What if we have a very low volatility? In the printscreens a changed up and down factors from (1,1;0,9) to (1,01;0,99) respectively. I got funny risk-neutral probabilities (see the printscreens). With such nubers our trees dont seem to work properly? How do you think?
Hi @Maxim Rastorguev You are right, I've never noticed this before: it's true even when we use the volatility, σ, input; e.g., σ < 3.0% returns p > 1.0. The formulas are correct (obviously, as all of the examples match Hull!) ... I'm not immediately sure how to explain this ... will update if/when I figure it out. Thank you for sharing the outcome
Yes, David the formula is ok. I succesfully checked it all by construction trees through replicating cash flows of options via borrowing and buying spot stock (didnt use risk neutral probabilities at all). All the results were totally in correspondence with Hull.
Your parameters violate the assumption u > a.
Your jump up must be at least as big as the gain from the risk free rate.
That is because per definition the average gain of the asset under the risk neutral probabilities must equal the risk free rate. That is not possible if u is chosen too small.
David, I cant get how you calculated the average return in the binominal trees learning spreadsheet. I changed the returns as shown leaving all the formulas for total returns as they are. For me the number zero of average return doesnt look logical (see the picture please)
Hi @Maxim Rastorguev Yes, your period (annual) returns are +15%, -15%, +15%, -15%, 0% such that the arithmetic return is given by (+15%, -15%, +15%, -15%, 0%)/5 = 0. This is meant to illustrate the same distinction explained in Hull's Business Snapshot 15.1 (see below). Just as you are suspicious of the zero, Hull says "What average return should the fund manager report? It is tempting for the manager to make a statement such as: ‘‘The average of the returns per year that we have realized in the last 5 years is 14% [i.e., his arithmetic average]’’ Although true, this is misleading. It is much less misleading to say: ‘‘The average return realized by someone who invested with us for the last 5 years is 12.4% per year. [i.e., his geometric mean, analogous to your -0.91%]’’ In some jurisdictions, regulations require fund managers to report returns the second way."
Similarly, the standard deviation of your series is about 13.4% and--this is always interesting to me--we can see that the geometric average of -0.91% is, in fact, less than zero by 1/2 the variance: 0% - 13.4%^2/2 ~= -0.91%. Put another way, volatility erodes returns. Thanks,
Interesting! of course I knew the geometric was less than arithmetic but have not thought of this special case when it is in fact a loss while the arithmetic says it is status quo. May be it is even possible to construct a case with arithmetic gain on the back of actual loss. Very misleading basically.