Risk Latte - Betting on Anything - It All depends on the Dispersion

Betting on Anything - It All depends on the Dispersion

Rahul Bhattacharya
Aug 01, 2005

Consider a very simple bet: if, after a year from now, the temperature of New York is equal to (or more than) what it is today then your friend will pay you $100. If it is less than what it is today then there will be no payment (pay off). How much should you pay your friend today for this bet? When we asked this question to our office manager (who is not a finance professional but an amateur gambler), she immediately answered $50. Is the answer correct?

Let's make two rather realistic assumptions: (a) the cost to your friend (whatever that may be) for executing this bet as well as any interest is quite negligible; and (b) the temperature of New York is a random variable and follows a stochastic path such that its movement in time is totally (and only) governed by its standard deviation (i.e. its dispersion from the mean value over the one year in question);

As a corollary to the second point, it is fair to assume that the current value of the temperature, i.e. today's temperature is the best estimate of the mean value of all daily temperatures from today until one year.

Let us assume that the dispersion around today's value of the temperature is 10% annually. This means though the expected temperature a year from now is the temperature that prevails today in the intervening period the temperature can fluctuate up or down.

Now let us make a totally unrealistic assumption: say, that all gamblers in this world, which includes you and your friend, only earn an annual return (profitability) on all kinds of bets which is absolutely certain, unchanging and known in advance. It could be 1%, 3%, 5% or whatever, but it is certain to happen and known in advance. Further, assume that this rate of return is presently so low that your friend can almost ignore it. In other words this rate of return is 0% (Of course if this is the case then there is absolutely no incentive for you and your friend to enter into a bet but still you guys go ahead just for the sake of friendship).

If all the above hold true then the pricing of this bet collapses to a simple proposition: the value of the temperature today is the best estimate of the temperature that will prevail in a year's time. Now, we have said above that the temperature can go up and down but in exactly one year's time it will most likely be where it is today. So, if the temperature today is T then can write a simple equation of the temperature movement as a function of its probability of exactly hitting the value what it is today as:

p*T*u+(1-p)*T*d = T
where,
p = probability
u,d = magnitudes of up and down moves in the temperature

However, the most important thing to note in the above equation is that both the values of u and d would depend on the dispersion (deviation) of the temperature from today's value which is assumed to be 10%. If that is the case (along with all the above assumptions holding true) then the value of p, the probability that the temperature will be the same as it is today is 47.50%. This would be the price of the bet.

A very simple back of the envelope answer to the original question would have been be $50. Since in the real world there is a 50:50 chance of temperature going up or down (or in other words there is a 50% chance of the temperature in one year being what it from what it is today and 50% chance of it being different) the best guess for a price of this bet is $50. However, this 50:50 assumption does not take into account the dispersion of the value of the temperature around its average value, which in this case is today's value.

In fact, it is one of the greatest insights into betting that the real world probabilities of 50:50 never hold. The reason is the dispersion or the deviation of the value of the bet around its mean.

The true value of a bet is always governed by the probabilities of an up move from the current value and the down move from the current value (where the world is binomial) which are in turn determined by the magnitudes of the up move and the down moves, u and d in the above equation, and which in turn are totally determined by the dispersion (deviation) of the value of the bet from its average value. The fair value of the bet is never 50%.

And it holds true for all kinds of bets in this world. Thus in this case if you paid $50 for the above bet to your friend then you overpaid for your bet!


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