Choosing Drift Parameters in Binomial Trees for Option Pricing

Summary

The document explains why binomial-tree formulas for the up factor can differ across constructions. In the Cox–Ross–Rubinstein tree, the up and down factors are set symmetrically around zero log drift, with volatility determining their size. The answer notes that other fixed drift choices can also produce the same limiting prices, so the factors need not follow one universal formula.

It identifies alternative drift choices intended to improve convergence, including choices based on interest rates, dividends, volatility, or the option’s strike relative to the initial asset price. The answer also places these formulas in the broader history of tree methods, noting that many variants exist and that some achieve higher-order convergence for European options. It offers no derivation, numerical comparison, or detailed conditions for the stated convergence claims, so readers should treat it as a concise conceptual explanation rather than a construction guide.

Key ideas

• Different binomial-tree methods can use different drift terms in their up and down factors.
• The CRR tree uses symmetric log moves determined by volatility.
• Several alternative drift choices may preserve limiting prices while improving convergence.
• Some more sophisticated trees are designed for higher-order convergence in European option pricing.

Tags

# Difference in formulas for u & d in Binomial trees


# Difference in formulas for u & d in Binomial trees












For a binomial tree, everywhere in Hull and other literature, we have found the formulas for

$$u = \exp(\sigma \sqrt{h})$$

but for binomial trees based on forward prices, we get a different formula

$$u=\exp((r−\delta)h+\sigma\sqrt{h})$$

Could anyone please provide an explanation of why there is this extra term of $\exp(r-\delta)$ multiplied here?

I understand that $\delta$ is for the constant dividend yield but why is there a difference in formulas for $u$ when binomial tress are constructed using forward prices?

## Answer by Mark Joshi (score 4, accepted)

https://quant.stackexchange.com/a/32525

there are many different trees. The first one, the CRR tree, used $$ u = e^{\sigma\sqrt{h}} $$ and $d = 1/u.$ However, you can take any real-world drift and still get the same prices in the limit so you can put $$ u = e^{\mu h +\sigma\sqrt{h}}, \text{ and } d = e^{\mu h -\sigma\sqrt{h}} $$ for any fixed $\mu.$

$\mu = 0$ is a poor choice for convergence. Better choices are $$ \mu = r - d - 0.5\sigma^2 $$ and $$ \mu = \frac{1}{T}(\log K - \log S_0). $$

There has been a huge amount of work on binomial trees in the last 40 years and there is now over 30 of them. More sophisticated trees achieve higher order convergence for European options.

I give a comprehensive survey in my book, More Mathematical Finance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.