Building a Recombining Trinomial Tree by Matching Moments
Summary
The document explains how to specify a three-branch asset-price tree when a single-period move can be up, middle, or down. The three branch probabilities and three move sizes initially leave six parameters. Probability normalization and matching the expected asset value to the forward price impose two constraints, while choosing reciprocal up and down moves and a middle move of one creates a recombining tree and reduces the remaining freedom.
The modeler can then match the variance to the target volatility over the time step, leaving one degree of freedom to choose. The response identifies a conventional choice for the up move and describes the remaining choice as a trade-off between convenience and approximation accuracy; it also notes that a multidimensional numerical solver can determine parameters directly. The discussion is a construction recipe, not a complete treatment of contingent-claim valuation or hedging. In particular, the original question's concern about incompleteness is not resolved in detail, and the stated parameterization depends on the chosen constraints and moment targets.
Key ideas
- A trinomial step begins with three probabilities and three possible asset multipliers to determine.
- Probability normalization and matching the forward expectation provide two constraints.
- Reciprocal up and down moves with a unit middle move produce a recombining tree.
- Matching variance leaves one model choice that can target approximation quality or a higher moment.
- A numerical root solver can handle the parameter equations directly.
Tags
Full text
# Extension of CRR model
# Extension of CRR model
I'm considering an extension of the binomial model where the risky asset can take three values at each node, that is $ S_{t+1}=\left\{ \begin{array}{ll} S_t\cdot u\\\nonumber S_t\cdot c\\ S_t\cdot d \end{array} \right.$
with $0<d<c<u$
If we consider $r\in]d,u[$ the market is arbitrage free but for sure it is not complete. I don't think I can find a unique price for the contingent claim so my question is what is possible ? I tried to solve the system by backward induction to find a hedging strategy but the system has no solution . By the way, the fact that we add a third value invalid all we have about the price of the contingent claim as an expectation since the binomial representation is broken ?
Thank you a lot
## Answer by Kermittfrog (score 2)
https://quant.stackexchange.com/a/71181
In all brevety: The model has 6 degrees of freedom:
$$p_u,p_c,p_d, u,c,d$$
We have the following two 'natural' constraints:
$$ p_u+p_c+p_d=1,\quad\quad E^{\mathbb{Q}}(S_{t+\Delta t})=F_{t+\Delta t} $$
leaving four d.o.f. Adding the constraints $u=1/d$ and $c=1$ induces a recombining tree that grows polynomially instead of exponentially. This leaves us with two degrees of freedom. Commonly, we want to moment-match the not only the first but also the second moment of the distribution of $S_{t+\Delta t}$,
$$Var(S_{t+\Delta t})=\Delta_tS_t\sigma^2$$
This leaves the modeler with one degree of freedom. You may close this d.o.f. so that it solves one additional requirement, i.e. the quality of the variance approximation or a higher moment of the distribution. Canonically, $u=e^{\sigma\sqrt{2\Delta t}}$ is chosen.
Note that this choice for $u$ results from a trade-off between convenience and accuracy. At any point, you can simply solve for the four degrees of freedom directly using some numerical multidimensional root solver.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.