Fair Value Formula for Option Trading

Fair Value Formula

We can find the same answer mathematically by multiplying the probability of winning by the reward and adding that to the probability of losing multiplied by the amount wagered. In this example:

This top line of this calculation is saying there is a 50% chance of winning $1.00, which means that has a value to you of 0.50 * +$1.00 = 50 cents. But the second line shows you also have a 50% chance of losing $1.00, which has a value of 0.50 * -$1.00, which equals -50 cents. The positive and negative 50 cents cancel out and you’re expected to walk away with nothing after many flips. We can even run a computer simulation to see if we are right. Figure 3 shows a computer model with the total profit or loss on the vertical axis and the number of tosses on the horizontal axis:

The $0 mark on the vertical axis represents the break even point since that is the point where zero profits or losses are incurred. You can see that after 500 tosses, you are about back at break even. However, prior to that, you can cer­tainly end up winning or losing due to chance. But in the long run, you would just expect to just break even. The $0 horizontal line acts like a magnet for a fairly valued bet. The profit and loss line does not get too far above or below break even without being pulled back toward zero. Although tire profit or loss line can stray from zero, it cannot move away from it indefinitely. The profit and loss line just tends to oscillate sideways around zero if the bet is fairly priced.

Lets use the fair value formula to see what it says about paying $1.50 for the $1 reward:

The formula shows that we are expected to lose 25 cents per flip, which is exactly what we reasoned earlier. Paying $1.50 for this bet is therefore too high of a price. Figure 4 (following page) shows that a computer simulation agrees with the formula:

After 500 flips, we are down roughly $90. Mathematically, we would expect to have lost 25 cents per flip multiplied by 500 flips for a total of $125. Of course, there will be fluctuations around that $125 figure and the chart shows that we are not too far off.

Notice that even though we are paying $ 1.50 for this bet (above fair value) that it is still possible for us to end up on the winning side in the short run. Figure 4 shows that we ended up on the winning side even after 100 flips. But after hundreds more, the profit and loss line does not tend to get pulled toward zero. Instead, it moves into a definite downward path and never returns.

Let’s see what the formula has to say about wagering 50 cents for the $1 reward:

As we found before, wagering only 50 cents to win $1.00 at the flip of a coin is a good deal as you would expect to win about 25 cents per flip. Figure 5 shows a computer simulation of this arrangement:

Notice that the profit and loss line is not pulled toward zero in the long run. Instead, it heads into unlimited gains. Mathematically, we would have expected to be up 25 cents * 500 flips = $125 and Figure 5 shows that we’re up about $100 so, once again, the computer simulation is not too far from what we’d expect.

Figure 5 shows we actually lost money after about 50 flips even though tire odds were on our side. So even though we could certainly lose in the short run, we will end up on the winning side after hundreds of flips, which is con­firmed in Figure 5.

Only when the price of the bet is $1.00 can we say that it is “fair” for both parties. As a reminder, just because tire bet is fair does not mean that someone cannot end up on the winning side (and the other on the losing side). The fair value just means that, over the long run, neither side is expected to end up on the winning or losing side.