day-15 NIFTY levels are 25,929.09 and 24,104.20. The expiry levels are 26,892.72, 25,000 and 23,240.49. We refined the model. Expiry did not move.

Up then down and down then up both land on 25,000. A tree where routes meet like this is called recombining. Fixed multipliers can be applied in any order, since u × d equals d × u. CRR goes one step further and sets u × d = 1, so the middle node equals the starting level.

The up branch gets a weight p and the down branch gets 1 − p. These risk-neutral weights are the ones that make the underlying grow at the model’s risk-free rate. They are not the chance that NIFTY rises, that an option finishes ITM or that a trade profits. The name describes a valuation tool, not how real traders feel about risk.

The growth rate depends on carry: financing cost minus the income from holding the underlying. With a continuous dividend yield q, the price grows at r − q under these weights, so p = [exp((r − q)Δt) − d]/(u − d). Our one-step inputs give p = 0.526984 and 1 − p = 0.473016. Reading that as “a 52.70% chance NIFTY rises” would turn a pricing weight into a forecast. In the two-step tree, the two mixed routes together carry weight 2p(1 − p). A model probability of finishing ITM must add up the right terminal routes, and it still is not a real-world probability.

The weights make sense only when d < exp((r − q)Δt) < u, which gives 0 < p < 1. This is the no-arbitrage condition: the underlying must be able to beat and to lag its carry-consistent growth. If p falls outside that range, change the interval or the parameters. Forcing p to 0 or 1 changes the calculation instead of fixing it.

Replication is why this works. If a portfolio reproduces an option’s cash flows in every state of the model, the portfolio and the option must start at the same value. No forecast is needed to say so. Delta, a local price sensitivity, is also not a forecast or a win rate; the Delta lesson develops that.

Start at expiry, where the payoffs are known, and step back one node at a time. For a European option, V = exp(−rΔt)[pVup + (1 − p)Vdown]. Vup and Vdown are the option values at the next step. The bracket is their weighted average, and exp(−rΔt) discounts it back one interval. The dividend yield changes p, but cash is still discounted at r. Discounting at r − q would be wrong here.

A separate, smaller example makes the idea easy to see. Take a hypothetical SBIN share and Call, both at ₹800, with one 30-day step, no dividends and a zero interest rate. Choose u = 1.05 and d = 1/1.05. That matches an annual volatility of ln(1.05)/√(30/365), about 17.02%, not the NIFTY example’s 18%. The expiry prices are ₹840 and ₹761.90, and the Call pays ₹40 or zero.

A European option can be exercised only at expiry. An American option can be exercised earlier, as the contract allows. At each such node the tree compares two numbers: the exercise value now, and the continuation value, the discounted value of keeping the option. An American option takes the larger. A European option simply continues until expiry. Exercise value is money at that node today, so do not discount it again.

Consider a separate conceptual American Put, expressed in rupees for easy reading. This is a foreign-style exercise illustration, not an NSE equity contract: spot and strike ₹100, one year remaining, 10% interest, no dividends, 20% annual volatility and two steps. At the first down node the underlying is ₹86.812345. Exercising gives ₹13.187655 now; discounted continuation is ₹8.310598. The American node takes ₹13.187655. That possibility raises its model price today to ₹4.448634, compared with ₹2.803441 for the otherwise identical European Put.

Not steadily. Go back to the NIFTY inputs and keep expiry at 30 days. One step gives a Call of ₹694.28, two give ₹507.43 and three give ₹609.35. The price does not move in one direction, because a coarse tree places its expiry nodes differently around the strike, where the payoff bends. Finer grids reduce the error, but the approach can swing from side to side.

For NIFTY, no early-exercise test is needed. NSE’s settlement rules for index options and its specifications for individual-security options describe European-style contracts. NIFTY index options settle in cash. Individual-stock options are physically settled. Selling an option before expiry is a closing trade, not early exercise, and it depends on liquidity and price.

An ITM index option settles its positive exercise amount in cash. A stock option held into exercise can bring a full share payment or delivery obligation and delivery margins. Paying the premium of a bought option does not remove that funding need. Check the contract and the current clearing rules; the SBIN sketch in this lesson does not describe real stock-option settlement.

The tree still earns its place for a European NIFTY option, because it explains replication, discounting, carry and numerical approximation. Index levels are in points and premiums are in rupees per unit. A real contract amount needs the lot quantity, which is not assumed here. The Settlement lesson turns a model payoff into an obligation.

Before you trust a number, name the underlying reference, the expiry, the exercise style, the dividend treatment and the units. Keep a spot-based tree with a dividend yield apart from a futures-based formulation. Putting a futures price into an unchanged spot formula can count carry twice.