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I am reading the book "Principles of Corporate Finance" 12th edition by Brealey, Myers and Allen. In the 21st chapter on Option pricing, they discussed the General binomial method for option pricing. The authors posed the question " How do we pick sensible values for the up and down changes in value?" and went on to give the following formulas.

$$1+\text{upside change} =u=e^{\sigma \sqrt{h}}$$ $$1+\text{downside change} =d=\frac1u$$

where $\sigma$ is the standard deviation of continuously compounded stock returns and $h$ is the interval in the binomial method as a fraction of a year.

Is there any explanation as to why these formulas for upside and downside change in the stock price make sense in the binomial method of option pricing.Under what conditions these formulas may not work? Thanks.

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Option pricing in general is about building a hedging portfolio where, if you have a given amount of money and follow a given strategy, you'll perfectly replicate the payoff of the option. Binomial option pricing is a particular way of building this strategy that has the benefit of being fairly intuitive to understand.

From a practical perspective, having $u = \frac{1}{d}$ is pretty important. Naively, you can make $u$ and $d$ whatever you like (and you can even make $u_t$ and $d_t$ time dependent), but this would lead to $2^t$ nodes at time $t$, while a tree with $ud=1$ can be designed in such a way that at timestep $t$ there are $t+1$ nodes. This brings the pricing algorithm from $O(2^N)$ to $O(N^2)$ -- an appreciable speedup on a trading floor if you have maybe 20 timesteps, downright necessary if you have 365.

As for the exact form of $u = e^{\sigma \sqrt h}$, perhaps someone has a more rigorous answer than mine, but this formula has the nice property that it relates the upward move to 1) the length of the timestep over which this move is observed, and 2) the volatility of the stock.

For the question of why a volatiltiy equalling the standard deviation of past returns makes for a fair value of the option, it doesn't necessarily. In options markets you can think of the implied volatility as the parameter $\sigma$ that equates the market price of an option with the value give in a model such as Black Scholes. I would personally see the binomial model with $\sigma$ chosen as you've described to be a starting point in the valuations process and a parameter one could "tune". (This is a personal opinion, implied volatilities for options you can't directly observe is a big topic that I'm no expert on.)

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