Boyle Trinomial Tree Probabilities as the Spacing Parameter Grows

Summary

This note examines how the risk-neutral probabilities in Boyle’s recombining trinomial tree behave as the parameter controlling the size of up and down moves becomes very large. Boyle’s parameterization uses a larger up factor than the basic CRR choice to help avoid negative transition probabilities or probabilities above one for reasonable inputs. The response gives the limiting probabilities: the up-move probability tends to zero, while the other two probabilities approach values determined by the per-step growth factor.

The interpretation is that an infinitely large up-jump becomes an event with zero probability. In this limiting case the trinomial tree effectively reduces to a binomial structure. The stated non-negativity and boundedness apply to reasonable parameter values, such as small time steps and rates; they are not a general guarantee for arbitrary inputs. The note offers a limiting explanation rather than a full derivation or a numerical comparison across parameter choices.

Key ideas

• Boyle’s parameterization increases the up factor to help avoid inadmissible probabilities in a trinomial tree.
• As the jump-size parameter grows without bound, the probability of an up move tends to zero.
• The remaining transition probabilities approach limits governed by the per-step growth factor.
• In this extreme limit, the trinomial tree behaves like a binomial tree.
• Probability bounds depend on reasonable model inputs and are not asserted for every parameter choice.

Tags

# Boyles Model for Trinomial Tree


# Boyles Model for Trinomial Tree












I know that the risk neutral probabilities in Boyle's Model for the Trinomial Tree by recombining where $m=1, u.d=1$ and $u=e^{\lambda\sigma \Delta t}$

$p_u=\frac{u(V+M^2-M)-(M-1)}{(u^2-1)(u-1)}$

$p_d=\frac{u^2(V+M^2-M)-u^3(M-1)}{(u^2-1)(u-1)}$

$p_m=1-p_u-p_d$

where $V=e^{\sigma^2\Delta t}$ and $M=e^{r\Delta t}$

When $\lambda \rightarrow \infty$ how would it impact the risk neutral probabilities?

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/51489

Note that Boyle (1988) introduces $\lambda$ because the CRR parameterisation $u=e^{\sigma\sqrt{h}}$ yielded negative probabilities (and probabilities above one) for reasonable parameter values. Instead, he uses $u=e^{\lambda\sigma\sqrt{h}}$, where $\lambda>1$ and $h=\frac{T}{n}$ is the length of one time step.

If you perform the limits, $p_u\to0$ and $p_d\to1-M$ causing $p_2\to M$ as $\lambda\to\infty$, where $M=e^{rh}$. Again, for reasonable parameter values, (i.e. $h$ and $r$ small, below $1$), the probabilities are non-negtaive and bounded above by one. So everything is fine.

However, you'll notice that up-jumps do not occur anymore: $\lambda\to\infty$ implies $u\to\infty$ and the event that the stock increases by an infinite number over one period should indeed have probability zero. So really, in this extreme case, the trinomial tree collapses to a binomial tree.

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