Converting Interest Rates for a Discrete Binomial Tree
Summary
The document clarifies how continuously compounded and annual-effective risk-free rates relate to the per-step rate in a discrete binomial model. Even when the asset tree advances in discrete steps, the risk-free rate may be quoted continuously; for a step lasting a fraction of a year, the corresponding discount factor is determined by the continuous rate and step length. This lets the model discount next-node payoffs consistently with the quoted rate basis.
It also gives the conversion from a continuously compounded annual rate to its annual-effective equivalent, and describes how a rate compounded quarterly can be converted to an annual-effective basis. Bringing rates quoted under different compounding conventions to a common annual-effective basis makes their implied accumulation comparable. The post is a conceptual explanation rather than a numerical solution to the stated tree exercises; applying it still requires knowing each step’s time length and expressing the rate on a matching basis.
Key ideas
- A discrete-time binomial tree can use a continuously compounded risk-free rate for discounting.
- The step discount factor depends on both the quoted continuous rate and the duration of each step.
- A continuously compounded annual rate converts to an annual-effective rate through exponential accumulation.
- Rates with different compounding frequencies can be compared after conversion to a common annual-effective basis.
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# Definition of interest rates in binomial tree model
# Definition of interest rates in binomial tree model
I'm studying financial mathematics from Shreve's text. I have two problems.
1) "for a binomial tree with three steps, where $S_0=20$, $u=1.05$, $d=.95$ and continuously compounded risk-free interest rate is 5% (note, we need to transfer continuously compounded interest rate first to be equivalent to interest rate compounded annually)..."
2) For a three step binomial tree in which $S_0=10$, $u=1.15$, $d=.9$ and the annual-effective risk free rate is seven percent...
I'm not exactly what these interest rates mean mathematically. I'm familiar with the very basic setup for a binomial tree model, where we assume there is no arbitrage and $d<1+r<u$, and this $r$ is the interest rate one accumulates on investment in the money market for each step in time. In quest 1, what does it even mean for interest to be compounded continuously if the binomial model is discrete time? For the second question, I don't know what the annual-effective interest rate means or how it relates to this $r$ (I don't believe it's defined in the text). Any help on clarifying these definitions would be greatly appreciated.
## Answer by thetableed (score 2)
https://quant.stackexchange.com/a/48740
Even though the steps are discrete, we can still express the risk free rate $ r $ as a continuously compounded rate - all this implies is that, if each time step is $ \tau $ units of time long, the discount rate you need to apply to the payoffs at the next node is $ e^{-r \tau} $
We can also describe the risk free rate as its annual effective rate, typically denoted $ R $, which is just the simple annual rate that implies the same force of interest as a compounded rate. If $ r $ is the continuously compounded rate, then we have $ 1 + R = e^{r} $, whereas for a quarterly compounded rate $ r_{q} $, $ R = (1 + \frac{r_{q}}{4})^{4} $. If you have rates quoted in different bases (e.g. monthly, quarterly, or semiannually), moving them to their respective annual effective rates lets you more easily compare the actual force of interest for each rate.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.