How do we know this model is arbitrage free? And why does it only mean revert if mu<0 and then it reverts to 0?
The SDE in the question describes the Vasicek short rate of interest with a long-term mean of zero, but only if mu < 0. The negative mu is required so that the increments in r(t) always drift towards zero. (When r(t)>0 then dr(t) has negative drift, when r(t)<0 then dr(t) has positive drift). A positive mu would repel the short rate away from zero. The Vasicek model is known to the arbitrage free, ie it leads to bond prices which cannot be exploited to make a risk-free profit. It is not obvious that this is case from the given SDE. We thank Vasicek for his efforts and then quote this as a result.
Thank you, Steve. Thank you, Vasicek. If we are given a non-standard model is it safe for us to quote that it is arbitrage free then?
We can't say that any given model is arbitrage free. However, if the non-standard model is a special case of a known model (as it the case here), then we can make the no-arbitrage claim.
