And in fact, Daniel Bernoulli, a famous mathematician,

resolved this paradox by introducing a so-called utility function.

The utility function has the following properties.

It measures how much utility or benefit you're paying from x units of wealth.

So u of x measures how much utility or benefit you obtain from x units of wealth.

Different people of course, have different utility functions.

The utility function should be increasing and concave.

It should be increasing to reflect the fact that people prefer

more money to less money.

And concavity is there to model the fact that getting an extra dollar

when your wealth is say, $1,000, gives you less

additional benefit than getting $1 when your wealth is $0.

In other words, going from $0 to $1 has more benefit

than going from $1,000 to $1,001.

And this idea is captured by using a concave utility function.

So, Bernoulli suggested using log utility function,

the log function is increasing and it's concave.

So, this is an example of a concave function.

It's like an inverted saucer.

Look, it's increasing and it's concave.

So the log utility function is what Bernoulli suggested.

And if we did that with the St. Petersburg game, we find the following.

The expected utility of the payoff is now the sum of the utility of the payoff.

So it's now log of 2 to the n if the first heads occurs on

the nth toss, times the probability of the first head occurring on the nth toss,

which is 1 over 2 to the power of n.

And if you recall, the log of 2 to the n, this is a property of logs,

equals n times the log of 2, well, we get this quantity over here.

And it's quite straightforward to show that this is, in fact, a finite number.

So this is how Bernoulli resolved the St. Petersburg Paradox.

He said that people don't compute values of gains by computing their fair value or

their expected value, but instead everyone has a utility function and

what they would compute is the expected utility of the payoff.

And from there you can determine how much an individual would be willing to pay

to play the game.

Okay.

So given this, you might think that all you need to do

is to figure out the appropriate utility function of an individual and

use it to compute the option price.

Well, maybe, but whose utility function?

The buyer's utility function, the seller's utility function, or

maybe some other utility function in the, in the marketplace?