Positional Option Trading (Wiley Trading)

by Euan Sinclair

  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

  23

  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

  24

  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

  25

  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

  26

  (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

  28

  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)

  31

  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.

  32

  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