Binomial pricing model

The binomial model answers "what is this option worth?" by refusing to be clever. It simply enumerates what could happen.

Break the life of the option into steps. At each step, assume the underlying can do exactly two things: move up by a fixed proportion, or move down by a fixed proportion, each with its own simulated probability. Draw every path. At the final nodes you know what the option pays, because at expiry an option's value is just its intrinsic value. Then work backwards through the tree, discounting, until you arrive at today.

The workbook credits the model to William Sharpe in 1978 and calls it the most flexible, intuitive and popular approach to option pricing. It is very accurate because it is iterative — and its implementation is complex and time consuming for the same reason.

Working backwards is the whole trick. At each node the option is valued as the discounted expected value of the two nodes it leads to, using risk-neutral probabilities rather than anyone's forecast of the market.

The risk-neutral probability is the one that makes the underlying itself grow at the risk-free rate in the tree. Once that is fixed, no view about the market's direction is needed to price the option — which is the deep result the model shares with Black-Scholes.

Where the binomial model earns its keep is flexibility:

• It handles American-style options, because at every node you can compare holding on with exercising now and take the larger. Black-Scholes cannot do this; it prices a European exercise only.
• It handles discrete dividends, changing rates and changing volatility, by varying the tree.

Its cost is arithmetic. A 50-step tree has thousands of nodes. The workbook contrasts it with Black-Scholes precisely on this axis: Black-Scholes is relatively simple and fast and does not rely on calculation by iteration; the binomial model does. As the steps get finer, the binomial price converges towards the Black-Scholes price for a European option.

Risk-neutral probability of an up move:

        e^(rΔt) − d
  p = ────────────────
           u − d

Value at a node = e^(−rΔt) × [ p × Value(up) + (1 − p) × Value(down) ]

where u = up factor, d = down factor, Δt = length of one step

A share trades at Rs 1,000. Over the next month it can rise to Rs 1,100 or fall to Rs 900. The risk-free rate is 6% p.a., and the option lot size is 100 shares. Price the one-month Rs 1,000 call on a single-step tree.

Step 1 — the payoffs at expiry:

Up node:   max(1,100 − 1,000, 0) = Rs 100
Down node: max(  900 − 1,000, 0) = Rs 0

Step 2 — the risk-neutral probability. With u = 1.10, d = 0.90, r = 6%, Δt = 1/12:

e^(0.06/12) = 1.005013

      1.005013 − 0.90
p = ─────────────────── = 0.105013 ÷ 0.20 = 0.5251
      1.10 − 0.90

Step 3 — discount the expected payoff:

Expected payoff = 0.5251 × 100 + 0.4749 × 0 = Rs 52.51
Call value      = 52.51 × e^(−0.06/12) = 52.51 × 0.995012 = Rs 52.24

One lot costs 52.24 × 100 = Rs 5,224.

Cross-check with put-call-parity, which must hold whatever model produced the price:

p = c − S + X·e^(−rt) = 52.24 − 1,000 + 995.01 = Rs 47.25

So the matching put is worth Rs 4,725 a lot. Two very different contracts, priced off one tree, and the arbitrage relation between them closes to the paisa.

Notice what never entered the calculation: anybody's opinion about whether the share will rise. The probability 0.5251 is not a forecast. It is the number that makes the share itself earn the risk-free rate inside the tree — and that is what makes the answer an arbitrage-free price rather than a guess.

Chapter 16.8 (Option Pricing Models) introduces the binomial model — Sharpe, 1978 — and sets it against Black-Scholes. Expect an attribution question, and a contrast question: which model is iterative (binomial), which is faster and closed-form (Black-Scholes), and which can price an American-style exercise (binomial).

• William Sharpe, 1978, for the binomial model; Fisher Black and Myron Scholes, 1973, for Black-Scholes. Both the names and the dates are examinable and easy to swap.
• "Accurate" and "slow" are the same property here. The iteration that makes it flexible is what makes it time consuming.
• The probabilities are risk-neutral, not real-world. They are not anyone's forecast of the market.
• Indian exchange-traded index and stock options are European style, so the model's American-exercise advantage is not available on them.
• More steps do not change the answer for a European option — they converge on the Black-Scholes value.
• The workbook's printed Black-Scholes formula carries OCR damage in the d₁ and d₂ lines. Work from the structure, not from the transcribed symbols.