Probably due to the difficulty of estimating the aggregate marginal
tax rate, little empirical work has been done on this issue, either.
Mason and Utke (2019) compared SPX and SPY options. ASPX
option profits are taxed 60% at the long-term capital gains rate
and 40% at the short-term rate. SPY options are taxed at the
higher short-term rate. After controlling for dividends and
American/European exercise features they concluded that there
was a persistent price difference where SPY options were cheaper.
Their conclusion was that this could be attributed to people being
less inclined to buy options with higher tax rates. However, the
effect was so small that it could be explained by dirty data or data
processing procedures (H. Contini, personal communication,
2019).
The Ability to Trade and Short the Underlying
The BSM formalism relies on the ability to hedge in the
underlying. Sometimes we will need to be able to short the
underlying. Sometimes this isn't possible. If we cannot short, then
the BSM hedging strategy clearly won't work. To see how pricing is
affected, think about pricing a forward, F. The current price of the
underlying is S, time until delivery is T, and interest rates are r.
If
, sell the forward contract and buy the
underlying for S with borrowed money.
On the expiration date, we deliver the underlying, and receive the
agreed forward price, F.
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We then repay the lender the borrowed amount plus interest. This
amount totals
. The difference between the two
amounts is the arbitrage profit. The principle of no-arbitrage
means this can't happen so
(A1.3)
But if
we can't form the arbitrage portfolio. In
this case we would need to buy a forward and short the
underlying, so we cannot rule out the inequality.
If we can't short the underlying, the forward price can be less than
the no-arbitrage value. Consequently, call values can be less than
the BSM value and put values can be more than the BSM values.
The put-call parity relationship is “shifted” as well. This means
that pricing in the risk-neutral world isn't possible. If we can't
trade the underlying at all, then we are even more lost.
It is unlikely that most traders will trade options on completely
untradeable options, but there are cases where the underlying is,
or becomes, illiquid, so it is worth knowing how to deal with an
untradeable underlying as a limiting case.
Before an exchange lists options, the underlying has to have a
certain amount of liquidity. For example, to have options on a
stock in the United States the company must be listed on either
the NYSE, AMEX, or NASDAQ. The company must have at least 7
million shares and the company must have more than 2,000
shareholders. But after the listing it is quite possible for the
underlying to become less liquid.
There are also instances where we want to price employee options
on a non-traded company. This stock is impossible to trade.
Finally, there are cases where the underlying becomes impossible
to short either through lack of a borrow or through a regulatory
change. Although this illiquidity is only on one side of the market
it is still enough to create problems.
One way to approach this problem is to use the “real option”
approach. Although the term real option is new, the idea isn't. In
1930, Irving Fisher explicitly wrote about the options that a
business owner had (although his credibility may have been
lessened due to his famous 1929 statement that “stocks have
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reached what looks like a permanently high plateau”). However,
the publication of the BSM model has enabled the concept to be
quantified.
An example of a real option is the decision to invest in a project.
Preliminary research costs are the price of the option. The exercise
price is the future profits. The payoff is the difference between
these two.
There are some important distinctions between real options and
financial options:
They are untraded in a conventional sense although they can
be “bought” or “sold” by buying or selling the business units
that create them.
The value of the option is directly dependent on the actions of
management because their actions control the initial premium.
These options are often more dependent on uncertainty than
on the measurable volatility of the underlying.
Some real options can be priced by using the BSM framework
because they involve an underlying that is a traded asset. An
example of this situation would be the option to start producing
oil. The premium would be the cost of equipment and exploration.
The underlying asset is oil. Because oil is traded, this option can
be hedged and the risk-neutrality arguments of BSM can be
applied.
But many real options are not on a traded asset. For example, if
the underlying is some sort of intellectual property, it will be
impossible to hedge, and hence the BSM formalism will be
inapplicable. The principle of no-arbitrage cannot be used. It is
common in these circumstances to assume the existence of a
traded security that is perfectly correlated with the underlying.
This is almost always a very unrealistic assumption.
However, if there are traded securities with imperfect correlation
to the underlying (obviously the higher the better), we can find
bounds on our real option prices. This was done by Capinski and
Patena (2003) (an application of the no-good-deal theory of Cerny
and Hodges, 2000), and, in the case of a perfectly correlated asset, their model gives the BSM option price.
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Their main assumption is that adding an option to a portfolio
won't change the Sharpe ratio. This isn't always true, but it is
generally a good approximation (and is the same assumption that
Black and Scholes used in their derivation). Unfortunately, the
bounds produced are too large for any practical valuation
processes (the range is generally from zero to the BSM price
assuming perfect correlation).
Hedging options with a correlated underlying produces a large
dispersion of P/L. In Figure A1.3 we show the effect on the P/L of
an option that is hedged daily with the underlying. The duration is
one year. Both realized and implied volatility are 30%, rates a
zero, and total vega is $1.000. One thousand paths of GBM were
used to find the dispersion.
In conclusion, when the underlying is untraded the best you can
do is find another product that should be similar (in the co-
integrated sense). This will let you value the option using the BSM
concept because you can now hedge. It probably won't be a good
hedge, but you need to do the best you can, accept the variance,
and ask for enough edge. You can't create information from
nothing, but you can get someone else to accept some of the
uncer
tainty through the hedge.
FIGURE A1.3 The standard deviation pf the P/L for an option hedged with an imperfectly correlated underlying.
Nonconstant Volatility
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The volatility of the underlying is not constant. The rolling 30-day
close-to-close volatility of the S&P 500 is shown in Figure A1.4.
Any nonconstancy of volatility creates a non-normal return
distribution. The simplest example is if the true return process is a
mixture of normal distributions with different volatilities. The
kurtosis of a mixture of normal distributions, with zero means, is
given by
FIGURE A1.4 S&P 500 30-day volatility from January 2000
through to the end of 2018.
(A1.4)
where px is the probability of being in the state with volatility, σx.
So, if there is a 50% chance of being in a state with a volatility of
20%, and a 50% chance of being in a state with a volatility of 80%,
the distribution will have a kurtosis of 5.3.
We could price options off the resulting distribution and then back
out the equivalent BSM implied volatilities to see how
stochasticity leads to a volatility smile, but we can also show this
by directly thinking about implied volatility as the stochastic
variable.
The implied volatility of an option is the forward-looking estimate
of the average underlying volatility over the lifetime of the option.
If volatility is not constant, an implied volatility smile will appear.
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Consider a case in which a $100 stock has a volatility with a 50%
chance of being 20% and a 50% chance of being 80%. So, half the
time the 30-day, 100 call will be worth $2.29 and half the time it
will be worth $9.13. The average value is $5.71. This corresponds
to an implied volatility of 50%, the average of the two possible
volatility states. Now consider a 120 call. In the low volatility state,
it is worth nothing and in the high volatility state it is worth $3.04.
In this case the average value is $1.52, which corresponds to an
implied volatility of about 62%.
This situation would create a symmetric smile, but extending the
idea so that underlying moves are negatively correlated to
volatility moves gives the asymmetric implied volatility skew
found in most products. Imagine we have the same two volatility
states (20% and 80%) but now the low volatility is associated with
a stock price of $102, and the high volatility state corresponds to a
stock price of $98. Going through the same exercise as just
described we find the 80 strike has an implied volatility of 68%,
the 100 strike has an implied volatility of slightly over 50%, and
the 120 strike has an implied volatility of 59%.
Nonconstant volatility creates the convexity of the smile. The
correlation between volatility moves and stock moves creates the
slope of the smile. Other causes also exist, but stochastic volatility
is clearly a contributor to the implied volatility structure.
Many pricing models that address this effect have been developed.
For example, as of January 4, 2019, ssrn.com listed 717 papers
with stochastic volatility in their title and 3,287 with those words
in their title, abstract, or keywords.
Market-makers also know about this effect. They typically use a
modified BSM model that has a different volatility for each strike,
which they periodically update. Is using a deterministic volatility
model and sporadically changing the parameter inferior to using
an explicitly stochastic volatility model?
A number of studies that compare the performance of the
modified BSM model to different correctly specified models have
reached the same conclusion, including Jung (2000), Engle and
Rosenberg (2002), Bollen and Raisel (2003), Yung and Zhang
(2003), Li and Pearson (2007), Jung and Corrado (2009), and Hull and White, 2017. A stochastic volatility model that captures the true stochastic process offers practically no improvement over
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the tweaked BSM model used by traders. In each case, the hedging
errors due to using BSM are economically insignificant.
A further consideration is that even a stochastic process that is
fairly effective at modeling the dynamics of the underlying will not
be able to exactly match all of the implied volatilities. The
variance-reducing effect of hedging at the implied volatilities
(Ahmad and Wilmott, 2005) will be reduced. Even if a stochastic
volatility model gives a higher terminal profit, it will always lead to
a more variable day-to-day PL.
Given that we won't ever know the true volatility generating
process, using the simpler, better understood, and non-
parametric-modified BSM model is probably the best solution.
Conclusion
Although the BSM model is based on many clearly incorrect
assumptions, most of these can be corrected for with simple
adjustments (some more ad-hoc than others). BSM has three large
advantages over more complex models. It is an industry-wide
standard that forms the basis for trader communication. Most
traders have based all of their learned intuition on the BSM
model. It is simple and implemented in all trading programs.
Given these advantages, there is no compelling reason to use a
more “correct” model over the BSM with the various adjustments.
Summary
BSM doesn't handle stochastic interest rates or interest rates
with transaction costs, but in practice this doesn't really matter
because the effect is typically small.
Adding dividends and carry rates to BSM is easy.
Very little research has been done on taxes and BSM, but this
is true of all other models as well.
It is possible to trade options on illiquid underlyings if there is
a correlated hedging instrument.
Volatility is stochastic, but BSM with periodically adjusted
volatility inputs works as well as stochastic volatility pricing
models.
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APPENDIX 2
Statistical Rules of Thumb
Just as a lot of popular trading rules have no basis in fact, the
generally accepted origin of the phrase “rule of thumb” is no more
than a myth. The story is that there was a 17th-century English law
that specified that the maximum width of a stick for wife beating
was the width of a man's thumb. No such law ever existed.
Nonetheless, these simple heuristics can be useful for quick
estimates or establishing a baseline.
Converting Range Estimates to Option Pricing
Inputs
In order to either subjectively price options or simulate outcomes,
you will need estimates of mean return and the variance of the
returns. Unfortunately, it is quite common for fundamental
analysts to give only a most-likely case and a range of possible
outcomes. It is probably best to use the common time series
methods to estimate volatility (refer to Sinclair, 2013), but the analysts' return information could be useful.
The approximations for doing this conversion are part of a class
called three-point-estimators. There are many such m
ethods that
differ due to the assumed shape of the approximating distribution.
First, convert the analyst's numbers to annualized percentages. So,
if she has a low stock price estimate of $90 on a $100 stock in the
next three months, that would convert to a negative 40% return. If
we assume the data is drawn from a triangular distribution, and
our low estimate is l, the median estimate is m and the high
estimate is h, the mean estimate is
(A2.1)
This is just an average of our data. This makes sense if we have no
idea of the historical accuracy of the analyst. But sometimes we
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want to more heavily weight the median. This would be the case if
we have higher confidence in the analyst. If we don't have much of
an idea how good the analyst is, we will have no reason to more
heavily weight the median (we should also lower our resulting
trade sizes in this situation).
A weighted three-point estimate of the mean is
(A2.2)
(This assumes a beta distribution for the true process.)
If we had confidence in the analyst, we would just use her median
value as the mean.
This method is far from ideal, but it could also be the only way to
use the information from an otherwise valuable source.
Rule of Five
This is an exceptionally simple heuristic that I read about in How
to Measure Anything: Finding the Value of Intangibles in
Business by Douglas Hubbard (2007). He gives no references and I've never seen it mentioned anywhere else.
If we are randomly sampling from a population, we can be 93%
certain that the population median lies within the range of five
measurements. For example, if I have a GBM with unchanging
parameters and I measure the volatility over five different periods
and get 30%, 20%, 26%, 18%, and 23%, I can be 93% certain that
the true median volatility is between 18% and 30%. The usual
caveats about what a random sample is apply. And often the
range, and hence confidence interval, will be wide but it is still
valuable to use even a very small sample to establish a base rate.
The only way the median could be outside the range is if all
measurements were either above the median or below it. Because
the chance of any observation being above the median is 50%, the
Positional Option Trading (Wiley Trading) Page 23
