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Hedged Option Positions 87 Discrete Hedging and Path Dependency 87
Life Cycle of a Trade 165
Introduction
In Chapter 3, we look at the normal shapes of the volatility surface, both over time and by strike. In order to distinguish our results from chance, we must carefully monitor the results of our trades.
Option Pricing
C is the value of the option. St is the underlying price at the time the number of stocks we are short. 2(St+1−St)2+θ+r(C−St) (1.4) where the second derivative is the option price relative to the underlying asset.
Volatility Measurement
As before, this should be calculated on an annual basis by multiplying it by the square root of the number of trading periods in a year. It is actually a weighted average of the Rogers, Satchell, and Yoon estimator, the close-to-open volatility, and the open-to-close volatility.
Implied Volatility Dynamics
You could check the actual volatility of ATM options, but that is very difficult. To find the implied jump, we compare the implied volatilities of first-month and second-month option ATMs. The volatility attributed to the event, σE, is the difference between the volatility of the first month and the volatility of the forward period.
Further, even if you are right, there will be significant volatility in trade results. For example, a trader likes to think of volatility as the linear slope of the implied volatility curve. We cannot directly attribute a shift in the shape of the curve to either skewness or kurtosis.
The slope and curvature of the implied curve are of decreasing variability (and therefore important).
Hedging
By plotting the certainty equivalent as a function of bet size, we can visualize our utility function. It is unique in that it has a constant absolute propensity to risk, since r=γ is independent of wealth, W. This is only true if we assume that the counterparty in an options trade has no way of predicting the direction of the underlying.
In particular, the middle of the no-transaction region does not coincide with the BSM delta. As return we choose the replication error: the total effect of all transaction costs. The relative quality of the different strategies depends on the degree of risk aversion and the level of transaction costs.
A number of researchers have concluded that market influence scales as the square root of trade size (Hasbrouck 1991; Madhavan and Smidt 1991; BARRA 1997).
Hedged Option Positions
An approximation for the value of an at-the-money option (put or call) is given by. We assume that we sell an option at an implied volatility,σi, and we then hedge at the realized volatility,σr. Note that we need to value the option because we are interested in the portfolio value as it is marketed, but must be valued at the realized volatility because that is how we have chosen to hedge.
This was the case for a short position consisting of 1,000 vega of one-year, at-the-money calls, sold at a volatility of 40 percent and hedged at the realized volatility of 30 percent until expiration. We go through the same analysis as before, but now all relevant variables are assessed by the implied volatility. This was the case for a short position consisting of 1,000 vega of one-year, at-the-money calls, sold at a volatility of 40 percent and hedged to the implied volatility until expiration.
Ahmad and Wilmott (2005) found expressions for the expected profit (as an integral) and the standard deviation (as a double integral) of an option hedged against the implied volatility.
Money
Management
The Kelly ratio in this case is 0.175, which corresponds to the top of the curve. The expected growth rate of an individual acting at a fraction, f, of the Kelly ratio is given by. To deal with these extreme traits, it is fairly common for Kelly followers to trade at a fraction of the Kelly ratio.
It shows the time evolution of the PDF bankroll when trading with the Kelly fraction. When deviations from fair value are larger, we will trade more. As with all strategies that increase the expectation of the logarithm of wealth, this is a very aggressive money management scheme.
We can see in Figure 6-17 that the peak of the P/L function is very close to the theoretical point of 0.75 standard deviations.
Trade Evaluation
In this case, it is important to first measure the performance of the driving strategy. While this could be the ideal, it is very unlikely to be the case for any trader. The Sharpe ratio is almost risky what BSM is to option pricing: practitioners are aware that it is imperfect, but it provides a clear framework for risk.
It is actually very easy to construct some P/L paths that demonstrate this weakness of the Sharpe ratio. It is important that the manager or lender is aware of the ways in which any measure of risk can be fooled. This could be done using analytic formulas based on the properties of the normal distribution, but it is usually much more instructive to look at the results of some simulations.
It is greater than 1, but the sample size is small and it is quite possible that the true population proportion is 1.
Psychology
Some of the latter have been known to protest that "there is no such thing as behavioral economics." For example, when we examine the fundamentals of a company, it is easy to become more convinced of the correctness of our position (see the discussion of confirmation bias later in this chapter). Brett Steenbarger (www.brettsteenbarger.com) writes amusingly of the attitude as "emotional socialism." This is the idea that we somehow just deserve to be confident, rather than developing skills and then learning to be confident in that skill.
Our minds will tend to remember the big picture at the expense of the more representative pictures. Ace Greenberg, the former chairman of Bear Stearns, put it this way: "The definition of a good trader is a guy who takes losses." This is a classic piece of trading lore. However, when classifying something as representative of a known situation, it is really important that it is actually representative of the known situation.
This can be very dangerous if we do not have a full understanding of the expected distribution of winning and losing days.
Life Cycle of a Trade
July implied volatility had been rising for several weeks, rising from the mid-1920s to the high 1930s. At this point, we were fairly confident that we had a head start in the spread between implied volatility and realized volatility, both on an absolute and relative level. Implied volatility had subsided, we had realized much of our expected P&L and we had few other positions in the portfolio.
We lost money over three days, but on the first of these we incurred costs by actively cutting the bid-ask spread in options, and on the second day we were comfortable enough to sell higher on the bid as implied volatility fell slightly higher bid. This is highly desirable and is a compelling reason to look for timed catalyst trades that could cause implied volatility to move in your direction. We execute these trades on the assumption that the implied volatility over the life of the option is false.
So we were actually unlucky, but not because we mispredicted volatility; we predicted it almost exactly.
Conclusion
This is the point where it is necessary to be aware of recent news stories, industry trends and behavioral psychology. Hedging makes volatility trading very path dependent: it is quite possible to correctly predict volatility and lose money. This means that each transaction is evaluated based on the expected return and risk, in the overall context of our objectives.
The ability to execute is a skill that enables us to participate in the deals we want to get involved in. In fact, it seems that more knowledgeable traders tend to be the most deficient in concentration. It is much better to be a trader in the dumbest pit than to grind in a very competitive business.
However, this will not guarantee success because there are other aspects that must also be present to be successful.
Model-Free
Implied Variance and Volatility
INDEX
Spreadsheet Instructions
Here we simulate the development of a hedged long option position of 1,000 vega (this can be changed in cell O18 on the historical return and cell N18 on the normal return) where we hedge each day at the close. This sheet uses least squares and Solver to extract the implied volatility, skewness and kurtosis consistent with the Corrado-Su model from the quoted call option prices. It should be set to minimize cell J2 (the sum of the squares of the differences between the market prices and the Corrado-Su prices) by changing cells C5 to C7.
The curve of implied volatility as a function of shock can be produced either by entering the implied volatilities directly in cells F3 through F9 or by using the Solution. For example, to obtain the implied volatility of shock 35, use Solver to minimize the difference between cells G3 and H3 by changing cell F3. The 'Simple' Normal Noise tab uses a modified rule to enter a trade where the basis is driven by an Ornstein-Uhlenbeck process with normally distributed innovations with a standard deviation of one.
The "'normal noise' exact" tab scales our input according to the optimal equation rule (6.28).
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