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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?

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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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