we can also see clearly that when things go wrong, selling teeny
options is far more painful than selling those with more premium.
In summary, when selling strangles or condors we want to take
advantage of the skew premium, but if we do this by selling the
option with the highest implied volatility, we expose ourselves to
horrendous risk. Once we have found our short put strike, we
choose the other call strike to give the delta or vega profile we
want.
TABLE 6.21 The Summary Statistics from Selling $1000
Vega of 260/335 SPY Strangles
Statistic
Maximum Implied Volatility
Strangle
Average
$3,980
Standard
deviation
$32,430
Skewness
−3.4
Excess kurtosis
15.1
Median
$20,092
10th percentile
−$37,100
Minimum
−$306,928
Percent profitable
76%
Choosing a Hedging Strike
129
The only reason we choose a short-volatility strategy is that we think implied volatility is too high. Consistent with this, any
option we buy will be one that we think is overpriced. This is true
even if there is no implied volatility skew. In the presence of a
skew we will be overpaying by even more. The long-hedging
options have negative expected value. They are a cost of doing
business. We want to choose the ones that give us our desired
amount of protection for the smallest amount of money.
As an example, consider the case in which we sell a 260 put for
6.28 and we don't want a possible loss of more than $5,000. Our
possible hedging options are shown in Table 6.22. The process is simple. We just find the hedging strike that gives the loss level we
are comfortable with for the lowest cost.
Maximum Loss = Profit of Hedge + Loss of Short Put − Hedge
Premium + Short Put Premium
So here, we would buy the 205 put for a hedge cost of $145.
Ignoring option premia, the worst P/L will occur right at the long
strike. Here we will have lost money on our short and have got
nothing from the long. Below the strike, the long put and short put
will cancel completely. Because our loss limit is $5,000 this would
put the lower put strike at 210. However, total premia received
will be $628 hedge premium, so the strike will be lowered by this
amount (divided by 100).
TABLE 6.22 Prices and Strikes of Possible Hedging Options for Our Short 260 Put Position
Strik
Market
Max Loss of Hedged
e
Price
Portfolio
180
0.64
$7,436
185
0.74
$6,946
190
0.88
$6,460
195
1.05
$5,977
200
1.23
$5,495
205
1.45
$5,017
210
1.68
$4,540
215
1.95
$4,067
220
2.25
$3,597
130
It can be tempting to buy lower premia, shorter-dated options as
hedges. This is almost always a bad idea. Generally, the
consecutively purchased short-dated options will have a higher
total premium than the single longer-dated option (a consequence
of total variance scaling with the square root of time). If we also
consider the extra transaction costs, the single option becomes
even more attractive.
We can illustrate this general rule and also show how exceptions
to it can occur by using the Brenner and Subrahmanyam (1988)
approximation. They show that for ATM options the BSM
equation is approximated by
(6.2)
Take this to be our hedging benchmark and compare it to the
alternative of buying two options consecutively. In that case, the
hedging premium would be
(6.3)
The two-option hedge is more expensive if the volatility term
structure is flat. However, if the volatility term structure is steep
enough, it could be worth rolling the shorter term hedges.
Specifically if the volatilities for each subperiod, σ 1 and σ 2, and the total volatility, σ, are related by
(6.4)
then it is worthwhile rolling shorter-dated hedges.
As an example, consider the situation in which we can either buy a
2-year option or a 1-year option and then another 1-year option
when the first one expires. The stock is $100, the strike is 100,
volatility is 30%, and rates are zero. The 2-year put is worth 16.8.
Each single 1-year put is worth 11.9. So here the 2-year option is
the cheaper alternative.
131
But consider the case where the 2-year option has an implied
volatility of 40% and the 1-year option has an implied volatility of
20%. Also assume that this term structure is constant in time, so
when the first 1-year option expires we can buy the next one at
20% as well. Now each 1-year option is worth 8.0 and the 2-year
option is worth 22.3. Here, we are better off buying a 1-year option
and rolling it later.
Expiration Choice
As I've emphasized throughout this chapter, there is more to
choosing an option to trade than simply finding the one that has
the highest expected return. Many other metrics are also relevant.
Median return, drawdown exposure, and percent of trades that are
profitable are also statistics to consider. However, it also seems
reasonable that our search can start with expected value and
expand from there. The purpose of trading is to make money. Risk
management should aim to protect expected value rather than
minimize risk. The safest position is no position. That will also
make no money.
The profitability of option trading is driven by the variance
premium. Many other effects are important but the variance
premium will always be dominant. It is to options what evolution
is to biology or what gravity is to physics. So when choosing an
expiration our prime consideration is which one has the most
variance premium. If we are selling, we want the expiration with
the highest premium. If we are buying, we want the expiration
with the lowest premium.
Israelov and Tummala (2017) studied this problem and wrote a
paper whose title is the perfect statement of our issue: “Which
Index Options Should You Sell?” By looking at S&P 500 option
performances from 1996 to 2015, they showed that short-dated
options had the highest variance premia. Their explanation for
this was this:
Option buyers seek to purchase insurance for their portfolio
and are typically concerned about monthly or quarterly
returns.… It is intuitive that the options which most directly
match these preferences are the most attractively
compensated for option sellers. (p. 14)
132
This is quite possibly true, but I think the more important reason is compensation for risk. Short-dated option risk is dominated by
gamma, and long-dated option risk is due to vega. The old story is
“vega wounds but gamma kills.” The sellers of short-dated options
are taking the most risk and they should be most compensated. A
good rule when looking for a variance premium is to look for
situations with the most risk. The variance premium is (a
mispriced) compensation for risk. The higher the risk, the higher
the mispricing.
This hypothesis is consistent with the results of Tosi and Ziegler
(2017). Using S&P 500 option data from 1996 to 2015 they showed
that the returns from shorting out-of-the-money put options were
concentrated in the few days preceding their expiration. Back-
month options generated almost no returns.
Their proffered reason for this was:
The concentration of the option premium at the end of the
cycle reflects changes in options' risk characteristics.
Specifically, options' convexity risk increases sharply close to
maturity, making them more sensitive to jumps in the
underlying price. By contrast, volatility risk plays a smaller
role close to maturity. (abstract)
And conclude:
Our results imply that speculators wishing to harvest the put
option premium should short front-month options only
during the last days of the cycle, while investors wishing to
protect against downside risk should use back-month options
to reduce hedging costs. (abstract)
Other studies that reach the same conclusion are by Andries et al.
(2015), Dew-Becker et al. (2014), and van Binsbergen and Koijen
(2015).
Conclusion
There is no “best” strategy. The choice of what to select is a matter
of personal risk preferences. Strangles win more often than
straddles but have less upside and more downside. Butterflies and
condors are more expensive than straddles and strangles in terms
of transaction costs. They will also realize their maximum possible
loss a significant amount of the time.
133
When choosing a short strike, the trader needs to balance
receiving the most edge by selling the options with the highest
implied volatility and the amount of risk that this produces.
Similarly, shorter-dated options will have more variance premium
than longer-dated ones but they also have more potential for
catastrophe.
Summary
Selling OTM option structures (strangles or condors) will give
higher median returns and a higher win percentage but this
can make it more difficult to distinguish between good trades
with expected value and good luck.
The highest volatility premium is in short-dated options. Long-
dated options have very little volatility premium.
The highest volatility premium is in far-out-of-the-money
puts. Out-of-the-money calls have very little volatility
premium.
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CHAPTER 7
Directional Option Trading
The genius of the BSM model is the idea that the direction of the
underlying doesn't matter when pricing an option. But although this
methodology leads to an arbitrage-free replication value, it is still possible
to trade options to make bets on the underlying direction. If one believes
the story of Thales and the olive presses, this was the original point of
options. Even now most traders use options directionally. Indeed, many
retail traders can only buy options, and directional trading is essentially
their only available tactic.
In this chapter, I will discuss directional option speculation, starting with
the theory of pricing with a directional view and then discussing the choice
of strikes, structures, and expirations. All examples will be given in terms
of long calls, assuming a bullish bias, but the ideas are trivially
generalizable to both puts and short option positions. I'm also going to
assume each option is on one share.
Subjective Option Pricing
Options offer many advantages over trading the underlying. The main
advantage is the ability to speculate on a more nuanced view than just “up
or down.” Also, leverage and the possibility of highly skewed payoffs can be
useful. However, these all add considerable complexity.
Here I'm going to (possibly optimistically) assume that the trader has a
valid prediction method for the underlying and show how she should
monetize this view. It is never easy to predict the direction of the
underlying. But when trading options, it is easy to be right in your
prediction of the underlying and still lose money. It is never good to solve
the hardest part of a problem and still fail.
The simplest directional option trading strategy is to buy a call if you think
the underlying will expire above the strike by more than the option
premium. This can be kindly characterized as “model-free directional
trading” and more realistically as “guessing.”
Consider this set of call prices on a $100 stock:
Strik Pric
e
e
95
8.7
100
5.6
105
3.2
135
Strik Pric
e
e
110
1.5
If we know the stock will expire at $120, our strike choice is trivial.
Investing $100 in each option would give the following profits:
Strik Profit
e
95
$187
100
$257
105
$369
110
$567
But the problems with trading are never about optimizing results when
predictions are correct. The real issue is how to control risk when we are
wrong. If the stock only goes to $106, our profits will be completely
different:
Strik Profit
e
95
$26
100
$7
105
−$68
110
−$100
These extreme differences illustrate the need for a better plan.
A Theory of Subjective Option Pricing
The dynamic hedging strategy used in the BSM model removes the need to
use a drift parameter. But that isn't to say that we can't include drift in our personal pricing. We won't be in the risk-neutral paradigm anymore and
our theoretical values will disagree with market prices, but that is to be
expected. If we agreed with market prices, we wouldn't be speculating.
BSM showed that the rate of return of the underlying is not relevant for
pricing options. But the underlying return obviously does affect the return of the unhedged option.
If we have a valuation model that explicitly includes drift, we can use it to
compare the theoretical values to market prices and find the most
attractive opportunities.
Luckily, several pre-BSM pricing models did include the return on the
stock. Boness (1962, 1964) found an option pricing model that is
functionally the same as BSM but is bas
ed in the real, rather than the risk-
neutral, world. It isn't arbitrage free, but it answers the most important
136
question a directional speculator has: If she thinks the stock is going up,
what option should she buy?
Instead of calculating the call value by taking the expectation of the payoff
in the risk-neutral world, Boness's result is the expectation in the physical
world.
With the normal notation:
(7.1)
(7.2)
where
(7.3)
(7.4)
To arrive at this formula, Boness had to make an assumption about
returns.
He needed to say that the stock return is the rate used to discount the
strike in the put-call relationship, that is, all cash flows would be invested
in the stock.
In the normal derivation of put-call parity, we form a portfolio that is long
a put, short a call, and long a share. At expiration this portfolio is worth the strike price, which means the portfolio is currently worth the discounted
strike value. Since the middle ages (where the idea was used as the basis for
mortgage lending) it has been known that the correct discount factor is the
interest rate. Drift is irrelevant. If anything other than the interest rate is
used as a discount factor, an arbitrage opportunity exists.
Even if we accept that drift is a real phenomenon, it is also reasonable to
include an interest rate as an alternative investment opportunity. The stock
appreciates at μ and cash is stored at r. It might appear that no investor would operate like this. If μ > r, why would he not invest all money in the stock and ignore the interest rate completely (Boness's model does this)?
In reality, people generally do split investments between assets with
137
different returns and risks. The stock has a higher return but also higher
risk, which is reflected in the volatility parameter.
It is possible to construct a pricing model that does this with a formal
argument from a modified BSM PDE. But this isn't necessary. Our model
needs to assume cash is invested at a risk-free interest rate and the forward
price of the stock is driven by a (physical world) drift.
If we slightly reinterpret some parameters, this model already exists: the
generalized BSM prices (European) options when the underlying pays a
continuous dividend yield. We use this, and the interest rate, to price
options of the forward value of the stock while also assuming that cash
Positional Option Trading (Wiley Trading) Page 15
