The need for Probability-Adjusted Payoffs As discussed in class, a Call’s payoff looks great, with its unlimited profit and limited loss profile. It can look even better if you shift your viewing window.
So to get a truer representation of the profit potential of a given option, we would need to consider how likely the underlying market is to move to a given price. In the above example, how likely is the underlying market price going to rise to say $55. By taking the basic calculated payoff from the option, being any positive intrinsic value minus the premium paid, and multiplying that by the probability of that given price level being reached, gives us the probability-adjusted payoff. For example, if the underlying market price is $55, then the $50-Call above would give us a payoff of $4.30, assuming we had paid $0.70 in premium for it. We would need to multiply this $4.30 by the probability of the underlying market being at $55, at the time of the option’s expiry; the market could be at other price levels above or below $55, so we need to know what the probability is of it being at $55 to calculated the payoff at $55. Assuming that probability was 5%, we would derive a probability-adjusted payoff of $0.215 (being 0.05 x $4.30) By adding all the probability-adjusted payoff’s at each price in the range of possible prices, we get the option’s probability-adjusted payoff.
Deriving Probabilities To be clear, the academics will argue that the probability of a given option expiring ITM can (and should) be derived differently, but the quick-and-dirty method would be to use delta as a good proxy. After all, even with using a statistically accurate, and computationally expensive, model, the inputs to the model are mere estimations of the future. As discussed, even taking a historically observed value, or a time series average, would require one to assume the future will reflect the past; so one estimate is as good as another. As such, Delta is often used in the trenches as a good proxy representation of probability.
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Once we know the delta of some of the options in the curve, we would be able to estimate the probability of the underlying market being at a given price at the expiry of the option. Think about it this way; the ATM option has a delta of 50. On a simplified level, that means that the option has a 50% chance of being ITM at expiry, which is logical, considering the market could go either up or down. That means that at the time of expiry of the option, 50% of the possible range of prices that the underlying market could be at are above the strike level (let’s call this the terminal values)
As for the 25-delta option, it means 25% of the possible terminal value are above that strike level.
This also means that 10% of the terminal values are between the 25-delta Strike and the 15-delta Strike. We can then derive the probabilities for each of the prices between 25-delta and the 15-delta.
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To keep things simple, we use linear interpolation to derive these values. Assuming there are 3-steps between the 25-delta and the 15-delta; Step A, Step B and the third step brings you to 15-delta. In this case, we simplistically take 10% divided by 3 steps, and assign a probability of 3.3333% to each step. So we say the probability of the terminal value being at A is 3.3333% and we can use that to calculate the probability-adjusted payoff at price point A. Summing all the probability-adjusted payoff’s at each price point in the range of possible terminal values, we get the option’s probability-adjusted payoff.
The range of terminal values can be estimated via the u and d values of the binomial tree model, or any other range-predictive models.
Worked Example Assume we have the following positions: Long $53-Call, Premium $1.00, 25-delta Short $56-Call, Premium $0.30, 15-delta The range of possible terminal values have been determined to be $39 to $62. Spot today is $50, meaning a 50-delta at $50, using the ATM concept. As such, we know that 25% of the terminal values will be between $50 and $53 (which gives us 3 x $1intervals). So linearly assigning probabilities across this range of terminal values, we get 8.333% for each $1-interval (25% divided by 3). Examining the range between the 25-delta and the 15-delta, we will have to divide 10% across 3 intervals, giving us 3.333% for each interval. And of course we do it for the range between the 15-delta and then limit of the estimated range.
The worked numbers are as follow: Terminal Value Delta Probability Payoff Prd-Adj
$50 50
$51
$52
$53
$54
$55
$56
$57
$58
$59
$60
$61
$62
25 15 0 8.3333% 8.3333% 8.3333% 3.3333% 3.3333% 3.3333% 3.0000% 3.0000% 3.0000% 3.0000% 3.0000% 0.0000% -$0.70 -$0.70 -$0.70 $0.30 $1.30 $2.30 $2.30 $2.30 $2.30 $2.30 $2.30 $2.30 -$0.0583 -$0.0583 -$0.0583 $0.0100 $0.0433 $0.0767 $0.0690 $0.0690 $0.0690 $0.0690 $0.0690 $0.0000
Of course the above has only considered the upside outcomes. On the downside, between $39 and $50, the same process would give us the probability-adjusted outcomes. In reality, we don’t need to tire ourselves working out those numbers, since we know that
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that range of terminal values represent 50% of the possible outcomes, and the payoffs on that side will all be -$0.70, so we can simply use -$0.35 as the total probability-adjusted payoff for that side.
Summing the payoffs from the upside (seem in the table above) and the -$0.35 for the downside outcomes, we get a probability-adjusted payoff of -$0.05. This compares with the Expected Payoff of $+0.30. Which do you want to present to your client? And if you were the client, which would you want to know?
Caveat: there will be variances to the results depending on the size of the intervals you use. The above example is not technically robust, but serves to illustrate the concept, which is what is required of this module. So for those of you without a statistics background, don’t freak out. Just understand and be able to apply this concept.
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