Using Volatility to Set Up a Binomial Option Price Tree

Summary

The document asks how to construct binomial trees for a six-month European call when annual interest and annualized stock volatility are given, and how to obtain option values and deltas at several tree resolutions. The answer addresses the first modeling step: use volatility and the time horizon to define reciprocal up and down multipliers, with the up factor expressed exponentially and the down factor as its inverse.

This gives a way to translate a volatility input into possible stock-price moves, but the response is only a brief starting point. It does not show how to adjust the time increment for trees with multiple periods, determine risk-neutral probabilities, discount expected payoffs, or calculate the call value and delta. In a multi-step tree, the move size depends on the length of each step, so the formula must be applied consistently to the per-step interval rather than mistaken for a complete pricing procedure.

Key ideas

• Volatility can be used to construct the up and down price multipliers in a binomial model.
• The suggested down multiplier is the reciprocal of the up multiplier.
• For a multi-period tree, the volatility scaling must reflect each step's time increment.
• The response gives no risk-neutral probability, option valuation, or delta calculation.

Tags

# How to construct the binomial model for European option?


# How to construct the binomial model for European option?












The annual interest rate is 5.3% and the annualized volatility of a non-dividend paying stock over the next six months will be 12.5% (annualized). i) Construct binomial trees of 5, 10 and 30 periods to calculate the value of a European-style 6-month call option with strike price 8.3% above today's spot price. ii)Calculate the corresponding Deltas in the three cases.

Well for the above problem, I stuck at how to construct the binomial tree model. For reference my knowledge of binomial tree model for stock option is the first 4 chapter of Shreve- Stochastic Calculus for finance.In his book, whenever he wants to construct a binomial model, he always have a up (u) factor, and down (d) factor for stock price, thus you can have the model pretty easily. In this problem, I am given the volatility, thus I have no idea how to start. Please help .

## Answer by TLP (score 1, accepted)

https://quant.stackexchange.com/a/9364

In the case that volatility is given, you're very close. You can calculate $u$ and $d$ using volatility:

$$ u = e^{\mbox{volatility}*\sqrt{T}} $$

$$ d = \frac{1}{u} = e^{-\mbox{volatility}*\sqrt{T}} $$

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