Binomial option pricing for the FRM® Exam Part I: trees and worked examples

Binomial option pricing values an option by working backwards through possible underlying prices, discounting risk-neutral expected payoffs at the risk-free rate. For a European option, replication gives the same price. For American options, each exercise node before expiry takes the greater of immediate exercise and continuation value. The probability used for valuation is a pricing weight, not a forecast of an actual upward move. (Back, Liu and Loewenstein, binomial trees (secondary source))

This guide supports study for the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP®). (GARP, FRM Exam information) Our questions complement GARP’s own study materials; they do not replace them. See the course page for context. All example inputs below are our own assumptions, not market observations.

What assumptions make a binomial tree work?

At each node, a share priced at $S$ moves to either $uS$ or $dS$ after one time step, with $u>d>0$. Our examples assume no dividends, no transaction costs, divisible shares, short selling and borrowing or lending at the same risk-free rate. These idealised trading conditions allow replication. (Back, Liu and Loewenstein, binomial trees (secondary source))

Let $R$ be the risk-free growth factor for one step. If the effective step return is $i$, then $R=1+i$. With an annual continuously compounded rate $r$ and step length $\Delta t$ in years, use $R=e^{r\Delta t}$ instead. The no-arbitrage condition is: (Back, Liu and Loewenstein, binomial trees (secondary source))

Otherwise, the share or the risk-free asset dominates the other across the two states, creating an arbitrage under these assumptions. Check the inputs before valuing the option.

How do you calculate the risk-neutral probability?

Choose the up probability $p$ so that the share’s expected growth equals the risk-free growth factor. The down probability is $1-p$. Solving for the weights gives: (Back, Liu and Loewenstein, binomial trees (secondary source))

The no-arbitrage condition places both probabilities strictly between zero and one. For next-step option values $V_u$ and $V_d$, the continuation value is:

No estimate of the share’s real-world expected return enters this equation. Risk-neutral pricing follows from replication and no arbitrage; it does not assume that investors are indifferent to risk. (Back, Liu and Loewenstein, binomial trees (secondary source))

One-step example: what is the European call worth?

Assume a share price of £72, strike £75, up factor 1.25 and down factor 0.80. Each step lasts six months, with an assumed effective risk-free return of 4% per step, so R = 1.04. This is a six-month return, not an annual rate. Our first call expires after one step.

The next share prices are £90 and £57.60. A call pays the positive excess over its strike, or zero: $C_T=\max(S_T-K,0)$. Its payoffs are therefore £15 and £0. (Back, Liu and Loewenstein, binomial trees (secondary source))

The call’s value is £7.69 to the nearest penny. Keep the probability as a fraction or retain full calculator precision until the final rounding.

How does a replicating portfolio produce the same price?

Hold $\Delta$ shares and a signed cash balance $B$ today. Positive B means lending; negative B means borrowing. Match both future option payoffs: (Back, Liu and Loewenstein, binomial trees (secondary source))

In our one-step call example, delta is 15 ÷ (90 − 57.60) = 0.462963 shares, approximately. The share holding costs £33.3333 and B is −£25.6410. Borrowing that amount leaves a net initial cost of £7.6923. The debt grows to £26.6667 at expiry. Using unrounded amounts, the up-state portfolio pays £15 and the down-state portfolio pays zero, exactly matching the call.

A different option price would allow buying the cheaper payoff and selling the identical, dearer payoff. Replication therefore explains the risk-neutral price. This portfolio replicates a long call; combined with a short call, it offsets the option obligation. (Back, Liu and Loewenstein, binomial trees (secondary source))

Two-step example: work backwards through the tree

Now give the same call two six-month steps to expiry, retaining all other inputs. The share tree recombines: an up move followed by a down move reaches the same £72 node as the reverse path. Calculate terminal payoffs, then discount back one step at a time. (Back, Liu and Loewenstein, binomial trees (secondary source))

The two-step European call is worth £9.86. As a check, its only positive payoff follows two up moves, so discount £37.50 multiplied by p² over two steps, using R². The middle terminal node has probability 2p(1 − p), because two paths reach it.

Replication now requires rebalancing. Initial delta is approximately 0.593542; after the first move it becomes 0.925926 at the up node or zero at the down node. Each delta uses the next two option values and share prices. See the option Greeks guide for the connection to hedging. (Back, Liu and Loewenstein, binomial trees (secondary source))

When should an American put be exercised early?

A put’s exercise value is $\max(K-S,0)$. A European put waits until expiry; an American put permits earlier exercise. At expiry, use the put payoff. At every earlier exercise node, including today, apply: (Naranjo, American options (secondary source)) (Back, Liu and Loewenstein, binomial trees (secondary source))

Use the same two-step tree and £75 strike. The table’s terminal put payoffs are £0, £3 and £28.92. At the first up node, continuation is (7/15 × £3) ÷ 1.04 ≈ £1.3462, exceeding zero exercise value. At the first down node:

Immediate exercise gives £75 − £57.60 = £17.40, so exercise there. Carry £17.40 back through the tree, rather than the smaller continuation value. At today’s node:

Waiting is preferable today because £8.4980 exceeds immediate exercise of £3. The European put, which retains continuation at both intermediate nodes, is worth £7.2036. The American tree value is £8.50, with an early exercise premium of approximately £1.29 over the European value.

An American option cannot be worth less than its otherwise identical European counterpart. However, early exercise is not automatically worthwhile whenever an option is in the money. With positive rates and no dividends, exercising an American call early is suboptimal: the call is worth at least the share price minus the discounted strike, which exceeds its exercise value. Dividends can change that conclusion. (Naranjo, American options (secondary source))

How do volatility and step length affect the model?

Our up and down factors were assumed. The Cox–Ross–Rubinstein specification instead links them to annualised volatility $\sigma$ and the step length: (Back, Liu and Loewenstein, binomial trees (secondary source))

Refining a fixed maturity into more steps requires recalculating the factors, probabilities and discounting. A coarse tree permits exercise only at its nodes, so its American price approximates the value of continuous exercise opportunities. More steps improve the time resolution, but prices need not converge monotonically. Check step units, the no-arbitrage condition and every exercise decision before trusting the result. (Back, Liu and Loewenstein, binomial trees (secondary source))

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