Binomial option pricing: where the risk-neutral probability comes from

CM2 · Financial Engineering & Loss Reserving · 12 min read

The result that does not sound possible

A share is at ₹400. In six months it will be at ₹500 or ₹320. What is a call option on it worth?

The obvious answer is that it depends on how likely the rise is, and the obvious answer is wrong. The option can be priced exactly without knowing that probability at all, and the reason is that the option's payoff can be manufactured out of shares and cash. If two things have identical payoffs in every state of the world, they must cost the same today, or there is free money available. That is the whole of option pricing theory, and the binomial model is the smallest setting in which it can be shown completely.

Replication: the argument that does the work

Hold Δ shares and ₹B of cash, and choose the two so that the portfolio is worth exactly what the option is worth, whichever way the share moves. Two equations in two unknowns:

where u and d are the up and down multipliers, h is the length of the step, and f_u and f_d are the option's values in the two states. Subtracting one from the other removes B immediately:

In words: the number of shares to hold is the change in the option's value divided by the change in the share price. That is the delta of the option, and it has arrived here as the answer to a hedging question, which is what it is.

With Δ known, either equation gives B, and the option must be worth whatever the portfolio costs today:

Nothing in that argument mentioned the probability of an up move. It could not have, because the two equations hold in both states separately — the portfolio does not need the share to go one way rather than the other. The option's value is fixed by the requirement that no arbitrage is available, and an arbitrage is available if the option trades at any other price.

Where the risk-neutral probability comes from

Substituting B back and rearranging the algebra — one line of tedium with no new ideas in it — the option value can be written as

In words: the option is worth the discounted average of its two possible payoffs, with the two outcomes weighted by q and 1 − q.

This is a rewriting, not a new method. It gives the same number the replication argument gave. What it does is make the arithmetic quick, because it avoids solving two equations at every node of a large tree.

The quantity q is the risk-neutral probability, and the single most important thing to understand about it is that it is not a probability of anything. It is a number between 0 and 1 that happens to make the discounted-expectation arithmetic come out right. It is determined by u, d, r and h, and the real probability of an up move never enters its definition.

What q does have is a defining property worth knowing, because questions ask for it. Under q, the share itself earns exactly the risk-free rate:

That is where the name comes from. q is the set of weights under which every asset — the share, the option, anything — grows at the risk-free rate, so that discounting at the risk-free rate is correct. In a world of risk-neutral investors the real probabilities would have that property; in the real world they do not, and q is the artificial substitute that restores it.

Why the real probability drops out

This is the conceptual question the topic exists to test, and it has a clean answer. The real probability of an up move does matter to the value of the share — a share more likely to rise is worth more, and the market has already taken account of that in the ₹400 price. The option is then priced relative to the share, by replication. All the information about real probabilities and risk preferences has already been absorbed into the share price, and the option price inherits it.

So the real probability is not being ignored. It is being used, once, where it belongs, and the option price depends on it only through today's share price. Using it a second time, by discounting a real-world expected payoff at the risk-free rate, double-counts it and gives the wrong answer — by ₹12.42 on the worked example below.

The condition for the model to make sense

In words: the share must be able to do better than cash and also worse than cash over the step.

If that fails, q falls outside the interval from 0 to 1 and there is an arbitrage in the underlying market before any option is written: a share that cannot do worse than cash should be bought with borrowed money without limit. So the condition is not a technical nicety, it is the statement that the market is sensibly priced — and checking it takes five seconds and catches a misread question.

More than one step

A two-period tree is the one-period argument used three times. Work backwards: price the option at each of the final nodes from its payoff, then use the one-period formula at each node of the previous layer, then once more to get back to today. The recursion is identical at every node, and the only thing that changes is which values are being averaged.

For a European option there is a shortcut. Since the only thing that matters is the payoff at expiry, the whole tree can be collapsed into one discounting step with binomial weights:

For an American option there is no shortcut, and this is the real reason the tree is worth building. At every node, compare the value of continuing with the value of exercising immediately and take the larger:

That single line is why binomial trees are used in practice when the Black-Scholes formula exists. The formula cannot handle a decision taken partway through; a tree can, because it visits every point at which the decision could be taken.

Where marks are lost

Using the real probability. The commonest and most expensive error in the topic. If a question gives you a real probability of an up move, it is either a distractor or the question is about something else, and saying so is part of the answer.

Getting the sign of the delta wrong. A call has a positive delta and a put a negative one. Replicating a put means shorting the share, and a positive delta on a put is a sign error that will survive to the final answer.

Using a simple interest or an annual effective rate where a continuously compounded one was given. The factor is erh when r is continuously compounded. With h = 0.5 and r = 6%, that is e0.03 = 1.0304545, not 1.03.

Forgetting the early-exercise test on an American option. Check it at every node, not only the ones that look likely. If the test is never applied the answer is the European price, and the whole point of the question has been missed.

Discounting over the wrong period. Each backward step discounts over one step length h. Collapsing a European option to a single calculation discounts over the whole term T. Mixing the two is a factor of erh adrift.

Questions students ask

Is a two-state share price not absurdly crude?

As a model of six months, yes. As a building block, no. Cut the six months into ever smaller steps, each with its own up and down move, and the distribution of the final price converges to the lognormal distribution of the continuous model — and the binomial price converges to the Black-Scholes price. The binomial model is best thought of as Black-Scholes with the limit not yet taken, which is also why it keeps working when Black-Scholes stops.

What is a state price deflator?

The same arithmetic packaged differently. Instead of multiplying by a risk-neutral probability and then discounting, you multiply each state's payoff by a single number for that state — its state price deflator — and add up. For a one-period tree the deflators are q e−rh and (1 − q) e−rh, so the two approaches are identical by construction. The deflator form is used because it extends neatly to real-world probabilities and to several periods at once.

Is an American call ever worth more than a European call?

Not on a share paying no dividends. Exercising a call early means paying the strike sooner than necessary and throwing away the remaining time value, so it is never optimal and the two prices are equal. An American put is different, and can be worth strictly more, because exercising early means receiving the strike sooner — and the interest on it is a real gain. The worked example below shows that happening.

Do u and d have to be reciprocals?

No. A question will give you whatever u and d it wants, and the method does not care. Setting d = 1/u is a convenience in multi-period trees because it makes the tree recombine neatly, and a common calibration sets u = eσ√h so that the tree matches a chosen volatility. Neither is a requirement of the model.

A worked example

A share that pays no dividend trades at ₹400. In each six-month period its price either rises by 25% or falls by 20%. The continuously compounded risk-free rate is 6% a year.

Price a one-year put option with a strike of ₹420, first as a European option and then as an American one.

The share, the two multipliers and the strike are invented for this example. They are not a real security.

Step 1: check the model is sensible, then find q

Confirm the defining property before using it: 0.5121212 × 500 + 0.4878788 × 320 = 256.0606 + 156.1212 = ₹412.18, and 400 × 1.0304545 = ₹412.18. Under q the share grows at the risk-free rate, which is what makes discounting at that rate legitimate.

Step 2: the tree and the payoffs

Put payoffs at expiry, max(420 − S, 0):

Step 3: the European put, collapsing the tree

Call that ₹46.18.

The second route: backward induction, node by node

At the up node, S = 500, with 0 and 20 ahead of it:

At the down node, S = 320, with 20 and 164 ahead of it:

And back to today:

₹46.175034 one way and ₹46.175039 the other — agreement to five decimal places, the sixth differing only because the three binomial weights in the first route were rounded to seven places before being used. Carry them unrounded and both give ₹46.175039. The two routes share no intermediate figure except q itself, so that is real evidence the arithmetic is right.

The third route: build the replicating portfolio

This is the one that proves the price rather than computing it. At time 0, with option values 9.4692 and 87.5871 ahead:

So the put is replicated by shorting 0.4339885 of a share and holding ₹219.77 in cash, for a net outlay of 219.770434 − 173.595400 = ₹46.175034. Check the portfolio does its job over the next six months:

Both match the option values at those nodes. The portfolio costs ₹46.18 and pays exactly what the option pays in both states, so the option is worth ₹46.18 and no other price can hold.

One node is worth inspecting because it shows what a delta of exactly −1 means. At the down node, S = 320, Δ = (20 − 164)/(400 − 256) = −1. The put is certain to be exercised from there, whichever way the share goes next, so it has become a short position in the share plus the strike held in cash: −320 + 407.587124 = ₹87.5871, with 407.587124 × 1.0304545 = ₹420.00, exactly the strike, next period. An option that is certain to be exercised has stopped being an option.

The American put

Repeat the backward induction, testing early exercise at every node.

Up node, S = 500. Intrinsic value is max(420 − 500, 0) = 0. Continuation is ₹9.469196. Hold.

Down node, S = 320. Intrinsic value is 420 − 320 = ₹100, against a continuation value of ₹87.587124. Exercise. The node is worth ₹100, not ₹87.59.

And at time 0 the intrinsic value is 420 − 400 = ₹20, well below ₹52.05, so hold today. The American put is worth ₹52.05.

Why the right to exercise early is worth ₹5.88 is visible in the one node where it was used. At S = 320 the put is deep in the money and has almost no time value left — its continuation value of ₹87.59 is below its ₹100 intrinsic value because waiting means receiving the ₹420 strike six months later, and the interest forgone on ₹420 is worth more than the remaining chance of the share recovering. Taking the ₹100 now is simply better. That trade-off cannot arise for a call on a non-dividend-paying share, because there exercising early means paying the strike sooner, and the interest runs the other way.

What happens if you use the real probability instead

Suppose you were told the share has a 60% chance of rising in each period, and you discounted the real expected payoff at the risk-free rate:

₹33.75 against the correct ₹46.18 — wrong by ₹12.42, which is 27% of the answer. And it is wrong in a way that cannot be patched by choosing a better discount rate, because the correct discount rate for the put would have to be negative: a put is a hedge, so an investor accepts a negative expected return to hold one. Finding that rate would require knowing the market's risk preferences. The risk-neutral calculation sidesteps all of it and gives the exact answer in one line, which is why it is the method and not a convenience.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

• Utility theory and risk aversion: non-satiation, A(w) and R(w)
• Measures of investment risk: variance, semi-variance, VaR and Tail VaR
• CAPM and the security market line: beta and the market price of risk
• Ito's lemma explained term by term, and what it does to a share price
• The chain ladder method: estimating outstanding claims from a run-off triangle
• The Bornhuetter-Ferguson method: the chain ladder with a second opinion
• The Black-Scholes formula and put-call parity, read term by term
• Ruin theory: the adjustment coefficient and Lundberg bound explained
• Mean-variance portfolio theory: the efficient frontier and why diversification works
• The term structure: spot rates, forward rates and par yields
• Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared
• The Merton model: company equity as an option, and where a credit spread comes from
• The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays
• The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in
• The average cost per claim method: developing numbers and severity separately
• Valuing a maturity guarantee by simulation, and checking it against the formula
• The Efficient Markets Hypothesis: the three forms, and how each one is tested
• Single-index and multifactor models: getting a covariance matrix out of a few betas
• Excess-of-loss reinsurance: the retained claim, and choosing the retention
• State price deflators: pricing with real-world probabilities
• What makes a risk insurable, and what to do when it almost is
• Two-state and intensity credit models: backing a default rate out of a bond price
• The martingale representation theorem, and why hedging is possible at all

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.