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For a CRR recombining Binomial Tree, let the underlying stock price be $S_0$ at $t=0$ and the time interval be $\Delta t$. The nodes at $t=\Delta t$ and probabilities reaching them can be written as:

$ \left\{ \begin{array}{**lr**} S_u = S_0e^{\sigma \Delta T},\ p_u=\frac{e^{r \Delta t}-d}{u-d}\\ S_d = S_0e^{-\sigma \Delta T}, \ p_d=1-p_u \end{array} \right. $.

And we will have $S_{ud}=S_{du}$ at $t=2\Delta t$ because $ud=1$.

Now, if I want to construct a $N$ step non-recombining Binomial Tree which is only limited to $d<e^{r \Delta t}<u$. How should I derive $u$, $d$ and $p$ under risk-neutral condition, except using real option prices to calibrate?

I've been going through literature reviews about numerous Binomial Trees proposed until now but failed to find a general method. Any textbook or paper link is welcomed! Thanks!

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    $\begingroup$ Can I ask for some context around the desire to use non-recombining trees and to not use real prices to calibrate? To my knowledge, recombining trees are used so that the calculation doesn't explode in size (with recombining trees, our algorithm is roughly $O(n^2)$, while without them it's roughly $O(2^n)$, where $n$ is the number of timesteps). And calibrating to real option prices isn't necessary per se, but ensures consistency with the market. $\endgroup$
    – Rylan
    Commented Sep 9, 2023 at 9:58
  • $\begingroup$ @Rylan Thanks for replying! The reason I want to use non-recombining tree is because I need each final node only to be reached by one unique path, so the calculation of greeks for each node will not have $p_u$ and $p_d$ involved. As for the possible exponential blowup, I'm thinking about only use $n \leq 5$, so the number won't be too large. $\endgroup$
    – Gull23
    Commented Sep 11, 2023 at 2:25
  • $\begingroup$ Assuming $r$ is fixed, then the general approach for trees is: choose $u$ and $d$ to match some dynamics of the market. $p_u$ and $p_d$ are just mathematical constructs to express a hedging strategy in terms of probability (in other words, they can't be interpreted as "the chance that the stock rises or falls"). Choosing $u$ and $d$ is analogous to looking for a vol for your options in a continuous time model -- in the absence of a liquid market, this is a difficult question lacking a "best" answer. $\endgroup$
    – Rylan
    Commented Sep 11, 2023 at 9:22
  • $\begingroup$ @Rylan Thank you very much! You are right, without calibration using price data from liquid market, determining appropriate $u$, $d$ and $p$ is very difficult. I found a paper using stock price historical distribution to construct non-recombining binomial tree, I will try this one out. $\endgroup$
    – Gull23
    Commented Sep 18, 2023 at 1:13

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