No-Arbitrage Bounds for a Continuously Compounded Binomial Model

Summary

The document asks how the one-period no-arbitrage condition in a binomial model changes when interest is expressed with continuous compounding and the option maturity is not one period. The response converts the gross risk-free return over the period to its continuously compounded equivalent. For a step of length Δt, the risk-free growth factor is exp(rΔt), and the risk-neutral up probability is formed by placing that growth factor between the down and up factors.

The probability lies between zero and one exactly when the risk-free growth factor lies between the two asset growth factors, giving the corresponding no-arbitrage bound. The explanation is concise and applies to a one-step model. It assumes consistent definitions of the up and down factors and the interest rate over the same interval; a multi-step tree must apply the condition at each step, and the document gives no worked numerical example.

Key ideas

• The one-period no-arbitrage condition places the risk-free growth factor between the down and up factors.
• Continuous compounding expresses the period’s risk-free growth as an exponential of the rate times the interval.
• The risk-neutral up probability is valid when it falls between zero and one.
• The interest rate and asset factors must refer to the same time interval.
• A multi-step binomial model requires checking the condition at each step.

Tags

# Binomial Model for options pricing with continuous compounding


# Binomial Model for options pricing with continuous compounding












I'm reading about Binomial Model on "Arbitrage Theory in Continuous Time" by Tomas Bjork. I found an important result which allow us to state that in a one period model $q_u$ and $q_d$ are actually probabilities. The following theorem holds:

"The model is arbitrage-free if and only if $$d\le(1+R)\le u."$$ At the same time I'm reading "The Bible" by Hull where the author deals with the model but using continuous compounding of interest rates. I'm wondering if an analogous of Bjork'sresult holds, such as

The model is arbitrage-free if and only if $$d\le e^{rt} \le u,$$ where $t$ is the option maturity (I'm taking into account maturities which are different from 1). Thanks in advance.

## Answer by LocalVolatility (score 0, accepted)

https://quant.stackexchange.com/a/32773

If the first statement is true then the second is also as you essentially just convert between the different compounding frequencies - i.e. from the one-period rate $R$ to the continuously compounded rate $r$. If the second result didn't hold then the up-probability

$$ p = \frac{e^{r \Delta t} − d}{u − d} $$

would not be in $[0,1]$.

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.