# Non-Negativity of up-factor and down-factor in Binomial No-Arbitrage Pricing Model

Consider a stock which is trading at $S_0$ at time $t=0$ and is expected to be trading at price $uS_0$ or $dS_0$ at time t=1 where $u$ and $d$ are up-factor and down-factor. The theory says that to rule out the arbitrage, we must assume that : $0<d<1+r<u?$ Can someone explain how does this assumption takes care of no-arbitrage?

• Hi QuantNut, welcome to quant.SE! Thank you for asking your question here. – Bob Jansen Jul 28 '14 at 13:51
• Thanks Bob.. I hope to take away a lot of learning and also to contribute in the ongoing discussions. – QuantNut Jul 29 '14 at 6:08

This sounds like the first chapter of Björks book am I right? It treats a single-stage model. Simply put, if $1+r \leq d$ you buy the stock and have $V_1\geq 0$ with positive probability of making a profit. If $1+r \geq u$ you want to sell the stock short and buy the bond from the proceeds. The result is the same.

Edit: To show that the condition is sufficient, we can follow Proposition 2.3 from "Arbitrage Theory in Continuous Time". In a two-asset, one-period world one can characterize all possible arbitrage portfolios (because $V_0 = 0$) by $x+yS_0 = 0$ ($x$ is the amount of money invested in bonds, $y$ in stocks) and thus write the value at time $1$ explicitly:

$V_1 = y S_0 (u- (1+r)), \text{if S goes up}$ and $V_1 = y S_0 (d- (1+r)), \text{if S goes down}$

Now, for an arbitrage portfolio with $y>0$ we need that $V_1 > 0$. That can only happen if $u>1+R$ and $d>1+r$. Similarly, for an arbitrage portfolio with $y<0$, whe get the other direction of the inequality.

• Clearly, there is arbitrage if the equation doesn't hold, so it is a necessary condition. Is the equation also a sufficient condition? – Bob Jansen Jul 28 '14 at 8:37

If the condition $$0<d<1+r<u$$ is not satisfied, the Binomial model (with $d<u$) would have immediate arbitrage opportunity:

1) $1+r\geq u$: Then the riskfree asset would yield least as much return as the stock in any state for any probability, so you could short the stock to buy the riskfree asset and end up with some riskless profit with positive probability (arbitrage).

2) $d\geq 1+r$: Then the stock would yield least as much return as the riskfree asset in any state for any probability, so you could short the riskfree assetto buy the stock and end up with some riskless profit with positive probability (arbitrage).

The condition $d,u>0$ is just to ensure positive stockprices in all states.