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Fair Value Depends on Perspective When Trading Options

Filed Under Beginner Options Trading 

We can even use computer simulation to see if we’re right. Figure 6-2 shows a computer model with the number of tosses on the horizontal axis and our total profit or loss on the vertical axis:
 
Figure 6-2: Computer Simulation of Fair Value (Paying $1 to Win $1)

Figure 6-2

You can see that after 500 tosses, we’re about back at breakeven. However, prior to that, we can certainly end up winning or losing due to chance. But in the long run, we’d expect to just break even. The “zero” horizontal mark in Figure 6-2 acts like a magnet for a fairly valued bet in that the profit and loss line doesn’t get too far from it. The profit or loss line can stray from zero but it cannot just move away from it indefinitely. The profit and loss line just tends to oscillate around zero.
 
Let’s use this same formula to see what it says about paying $1.50 for the $1 reward:
 
   (0.50) * +$1.00
+ (0.50) *-$1.50
 Expected value = -25 cents
 
The formula shows that we are expected to lose 25 cents per flip. Paying $1.50 for this bet is therefore too high a price, since we would expect to end up with certain losses over time. Figure 6-3 shows that a computer simulation agrees with the formula:
 
Figure 6-3: Computer Simulation Above Fair Value (Paying $1.50 to Win $1)

Figure 6-3

In fact, mathematically, after 500 tosses we would expect to end up at 500 tosses * -.25 cents = -$100 and that’s roughly where the computer simulation ended. Curiously enough, notice that even though we’re paying above fair value it’s still possible for us to end up on the winning side in the short run. Figure 6-3 shows that we ended up on the winning side even after 100 flips. But that is just due to some short-term good luck on our side. We had significant winnings to cover our losses after 100 flips. But if we stay in the game long enough, the profit and loss line does not tend to get pulled toward zero. Instead, it moves into a definite downward path and never returns. Once again, this shows that $1.50 is too high of a price to bet on this coin flipping game.
 
Let’s see what the formula has to say about wagering 50 cents for the $1 reward:
 
   (0.50) * +$1.00
+ (0.50) * -$0.50
            Expected value = +25 cents
 
Wagering only 50 cents to win $1.00 at the flip of a coin is a good deal for us, as we now expect to win about 25 cents per flip. Figure 6-4 shows a computer simulation of this arrangement:
 
Figure 6-4: Computer Simulation Below Fair Value (Paying 50 cents to Win $1)

Figure 6-4

Again, we would expect to have 500 tosses * +25 cents = $100 profit after 500 flips and that’s about where this computer simulation ends. Notice too, however, the chart shows we actually lost money after 75 flips even though the odds were on our side. That’s because the profit and loss line dips below zero up until the 75th flip mark. At that point, we head into uninterrupted profits. This profit and loss line is not pulled toward zero in the long run. Although we could certainly lose in the short run, we will end up on the winning side after numerous flips, which is confirmed in Figure 6-4.
 
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 the bet is fair does not mean you cannot end up on the winning or losing side. The fair price for both just means that, over the long run, neither side is expected to end up on the winning or losing side.
 
 
Fair Value Depends on Perspective
In the coin toss example, we calculated that $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 the fair value of any bet depends on our perspective; it depends on our views of the probability of winning.
 
For example, let’s assume that somebody offers to wager $1.50 for this bet. There are two ways we could look at it. First, we could assume there is a 50% chance of winning and losing and assume that is too high of a price since it results in an expected loss of 25 cents per flip:
 
   (0.50) * +$1.00
+ (0.50) * -$1.50
Expected value = -25 cents
 
However, we could also look at this bet another way. We could assume that it’s priced fairly since nobody should intentionally pay more than what they think is fair. If someone offers to pay $1.50, we could say that the gambler must think it is a fair price to pay. In order for that to be true, the gambler would have to think his chances of winning are 60% since that results in a fairly valued bet:
 
   (0.60) * +$1.00
+ (0.40) *-$1.50
            Expected value = 0
 
If a gambler were willing to pay $1.50 for this bet, we would say he is implying that his chances of winning are 60%. In other words, just by the fact he is willing to pay $1.50 for such a bet we can back into it mathematically and assume he believes his chances of winning are 60%; otherwise he would not bid so high.
 
This shows 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 a price. Second, we can assume the $1.50 is a fair price and adjust the probabilities to make the expected value equal to zero. We can back into this figure algebraically and, in this case, we’d say the gambler willing to pay $1.50 for this bet is implying that there is a 60% chance of winning the $1.00 prize and a 40% chance of losing the $1.50 wager.
 
Now, as gamblers, it’s up to us to decide which viewpoint is more realistic. Should we assume the chances of winning are 50% and be willing to pay only $1.00? Or is 60% a better assessment? Notice that if we assume 50% is the better guess we will be outbid by another gambler if he feels 60% is the more realistic probability. We 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 forgoing the bet and letting someone else make it. It’s better to miss out on some reward rather than lose our money.
 
Whether we should use 50%, 60% (or something else) to value this coin flip is an important question. It’s even more important when valuing options. However, few option traders ever check to see how the price of an option compares to their assessment of value. Failure to do so is the leading reason that option traders lose with options. In order to make that assessment, option traders need to use the Black-Scholes Model.
 
The Black-Scholes Option Pricing Model
We briefly mentioned the Black-Scholes Model in Chapter Five. There are many mathematical pricing models that can tell us what the price of an option “should be.” Naturally, there will be minor variations in the answers depending on the assumptions in the model. The most famous is the Black-Scholes Option Pricing Model named after Fischer Black and Myron Scholes. Its development was no small feat, as the model relies on complex mathematics and arbitrage pricing relationships to determine what the price of an option should be and is considered to be one of the biggest breakthroughs in the modern financial era. In fact, the 1997 Nobel Prize in Economics was awarded to Myron Scholes for its development (unfortunately, Fischer Black died in 1995 and the Nobel prize is not awarded posthumously).
 
According to the Black-Scholes Model, there are six factors needed to determine the price of a call and put option:
 
  • Stock Price
  • Exercise Price
  • Risk-Free Interest Rate
  • Time to Expiration
  • Dividends
  • Volatility
 
Notice the last factor, volatility. Of these six inputs, volatility is the most important for the fact that it’s the only true unknown factor. For example, assume the risk-free interest rate is 5% and hundreds of traders are trying to value a 30-day, $100 call option on a stock trading for $95. We’ll also assume the stock pays no dividends over the life of the option. Notice all of the factors are automatically determined except volatility:
 
  • Stock Price = $95
  • Exercise Price = $100
  • Risk-Free Interest Rate = 5%
  • Time to Expiration = 30 days
  • Dividends = 0
  • Volatility = ?
 
 

To be continued…..

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