And, in only trivial situations could the equations be analytically
solved. The Black-Scholes-Merton (BSM) model was meant to be
of this type.
But it isn't used that way at all.
The inventors of the model envisaged that the model would be
used to find a fair value for options. Traders would input the
underlying price, strike, interest rate, expiration date, and
volatility and the model would tell them what the option was
worth. The problem was that the volatility input needed to be the
volatility over the life of the option, an unknown parameter.
Although it was possible for a trader to make a forecast of future
volatility, the rest of the market could and did make its own
forecast. The market's option price was based on this aggregated
estimate. This is the implied volatility, which became the
fundamental parameter. Traders largely didn't think of the model
as a predictive valuation tool but just as an arbitrage-free way to
convert the quickly changing option prices into a slowly changing
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parameter: implied volatility. For most traders, BSM is not a
predictive model; it is just a simplifying tool.
This isn't to say that BSM can't be used as a pricing model to get a
fair value. It absolutely can. But even traders who do this will
think in volatility terms. They will compare the implied volatility
to their forecast volatility, rather than use the forecast volatility to
price the option and compare it to the market value. By using the
model backwards, these traders still benefit from the way BSM
converts the option prices into a slowly varying parameter.
We need to examine the effects of the model assumptions in light
of how the model is used. Although the assumptions make the
model less realistic, this isn't important. The model wasn't used
because it was realistic; it was used because it was useful.
Obviously, it is possible to trade options without any valuation
model. This is what most directional option traders do. We can
also trade volatility without a model. Traders might sell a straddle
because they think the underlying will expire closer to the strike
than the value of the straddle. However, to move beyond
directional trading or speculating on the value of the underlying at
expiration we will need a model.
The BSM model is still the benchmark for option pricing models.
It has been used since 1973 and has direct ancestors dating to the
work of Bachelier (1900) and Bronzin (1906). In terms of scientific
theory, this age makes it a dinosaur. But just as dinosaurs were the
dominant life form for about 190 million years for a reason, BSM
has persisted because it is good.
We want an option pricing model for two reasons.
The first is so we can reduce the many, fast-moving option prices
to a small number of slow-moving parameters. Option pricing
models don't really price options. The market prices options
though the normal market forces of supply and demand. Pricing
models convert the market's prices into the parameters. In
particular, BSM converts option prices to an implied volatility
parameter. Now we can do all analysis and forecasting in terms of
implied volatility, and if BSM was a perfect model, we would have
a single, constant parameter.
The second reason to use a pricing model is to calculate a delta for
hedging. Model-free volatility trading exists. Buying or selling a
straddle (or strangle, butterfly, or condor, etc.) gives a position
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that is primarily exposed to realized volatility. But it will also be
exposed to the drift. The most compelling reason to trade volatility
is that it is more predictable than returns (drift) and the only way
to remove this exposure is to hedge. To hedge we need a delta and
for this we need a model. This is the most important criterion for
an option trader to consider when deciding if a model is good
enough. Any vaguely sensible model will reduce the many option
prices to a few parameters, but a good model will let us delta
hedge in a way that captures the volatility premium.
In this chapter we will examine the BSM model and see if it can
meet this standard. By BSM model I mean the partial differential
equation rather than the specific solution for European vanilla
options. The particular boundary conditions and solution methods
aren't a real concern here.
Derivations of BSM can be found in many places (see Sinclair,
2013, for an informal derivation). Here we will look at how the
model is used.
Option Trading Theory
Here we will very briefly summarize the theory of option pricing
and hedging. For more details refer to Sinclair (2010; 2013).
An option pricing model must include the following variables and
parameters:
Underlying price and strike; this determines the moneyness of
the option.
Time until expiration.
Any factors related to carry of either the option or the
underlying; this includes dividends, borrow rates, storage
costs, and interest rates.
Volatility or some other way to quantify future uncertainty.
A variable that is not necessary is the expected return of the
underlying. Clearly, this is important to the return of an option,
but it is irrelevant to the instantaneous value of the option. If we
include this drift term, we will arrive at a contradiction. Imagine
that we expect the underlying to rally. Naively, this means we
would pay more for a call. But put-call parity means that an
increase in call price leads to an increase in the price of the put
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with the same strike. This now seems consistent with us being
bearish. Put-call parity is enough to make the return irrelevant to
the current option price, but (less obviously perhaps) it is also
enforced by dynamic replication.
This isn't an option-specific anomaly. There are many situations in
which people agree on future price change, but this doesn't affect
current price. For example, Ferrari would be justified in thinking
that the long-term value of their cars is higher than their MSRP.
But they can build the car and sell it at a profit right now. Their
replication value as a manufacturer guarantees a profit without
taking future price changes into account. Similarly, market-
makers can replicate options without worrying about the
underlying return. And if they do include the return, they can be
arbed by someone else.
The canonical option-pricing model is BSM. Ignoring interest
rates for simplicity, The BSM PDE for the price of a call, C, is
(1.1)
where S is the underlying price, σ is the volatility of the
underlying, and t is the time until expiration of the option.
Or using the standard definitions where Γ is the second partial
derivative of the option price with respect to the underlying and θ
is the derivative of the option price with respect to time,
(1.2)
This is then solved using standard nu
merical or analytical
techniques and with the final condition being the payoff of the
particular option.
This relationship between Γ and θ is crucial for understanding
how to make money with options. Imagine we are long a call and
the underlying stock moves from St to St+1.The delta P/L of this
option will be the average of the initial delta, Δ, and the final delta,
all multiplied by the size of the move. Or
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(1.3)
If this option was initially delta-hedged, the P/L over this price
move would be
(1.4)
Next note that
(1.5)
so that the profit in hedging over each time interval is
(1.6)
(Although equation 1.6 is only asymptotically true, if we worked
with an infinitesimal price change, this derivation would be exact.
This is the first term of the BSM differential equation. Literally,
BSM says that these profits from rebalancing due to gamma are
exactly equal to the theta of the option. Expected movement
cancels time decay. The only way a directionally neutral option
position will make money is if the option's implied volatility
(which governs theta) is not the same as the underlying's realized
volatility (which determines the rebalancing profits). This is true
no matter which structure is chosen and the particulars of the
hedging scheme.
If we can identify situations where this volatility mismatch occurs,
the expected profit from the position will be given by
(1.7)
This is the fundamental equation of option trading. All the “theta
decay” and “gamma scalping” profits and losses are tied up in this
relationship.
Note also that this vega P/L will affect directional option trades. If
we pay the wrong implied volatility level for an option, we might
27
still make money but we would have been better off replicating the
option in the underlying.
The BSM equation depends on a number of financial and
mathematical assumptions.
The underlying is a tradable asset.
There is a single, risk-free interest rate.
The underlying can be shorted.
Proceeds from short sales can be invested at the risk-free rate.
All cash flows are taxed at the same rate.
The underlying's returns are continuous and normally
distributed with a constant volatility.
Traders have devised various workarounds to address these
limiting assumptions (see Appendix One). The most important of
these is the concept of the implied volatility surface. If the BSM
were an accurate descriptive theory, all options on a given
underlying would have one volatility. This is not true. For the BSM
equation to reproduce market option prices, options with different
strikes have different implied volatilities (the smile) and options
with different maturities have different implied volatilities (the
term structure). These implied volatilities make up the IV surface.
An example is in shown in Figure 1.1.
The IV surface exists partially because the BSM is mathematically
misspecified. The underlying does not have returns that are
continuous and normally distributed with a constant volatility.
However, even a model that perfectly captured the underlying
dynamics would need a fudge factor like the implied volatility
surface. Some of the reasons for its existence have nothing to do
with the underlying. Different options have different supply and
demand, and these distort option prices. Because of this, there is
often an edge in selling options with high volatilities relative to
others on the same underlying (see the section on the implied
skewness premium in Chapter Four).
Equation 1.7 gives the average PL of any hedged option position,
but there is a wide dispersion of results for this mean, and the
spread of this distribution decreases with the number of hedges.
Figure 1.2 shows the PL distribution of a short straddle that is
never re-hedged, and Figure 1.3 shows the distribution when the
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straddle is re-hedged every day. The underlying paths were
generated from 10,000 realizations of a GBM. The implied and
realized volatility were equal so we expect an average PL of zero.
FIGURE 1.1 The implied volatility surface for SPY on September 10, 2019.
FIGURE 1.2 The terminal PL distribution of a single short one-year ATM straddle that is never re-hedged. Stock price is $100,
rates are zero, and both realized and implied volatilities are 30%.
29
FIGURE 1.3 The terminal PL distribution of a single one-year ATM straddle that is hedged daily. Stock price is $100, rates are
zero, and both realized and implied volatilities are 30%.
The dependence of the standard deviation of the PL distribution
on the number of hedges is shown in Figure 1.4.
FIGURE 1.4 The standard deviation of the terminal PL
distribution of a single one-year ATM straddle as a function of the
number of hedges. Stock price is $100, rates are zero, and both
realized and implied volatilities are 30%.
30
The reason to hedge less frequently and accept a wider standard
deviation of results is that hedging costs money. All hedges incur
transaction costs (brokerage, exchange fees, and infrastructure
costs). Costs like this are an easily forgotten drain on a portfolio.
Individually they are small, but they accumulate. To emphasize
this point, Table 1.1 compares the summary statistics of results for the daily hedged short straddle when there is a transaction cost of
$.10 per share and when hedges are costless.
The difference between these two cases is roughly equivalent to
misestimating volatility by two points.
In practice, aggressive re-hedging is done by market-making firms
and some volatility specialists. The vast majority of retail and buy-
side users seldom or never hedge. The relevant theory for those
hoping to approximate continuous hedging is discussed in Sinclair
(2013). In this book we will generally assume that no re-hedging takes place. These results are also applicable to those who hedge
infrequently. They can just assume that the original position has
been closed and a new one opened. So, a one-year position that is
hedged after a month would thereafter have the expected
distribution of an 11-month option.
TABLE 1.1 Statistics for the Short One-Year ATM Daily Hedged Straddle With and Without Hedging Costs (stock
price is $100, rates are zero, and both realized and
implied volatilities are 30%.)
Statistic
Costless
$.10/Share
Hedges
Hedges
Average
−$6.10
−$121.54
Median
−$49.85
−$111.68
Percent
profitable
44%
30%
Conclusion
The BSM model gives the replication strategy for the option. The
expected return of the underlying is irrelevant to this strategy. The
only distributional property of the underlying
that is used in the
BSM model is the volatility. A hedged position will, on average,
make a profit proportional to the difference between the volatility
implied by the option market price (by inverting the BSM model)
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and the subsequent realized volatility. The choice of the option
structure and hedging scheme can change the shape of the PL
distribution, but not the average value. These choices are far from
immaterial, but successful option trading depends foremost on
finding situations in which the implied volatility is mispriced.
Summary
Arbitrage-free option pricing models do not include the
underlying return. BSM includes only volatility.
Inverting the pricing model using the option's market price as
an input gives the implied volatility.
The average profit of a hedged option position is proportional
to the difference between implied volatility and the subsequent
realized volatility.
Practical option hedging is designed to give an acceptable level
of variance for a given amount of transaction costs.
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CHAPTER 2
The Efficient Market Hypothesis and
Its Limitations
A lot of trading books propagate the myth that successful trading
is based on discipline and persistence. This might be the worst
advice possible. A trader without a real edge who persists in
trading, executing a bad plan in a disciplined manner, will lose
money faster and more consistently than someone who is lazy and
inconsistent. A tough but unskilled fighter will just manage to stay
in a losing fight longer. All she will achieve is being beaten up
more than a weak fighter would.
Another terrible weakness is optimism. Optimism will keep losing
traders chasing success that will never happen. Sadly, hope is a
psychological mechanism unaffected by external reality.
Emotional control won't make up for lack of edge. But, before we
can find an edge, we need to understand why this is hard and
where we should look.
The Efficient Market Hypothesis
The traders' concept of the efficient market hypothesis (EMH) is
“making money is hard.” This isn't wrong, but it is worth looking
at the theory in more detail. Traders are trying to make money
Positional Option Trading (Wiley Trading) Page 3
