Building a Trinomial Option Tree with a Deterministic Rate Curve
Summary
The document considers extending a constant-volatility trinomial stock option model to use a deterministic, time-varying interest rate curve. The suggested approach is to set the rate for each time step so that its discount factor matches the curve over that step, then discretize the log-price process using a time-dependent drift and a trinomial approximation to Brownian motion. Option values can then be calculated by backward induction.
The discussion frames this as one modeling choice among many, rather than a uniquely prescribed construction. It notes that trinomial and binomial trees admit multiple discretizations, so the details of the drift, transition probabilities, and tree structure matter. The answer does not provide a full implementation or demonstrate that every choice of step size and parameters yields valid probabilities or a recombining tree. Those properties must be checked for the particular scheme being implemented.
Key ideas
- A deterministic rate curve can be represented by rates chosen separately for each time step.
- Step rates can be selected to match the discount factor over each interval.
- The log-price process can use a time-dependent drift with constant volatility.
- A trinomial discretization is one of many possible tree constructions.
- Validity of probabilities and tree behavior depend on the chosen discretization.
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# Trinomial model for stock options with deterministic interest curve
# Trinomial model for stock options with deterministic interest curve
I am implementing a basic trinomial model with constant volatility right now. I want to do an extension that does not take a constant riskfree rate as input, but interpolates between different given interest rates. I thought this would just change the following: 1) the discount factor at each step 2) the risk neutral probabilities at each step So I could calculate all the interpolated riskfree rates and all the risk neutral probabilities, use those for backward induction and get my option price. Because the up and down movements would stay the same the tree would still recombine and I assume I would still end up with positive probabilites that sum up to one (can´t see why not!).
However, I´m not so sure this approach works anymore. The main reason is I couldn´t find any papers or implementations for this approach. That got me puzzled whether I am missing something. Maybe it´s the assumption that the volatility stays the same with a changing riskfree rate.
Google and library search couldn´t really help me, because with all the search words I ended up at articles dealing with interest rate models, which I am not trying to delve into right now, because they are too complex and I only try to implement a deterministic interest rate structure.
Reframing the question: would my approach work or is there some part missing that I didn´t think of.
Thanks!
## Answer by Mark Joshi (score 1)
https://quant.stackexchange.com/a/27711
you could do it that way. Or you could take a time-dependent drift and discretize
$$ d \log S_t = (r(t) - 0.5\sigma^2) dt + \sigma dW_t.$$
$r(t)$ is chosen to be constant across each step so that the df is correct across each step. Take $W_{t+\Delta t} - W_t$ to be $\sqrt{3 \Delta t} X$ with $X \in \{-1,0,1\}.$
There are many ways to do a trinomial tree. Eg There are over 30 binomial trees in the literature. See my book "More mathematical finance" for a discussion of the modelling choices.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.