# 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?

## 1 Answer

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.