We are just learning about binomial option pricing, and how the up-factor and the down-factor must match the risk-neutral price.

p * u + (1 - p) * d = continuous risk free rate compounded

CRR proposed that u = 1/d as well.

What happens if that risk-neutral pricing does not hold? How would you exploit this type of arbitrage? Would you invest in an instrument that grows at the risk-free rate and short/long whatever instrument if u or d is higher?

Also might as well ask this question: Which options are most liquid? OTM, ATM, or right outside the money?


3 Answers 3


To rule out arbitrage in the one-period model, we must assume $$ 0 < d < 1+r < u, $$ where $u$ is the up-factor, $d$ is the down-factor and $r$ is the risk-free interest rate. This chain of inequalities is the no-arbitrage condition.

To see what happens if it doesn't hold, consider the case in which $$ 0 < 1+r < d < u. $$ Let $S$ denote the initial stock price. To exploit this situation, borrow $\$S$ from a risk-free money market account and buy the stock for $\$S$. Note the initial cost of this portfolio is $0$. At the final time: $$ \begin{array}{c|c|c|} \text{up state} \\ \hline & \text{receive:} & $uS \\ & \text{pay:} & $(1+r)S \\ & \text{profit:} & $(u - (1+r))S > 0\\ \text{down state} \\ \hline & \text{receive:} & $dS \\ & \text{pay:} & $(1+r)S \\ & \text{profit:} & $(d - (1+r))S > 0\\ \end{array} $$ Thus you've started with $\$0$ and profited in every state of the world - arbitrage.

If we instead have $$ 0 < d < u < 1+r, $$ long the money market account and short the stock. Can you work out the sure profit in this case? This is the standard argument for exploiting the arbitrage condition, and can be found in Shreve I, page 2.


ATM options are always more liquid. Options with shorter maturities are also more liquid. Best way to learn more is to open a brokerage account that doesn't have any minimum amounts or monthly fees and you can watch some delayed live option quotes across a whole chain of strikes and maturities .

The first question you are asking is really how to profit if the actual drift is considerably different from the risk-free rate.

Here's one good vanilla way - buy call (put) if the actual drift is considerably higher (lower) with the following properties : long maturity (e.g. Leaps) so that the theta decay is small, always either deep OTM or deep ITM so that the option is minimally affected by vega. The choice of deep OTM vs deep ITM is a personal one, because in deep OTM the break even profit may not be reached and you can lose 100% of premium, while in deep ITM, the leverage (gearing) you can get is low and the profits can be much subdued. In practice, since this strategy will be operating at longer maturities and away from the ATM - you will be operating in the illiquid part of the option surface. So to minimize the option bid-ask, from practical experience, you really have to limit yourself to underlying that trade > 5 millions shares / day and the price of underlying should be > 20 if you want to buy puts or <40 if you want to buy calls .

This is a vanilla trade construction. You can always explore more complicated structures. The best structures I have found are OTC options on multiple equity indices (max/min of 3 asset returns, etc.,) if you have a drift view on 3 similar equity indices and a lot of big brokers will quote you max/min of N asset options on super liquid underlyings, or OTC average price options (TAPOs) in the case of commodities which are usually not that liquid.


You ask what to do "if risk-neutral pricing does not hold". By this I assume you mean that the price of an option is not equal to its expected value under the risk neutral probabilities (these are the probabilities that are calculated by enforcing the condition tha the expected return on the stock is the risk free rate, which can only exist if d <1+r< u). So the answer is that there is a combination of one option and k stocks, where k is a number to be calculated, which produces a guaranteed return different from the risk free rate. If the option price is too high you would sell the option versus the stock hedge, and if the option price is too low you would buy it versus a stock hedge.


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