flows are discounted at the risk-free rate. Traders use this model now and
interpret the dividend yield and interest rate to be (imperfectly known)
objective variables. (Note that in practice different traders will have
different dividend estimates and marginal rates. This could, and in rare
cases does, allow arbitrage.)
We turn this into a model that incorporates drift by reinterpreting the
dividend yield as a subjective drift estimate. We use our drift estimate to
give a subjective estimate of the underlying's forward price. This model will
also allow arbitrage, but it will be consistent with our opinions of the real
world.
The prices of calls in this model are
(7.5)
where
(7.6)
(7.7)
As with any other pricing variable or parameter, it helps to have a greek to
measure the impact of an incorrect estimate. The partial derivative of the
subjective option price with respect to the drift is given by
138
(7.8)
Now we can derive subjective values for options and see which are most
mispriced. In this example, we consider 1-year options on a $100 stock,
with a volatility of 30%, a drift of 10%, and zero interest rates. The risk-
neutral BSM prices and the subjective prices are shown in Table 7.1.
As the drift is primarily going to be a delta effect, it should be no surprise
that the greatest absolute difference in values is in the lower strikes. If an
investor wants to buy a fixed number of options, these might be the best
choice, but in this case, it is probably better to just buy the underlying
because the discrepancy would be greatest there. The largest percentage
edge (“bang for the buck”) is in the highest strikes, so these appear best for
a trader who wants to invest a certain dollar amount.
TABLE 7.1 A Comparison of Risk-Neutral and Subjective Option Prices
Call
BSM Subjectiv
Difference in $
Difference as % of
Strike Price
e Price
from BSM Value
BSM Value
80
22.53
32.53
8.99
0.38
85
20.09
28.55
8.46
0.42
90
17.01
24.88
7.86
0.46
95
14.29
21.52
7.23
0.51
100
11.92
18.49
6.57
0.55
105
9.88
15.79
5.91
0.60
110
8.14
13.41
5.27
0.65
115
6.67
11.32
4.65
0.70
120
5.44
9.51
4.07
0.75
However, this analysis doesn't consider the different risk characteristics of
options with different strikes.
Now that we have a theoretical model, we can talk about the distribution of
subjective option returns (assuming our drift and volatility estimates are
correct).
Distribution of Option Returns: Summary Statistics
(7.9)
(7.10)
139
is the true probability of expiring in the money, as opposed to the
incorrect but frequently stated number,
, which is the risk-neutral
probability. And the drift is a large determinant of whether an option
expires in the money. As an example of the size of the difference, consider a
1-year 110 strike call on a $100 stock, with volatility of 30%, a drift of 10%,
and an interest rate of zero. The risk-neutral probability of exercise is 32%,
whereas the subjective probability is 45%. At very short timescales the
volatility will overwhelm the drift, but in general it is bad to assume that
the risk-neutral probabilities are indicative of anything in the real world.
Figures 7.1 and 7.2 show how N( d4) depends on the drift and how it varies with strike.
FIGURE 7.1 Probability of the 3-month 150 strike call expiring in the money. Stock is $100, volatility is 30%, and rates are zero.
FIGURE 7.2 Probability of the 3-month calls expiring in the money when the return is 20%. Stock is $100, volatility is 30%, and rates are zero.
140
Chance of Profit = N(d5)
(7.11)
Also, by assuming the underlying price is lognormally distributed, we can
easily calculate the median intrinsic value and hence profit.
(7.12)
(7.13)
Further,
(7.14)
(7.15)
An example is shown in Figure 7.3.
It is possible to calculate the moments of the options returns (Ben-Meir
and Schiff, 2012; Boyer and Vorkink, 2014; Sinclair and Brooks, 2017), but
the equations are complicated and give no real insight. The most important
fact is that options have significantly positive skew. It is also important to
note that this extreme skewness occurs even when the underlying has
normally distributed returns. This skewness is intrinsic to options and is
not inherited from skewness of the underlying. If the underlying has non-
normal returns, the effects on the options will be magnified. In this case,
there won't be analytic expressions for the option moments and
simulations will be necessary to understand the option return moments.
141
FIGURE 7.3 Ninetieth percentile of the profit of the 3-month 100-strike call. Stock is $100, volatility is 30%, and rates are zero (the risk-neutral call value is $5.98).
Strike Choice
All of the individual risk measurements given above need to be considered.
None is sufficient on its own to determine that a particular option is “best.”
And this is probably a good thing, because “best” or “optimal” is only
optimal with respect to a given criterion, and trading decisions need to be
based on more than just one criterion.
In the specific case of strike selection (or investment selection in general) it will be impossible to give a single optimal solution. Some problems can't
sensibly be solved this way. Think about the question, “What is the best car
in the world?” Here, there are many plausible definitions of best. Does best mean fastest? Most luxurious? Safest? Greenest? Cheapest to buy?
Cheapest to run? Most reliable? Within each category it is possible to make
valid comparisons. A Ferrari is better than a Lamborghini. A Toyota is
better than a Geo. But the argument over whether a Ferrari is better than a
Toyota is unresolvable, because different car users have different goals and
preferences.
In some other situations, it is possible to define a clear and unambiguous definition of best. Consider baseball. The goal of a baseball team is to win games. The best player is the one who most helps his team to do this. The
individual skills of hitting, throwing, catching, and running are now seen as
components of the ability to create wins, rather than unrelated goals in
themselves. So here, a statistic that aggregates the component skills by
putting them on a consis
tent scale and converting them into a measure of
wins created is very useful. You can then meaningfully compare a power-
hitting catcher to a fast, agile shortstop. But this introduces the danger of
overreliance on the power of this single statistic. This number will still have
142
methodological issues, and there will always be sampling problems with the individual component measurements. Even the best composite statistic
should be a starting point rather than a definitive answer.
Trading is somewhere in between these situations. The goal of making
money is absolute but risk is personal, both in terms of how it relates to a
given trader's edges and abilities but also in terms of risk tolerances and
aversions. It should be obvious that different people have different levels of
risk aversion. This could be personal or it could be because of an external
mandate. Anyone trading someone else's money will have to conform to
the risk preferences of the capital provider. But it is also important to
remember that risk depends on the skills of a specific trader. Risk is all of
the things outside our control. So, traders with different sources of edge
will have different remaining risk factors. If one trader has an edge in
volatility prediction and another doesn't, volatility is an edge for the first
and a risk to the second. This is true in most of life. For a heart surgeon,
doing a bypass is a low-risk operation. For a random person, it would be
murder.
So, a composite statistic for comparing risk and reward will be useful, but
we should also not expect too much of it.
The most well-known of these statistics is the Sharpe ratio, the ratio of
(excess) return to the volatility of the return. It is also well-known that the
Sharpe ratio is not perfect. It has a large sampling error, generally
comparable in size to the estimate. It doesn't distinguish between downside
and upside volatility. It doesn't take higher-order moments into account at
all. These are all problems, but the specific option-related issue is that we
will be dealing with heavily skewed returns. (Skew can be a feature, not a
bug. Positive skew is a good reason to include long options in a portfolio.)
The first work to address this failing was done by Hodges (1998), who
showed the nature of the problem with a very simple example. We have two
probability distributions, A and B, of excess returns.
Distribution A
Return
−25 −15 −5 5% 15 25 35
%
%
%
%
%
%
Probabilit
0.2 0.0
y
0.01 0.04 0.25 0.4
0
5
4 0.01
Distribution B
Return
−25 −15 −5 5% 15 25 45
%
%
%
%
%
%
Probabilit
0.2 0.0
y
0.01 0.04 0.25 0.4
0
5
4 0.01
Summary Statistics
143
Distribution
A
B
Mean return
5.00% 5.10%
Standard
10.00 10.34
deviation
%
%
Sharpe ratio
0.50 0.493
Clearly distribution B is better than distribution A. The only difference is
that the outcome of 35% has been increased to 45%. But this (good) change
has increased the standard deviation more than the return, so the Sharpe
ratio of distribution B is lower than that of distribution A.
Hodges derived a generalized Sharpe ratio (GSR) for an investor with
exponential utility, but it was necessary to know the complete distribution
of payoffs to make the calculation.
Pézier (2004) applied similar reasoning to create a GSR that only requires
the moments of the distribution (see Maillard, 2018, for a full derivation
and discussion). His GSR is
(7.16)
TABLE 7.2 Projected Performance Numbers for Long Positions in Different Strike 3-Month Calls on a $100 Stock with a Drift of
10%, Volatility of 30%, and Zero Interest Rates
Strik Averag
Average
Median
90th
Probabilit GS
e
e
Percentag Percentag Percentile y of Profit
R
Dollar
e Return
e Profit
Percentag
Profit
e Profit
80
$2.41
11.8%
10.4%
116.9%
52%
0.34
85
$2.26
14.1%
9.7%
145.7%
50%
0.33
90
$2.04
16.9%
4.2%
185.0%
48%
0.32
95
$1.75
20.2%
−13.1%
237.6%
44%
0.31
100
$1.43
23.8%
−57.6%
305.8%
37%
0.27
105
$1.10
27.8%
−100%
387.8%
31%
0.25
110
$0.80
32.0%
−100%
470.4%
24%
0.23
115
$0.56
36.6%
−100%
509.0%
17%
0.20
120
$0.37
41.3%
−100%
378.3%
12%
0.16
where SR is the standard Sharpe ratio, λ3 is the skewness of returns, and λ4
is the kurtosis. For normal returns, the GSR reduces to the Sharpe ratio.
Positive skewness increases the GSR. Negative skewness lowers the GSR.
Any kurtosis lowers the GSR.
144
Table 7.2 gives the various statistics for different 3-month call options on a
$100 stock with a return of 10%. Both realized and implied volatilities are
30% and rates are zero. Each trader needs to choose the strike that most
closely matches what they are looking for.
This analysis assumes we have paid the correct volatility level for the
options. If we pay too much, our results look much worse. Even when
trading purely directionally, implied volatility is very important. This is
shown in Table 7.3 where we assume that the implied volatility was 30%
but realized volatility was only 22% (this roughly corresponds to the typical
variance premium). This effect needs to be considered if the implied
volatility of the strike under consideration is very different from the ATM
volatility.
(Interestingly, the GSR of the 120 strike is better than that of the 105, 110,
and 115 strikes. This is because of the extreme skew of the results.)
TABLE 7.3 Projected Performance Numbers for Long Positions in Different Strike 3-Month Calls on a $100 Stock with a Drift of<
br />
10%, Implied Volatility of 30%, Realized Volatility of 22%, and
Zero Interest Rates
Strik Averag
Average
Median
90th
Probabilit GSR
e
e
Percentag Percentag Percentile y of Profit
Dollar
e Return
e Profit
Percentag
Profit
e Profit
80
$2.16
10.6%
10.4%
86.5%
54%
0.37
85
$1.74
10.8%
9.7%
106.8%
53%
0.33
90
$1.12
9.3%
4.2%
133.4%
50%
0.24
95
$0.43 4.9%
−13.1%
166.3%
43%
0.10
100
−$0.17
−2.6%
−57.6%
201.9%
36%
−0.0
4
105
−$0.53
−13.3%
−100%
230.6%
26%
−0.1
9
110
−$0.65
−26.0%
−100%
222.1%
18%
−0.21
115
−$0.60
−39.8%
−100%
100.7%
11%
−0.21
120
−$0.47
−53.1%
−100%
−100%
6%
−0.0
6
Fundamental Considerations
So far, we have assumed our forecast was only of the mean and variance.
Sometimes we may have a more complex view. For example, this is
145
common in the Eurodollar market. Traders tend to forecast in discrete
increments; for example, a 25 bp cut has a 40% chance of occurrence,
instead of continuous outcomes, that is, a mean return of 5%.
In these cases, each strike should be evaluated with a different subjective
drift parameter. Although, given the trader's bias toward a certain
probability distribution, the analysis will probably confirm only preexisting
opinions (opinions in, opinions out).
Traders in most other products should be careful to ask themselves if their
forecasts of the distribution lead to expected value. It is hard to predict
volatility. It is harder to predict return. Predicting the full distribution is
probably a manifestation of overconfidence.
Conclusion
There is no simple answer to the question, “What strike should I buy?”
Positional Option Trading (Wiley Trading) Page 16
