Fair Value Depends on Perspective

Fair Value Depends on Perspective

In the coin toss gamble, we calculated drat $1.00 was the fair value of the bet. However, that result is due to our assumption that the chance of winning (and losing) is 50%. Obviously, if we used different probabilities, we would get dif­ferent results. This means that the fair value of any bet depends on our per­spective; it depends on our views of the future probability of winning.

For example, let’s assume that somebody offered to pay $1.50 for this coin toss game. There are two ways we could look at it. First, we could assume there is a 50% chance of winning and losing and say that it is too high of a price since that results in an expected loss of 25 cents per flip:

However, we could also look at this bet another way. We could assume that the $1.50 price is a fair price since nobody should intentionally pay more than what they think is fair. If we could assume that it is a fairly valued bet, we would need to adjust the probabilities so that they balance to zero. It works out that if there is a 60% chance of winning the dollar (and therefore a 40% chance of losing your wager) then $1.50 is a fair price to pay for the game. If somebody were willing to pay $1.50 to play this game, we would say that the gambler is implying that his chances of winning are 60% (and the chances of losing are 40%) since those probabilities result in a fairly valued bet:

Although the gambler may not specifically state that he believes the probability of winning is 60%, the mere fact that he is willing to pay $1.50 for such a bet implies a 60% probability of winning if we assume the bet is fair.

This shows that there are two ways of looking at any bet. First, if we believe there is only a 50% chance of winning then paying $1.50 is too high of a price. Second, we can assume the $1.50 is a fair price and adjust the prob­abilities to make the bet fair. So is $1.50 a fair price to pay for this game? It all depends on how we view the future probability of winning.

Now, as the gambler, it is up to you to decide which viewpoint is more realistic. Should you assume the future chances of winning are 50% and only be willing to pay $1.00? Or is a 60% chance of winning more realistic? Notice that if you assume that 50% is the correct probability that you will be outbid by another gambler if he feels 60% is the more realistic probability. You would only be willing to bid up to $1 for the bet while he would be willing to pay up to $1.50. It is critical that we are confident in our assessments. If 60% sounds like too high of a probability, we’re probably better off not taking the bet. It’s better to miss out on some reward rather than to lose some or all of our money.

You can see from this example that your probability assessment of 50% or 60% (or something else) to value this coin flip game is an important assump­tion if you are to make money. It is even more important when valuing options. However, as we said before, very few option traders ever check to see how the price of the option compares to their assessment of value. In order to do that, option traders need to use the Black-Scholes Model before stepping into any trade.