Trinomial model for stock options with deterministic interest curve

Question

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 1

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.