How to Determine Parameters in a Non-recombining Binomial Tree for Option Pricing
How to Determine Parameters in a Non-recombining Binomial Tree for Option Pricing
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Gull23 · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: Gull23
Original post: https://quant.stackexchange.com/questions/76625
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
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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Hritabrata Das · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: Hritabrata Das
Original post: https://quant.stackexchange.com/a/85752
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
A non-recombining binomial tree basically drops the assumption of up-down movement equals down-up movement.
We would need to specify two path dependent volatilities so that at step 2 your stock prices are different and they do not convergence.
Once done , use standard risk neutral formula and your p's should be reflected.
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Quoted from Forex.com.bd-Editorial External question — Quantitative Finance Stack Exchange Author: Gull23 Source score (net votes, not local likes): 0 Original post: https://quant.stackexchange.com/questions/76625 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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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