Basing the decision on maximizing average return, median return, or
probability of profit will lead to different answers. And there are many
other statistics that could be sensibly considered. It is also quite likely that a trader's criteria will change based on the rest of her portfolio. The
decision needs to be made based on personal utility and on a case-by-case
basis.
Summary
By interpreting the dividend yield term in the generalized BSM model
to be a drift parameter we can get subjective option values that depend
on return.
These prices are not arbitrage free but can be used to derive real-world
statistics (as opposed to risk-neutral statistics).
Different evaluation criteria will suggest very different “optimal”
strikes.
146
CHAPTER 8
Directional Option Strategy
Selection
In addition to choosing a strike and expiration, the trader needs to
decide what strategy to employ. There are many option structures
that could be used to speculate directionally, but here we will
confine ourselves to those that could be considered the
fundamental building blocks of the others: the long call, the long
call spread, the short put, the short put spread, and the risk
reversal.
It is possible to construct a matrix that constructs optimal
positions using various risk measures such as the Sharpe ratio, the
generalized Sharpe ratio (GSR), or the Kelly ratio. Doing this isn't
stupid but here are several problems with the approach:
Different criteria will recommend different structures, and
none will express the investor's real utility. All the criteria are
useful guides, but none are definitive.
Often the difference between strategies will be minimal using
this method.
The various risk ratios assume that our forecasts are correct. It
is more important to understand what happens if we are
wrong. Although it is possible to calculate derivatives of the
ratios (e.g., the derivative of GSR with respect to return), it is
more instructive to again run simulations.
Long Stock
This is our baseline. We buy 100 shares of a $100 stock. We expect
a 20% return and realized volatility is 30%. If our return estimate
is correct, the (lognormal) PL distribution after 1-year is shown in
Figure 8.1 and summary statistics are shown in Table 8.1.
147
FIGURE 8.1 The PL distribution for 100 shares of a $100 stock with a 20% return; volatility is 30%.
TABLE 8.1 Summary Statistics for 100 Shares of a $100
Stock with a 20% Return (Volatility is 30%.)
Average
$2,640
Standard deviation
$3,736
Skewness
1.12
Excess kurtosis
2.73
Median
$2,214
90th percentile
$7,940
Maximum (in a 10,000-path
simulation)
$27,220
10th percentile
−
$1,660
Minimum (in a 10,000-path
−
simulation)
$6,260
Percent profitable
75%
Long Call
Consider a long ATM 1-year call on a $100 stock when rates are
zero. We expect a 20% return and both implied and realized
volatility are 30%. If our return estimate is correct, the PL
distribution from a simulation of 10,000 paths is shown in Figure
148
8.2 and summary statistics are shown in Table 8.2. The initial
value of the call is $11.92.
FIGURE 8.2 The PL distribution for a 1-year ATM call option on a $100 stock with a 20% return. Both implied and realized
volatilities are 30% and rates are zero.
TABLE 8.2 Summary Statistics of the PL Distribution for a 1-Year ATM Call Option on a $100 Stock with a 20%
Return (Both implied and realized volatilities are 30%
and rates are zero.)
Average
$1,516
Standard
deviation
$3,198
Skewness
1.66
Excess kurtosis
4.12
Median
$538
90th percentile
$5,843
Maximum
$23,06
8
10th percentile
−$1,192
Minimum
−$1,192
Percent profitable
58%
If we had bought 100 shares of stock, our average median profit
would have been $2,200. The option premium pays for the
149
leverage relative to the shares and the limited downside. A call option is similar (but not identical) to a long stock position and a
stop-loss order at the strike. However, stops will kill some trades
that would eventually have recovered. Options won't do this, and
this is the benefit of paying the premium.
Long Call Spread
We buy the 1-year ATM call and sell the 20-delta call (the 135
strike). The PL distribution from a simulation of 10,000 paths is
shown in Figure 8.3 and summary statistics are shown in Table
8.3. The initial value of the spread is $9.04.
This P/L distribution is similar to a long position with a stop and a
profit target. Although returns are far from normal, the extreme
values have been eliminated along with the skewness.
In indices, it is quite possible that the implied skew means you will
be selling the short strike at a discount to the ATM volatility. In
many other products, you will receive a premium for the short
strike. This changes the initial premium and hence the profits but
won't change the shape of the terminal distribution.
One benefit that the call spread offers over the call is related to
psychology. When holding a call, particularly an OTM call, you are
paying for the extreme upside. This means you need to continue to
hold the option. A lot of traders have trouble with this (in my
experience, amateurs can't take losses and professionals are too
inclined to take profits). Instead of fighting this tendency, it may
be better to buy a call spread instead of a call. The short strike will
be the profit target and the position won't have cost as much to
initiate.
150
FIGURE 8.3 The PL distribution for a 1-year ATM/20-delta call spread on a $100 stock with a 20% return. Both implied and
realized volatilities are 30% and rates are zero.
TABLE 8.3 Summary Statistics of the PL Distribution for a 1-Year ATM/20-Delta Call Spread on a $100 Stock with
a 20% Return (Both implied and realized volatilities are
30% and rates are zero.)
Average
$819
Standard
deviation
$1,502
Skewness
0.01
Excess kurtosis
−1.76
Median
$759
90th percentile
$2,596
Maximum
$2,596
10th percentile
−$904
Minimum
−$904
Percent profitable
58%
Short Put
151
We sell the 1-year ATM put. The PL distribution from a simulation
of 10,000 paths is shown in Figure 8.4 and summary statistics are shown in Table 8.4. The initial value of the put is $11.92.
Choosing a short put instead of a long call is really about
preferring a high probability of a smaller profit to a larger average
profit and positive skewness.
Covered Calls
A covered call consists of a long position in a stock and short
position in a call on that stock. In exchange for receiving the
option premium, the investor has her upside capped. Figure 8.5
shows the covered call payoff at expiration when a 100-strike call
option is sold for $5.
FIGURE 8.4 The PL distribution for a short 1-year ATM put option on a $100 stock with a 20% return. Both implied and
realized volatilities are 30% and rates are zero.
TABLE 8.4 Summary Statistics of the PL Distribution for a Short 1-Year ATM Put Option on a $100 Stock with a
20% Return (Both implied and realized volatilities are
30% and rates are zero.)
Average
$706
Standard
deviation
$986
Skewness
−2.22
152
Excess kurtosis
4.53
Median
$1,192
90th percentile
$1,192
Maximum
$1,192
10th percentile
−$8,43
Minimum
−
$5,172
Percent profitable
78%
FIGURE 8.5 The payoff of the covered call as a function of stock price at expiration.
Synthetically a covered call is the same as a short put. Instead of
selling a call against an established long stock position, investors
sometimes sell a put and hold enough cash to be able to purchase
the stock if they are assigned. Synthetically, this position is the
same as a covered call with the same strike. However, there are
some differences in how and why these strategies are used:
Some investors are prohibited from put selling, but they can
write covered calls.
The chosen strikes tend to be different, with both strategies
generally implemented with out-of-the-money options.
Selling out-of-the-money put options means the trader usually
benefits from selling at an implied volatility premium.
153
The psychological effects on the trader are also somewhat
different. The holder of a covered call tends to be happy with
rallies, whereas the seller of the put often feels she has missed out
in the case of a large rally. This is because we frame the situations
differently. Covered calls are framed as a situation in which we are
long and are prepared to sell, whereas short puts are seen as a
situation in which we are waiting to get long at a certain price.
Part of this is due to the different strike choices but the reasoning
is still specious. This shouldn't be a relevant consideration in
strategy selection, but in practice it is.
Covered calls have been popular with retail traders due to the
argument that “I would sell the stock if it went to the strike price,
so why not get paid to do that?” This reasoning is poor, but
covered calls are also the rare example of a popular retail strategy
that works well and makes good sense. Over time they have
delivered equity-like returns with lower risk. For example,
consider the CBOE BuyWrite Index. This consists of holding the
SPX portfolio and selling slightly out-of-the-money 1-month calls
that are held until expiration. The performance is shown in Figure
8.6 and summarized in Table 8.5, together with the S&P 500
(including dividends).
We can see that the outperformance of the covered call strategy is
robust with respect to the exact implementation by looking at the
results of BXY (which sells 2% out of the money calls) and BXMD
(which sells 30-delta calls). These results are summarized in Table
8.6.
154
FIGURE 8.6 The performance of the CBOE BuyWrite Index compared to that of the S&P 500 index from June 1988 to
September 2019.
TABLE 8.5 Summary Statistics for BXM and the S&P 500
Statistic
BXM
S&P
500
Annual return
8.5%
7.7%
Volatility
12.6% 17.3%
Max
drawdown
40.1% 56.8%
Skew
−0.67
−0.29
TABLE 8.6 Summary Statistics for BXY, BXMD, and the S&P 500 from June 1988 to July 2019
Statistic
BXY
BXMD
S&P
500
Annual return
8.6%
10.3% 7.7%
Volatility
12.6% 14.7% 17.3%
Max
drawdown
40.1%
46.9% 56.8%
Skew
−0.67
−0.46
−0.29
155
Components of Covered Call Profits
The discussion of strike choice up until now has only focused on
the risk characteristics of the options. We also need to consider
the factors that drive option returns when making the choice. As
an example, we consider a covered call. This is the simplest
possible option position, but this analysis is quite general.
The reason that covered calls can provide equity-like returns with
lower volatility is that they are exposed to two profitable factors:
the equity market risk factor and the volatility premium. Selling a
call against an existing position reduces the portfolio's exposure to
the stock while adding a short volatility exposure. The lower
volatility of a covered call position is due to the diversification that
two factor exposures provide.
Consider a stock with a current price of $100. We assume that the
stock increases by 10% a year and has a volatility of 15%. We also
assume that dividends and rates are zero. We sell a 1-year call
option with an implied volatility of 20% against this position.
This at-the-money covered call has a delta of 0.47 when evaluated
at the realized volatility, so it will earn 4.7% from its exposure to
the stock's appreciation (to a first approximation). Also, this
option has a premium of $7.97 ($2.00 more than it would have
been worth at the true realized volatility). So, about 25% of the
option premium that the seller collected is harvesting the volatility
premium. That is, we expect to gain 2.0% a year from harvesting
the short volatility premium. Here the total expected return of the
covered call is 6.7%. Although this example has a lower expected
return than the stock, it also has lower volatility.
Obviously, this theoretical decomposition varies with respect to
the volatility premium, and the ex-post return is affected by both
realized volatility and return. Also, the strike choice and
expiration of the short call determines how much of the profit
comes from directional exposure and how much co
mes from
volatility harvesting.
First, we look at the decomposition as a function of a strike
assuming all strikes have the same variance premium. This is
shown in Figure 8.7.
We can see that although the equity premium is an increasing
function of strike, the volatility premium is peaked at strikes just
156
above the current stock price. This is where the option's sensitivity
to volatility (vega) is greatest, so it is also where exposure to the
volatility premium is maximized.
Note that if there is no volatility premium, the covered call earns
just the stock return multiplied by its delta (the exposure to its
underlying equity). This is the most important concept. Selling
options only makes sense if a volatility premium exists. Further, if
an investor is more confident in the existence of the volatility
premium, he should sell options just above the ATM. Conversely,
if he is more confident in the equity return, he should sell options
that are further out of the money. The fact that different investors
have different forecasting abilities means that they will also
choose different option structures.
FIGURE 8.7 The total profit of the covered call and how much comes from equity return and volatility premium.
Covered Calls and Fundamentals
It is well-known that value stocks, momentum stocks, low-beta
stocks, and small-cap stocks tend to be the best performers. So,
investors should preferentially own these. Sadly, with the
exception of momentum stocks, these classes of equities tend to
have the lowest variance premia. This means that the investor has
157
to choose between better delta performance or better variance
performance and make the strike choice accordingly. Again, this
choice will depend on the ability of the individual trader.
It is also important to note that the variance premium and
subsequent stock returns (the two sources of edge) are not
completely independent. High-variance premium predicts high
future stock returns. This effect has been extensively studied (see,
for example, Bollerslev and Zhou, 2007; Bollerslev and Todorov,
2011; Kelly and Jiang, 2014; Bollerslev et al., 2014) and exists for both single stocks and at the index level. This effect is strong
enough that it can be used as a timing signal: when the variance
Positional Option Trading (Wiley Trading) Page 17