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




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?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.