I
n Chapter 1 we stated the central result thatP/L=vega(σimplied−σrealized) (5.1) Actually this is only true on average. There are wide fluctuations around this amount. In this chapter we look at the causes and magnitude of these fluctuations so we know exactly what we can expect as the results of our trades.
DISCRETE HEDGING AND PATH DEPENDENCY
Imagine we buy one-year call options with a total vega of $1,000, at an implied volatility of 30 percent, where realized volatility over their lifetime is in fact 30 percent. We hedge delta each day on the close. The profit/loss (P/L) amounts for 10 realizations of this process are given as follows:
$(516)
$154
$(108)
$(537)
$331
$(1,741)
$(230)
87
–200,000 –150,000 –100,000 –50,000 0 50,000
0 50 100 150 200
Terminal Stock Price
Hedging Portfolio Value
FIGURE 5.1 One Hundred Realizations of the Hedging Strategy for the One-Year Option
$138
$423
$1,984
The average P/L is a loss of $10.2. This is reasonably close to the ex- pected value of zero, but there is a wide dispersion of values. No doubt in one universe there is a trader who is justifying his loss of $1,741 to his man- ager and another who is concocting a story about how his trading prowess made $1,984. What is really going on here? What will the actual distribu- tion of results look like? The central insight in the BSM argument is that an option can be replicated by trading the underlying. Figure 5.1 shows the results of 100 simulations, where we actually try to replicate this option.
As we are long a call, the replicating portfolio should look like a short call position.
So things are approximately what we expect. This indeed looks like the payoff from a short call position struck at 100, with the initial value corresponding to a vega of $1,000. But the replication isn’t perfect. There is significant dispersion around the true value. The dispersion is dependent on the final underlying price, with the most variability occurring when it expires near the strike.
The first thing to note is that we are not hedging continuously. Dis- crete hedging makes our nondirectional strategy very path dependent (this has nothing to do with the weak path dependence of American options as opposed to European options). Two sample paths with exactly the same realized volatility can lead to different P/Ls. Consider an extreme exam- ple where the all the movement of the underlying is due to one jump. In the first path the jump happens on the first day, as shown in Figure 5.2. In the second path the jump happens right before expiration (Figure 5.3).
Time
Price
FIGURE 5.2 Path 1: The Jump on the First Day
In each case the positionP/Lis the change in the call price minus the change in the value of the hedge position.
P/L=C(T)−C(0)−
(t)×[S(t)−S(t−1)] (5.2) But in this special case we only need to evaluate the delta immediately before the jump. For the case of an at-the-money call
Case 1: >0.5 Case 2:=0.5
The difference inP/Lin these two instances is
(P/L)=(1−2)×[S(T)−S(0)] (5.3) So if we are short this call we do better in the first instance because we are long more stock as a hedge. This is admittedly an extreme example but the general principle holds. The timing of a move is very important to the profitability of an option.
Time
Price
FIGURE 5.3 Path 2: The Jump Right Before Expiration
Frequency
0 2 4 6 8 10 12 14 16 18
–18,000 –16,000 –14,000 –12,000 –10,000 –8,000 –6,000 –4,000 –2,000 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 More
P/L
Percentage of Observations
FIGURE 5.4 The P/L Distribution for $1,000 Vega of Initially At-the-Money Op- tions When Hedged Once a Week
Let’s now look in more detail at the size and nature of this effect when we have more realistic sample paths. Take a stock initially trading at $100.
Assume rates, dividends, and the drift are zero. Again we buy one-year call options with a total vega of $1,000, at an implied volatility of 30 percent, and simulate 100 paths where the realized volatility over their lifetime is in fact 30 percent. Specifically we use geometric Brownian motion (GBM), so the stock path is given by
S(t+t)=S(t) exp
−σ2
2 t+σ√ tε
(5.4) whereεis drawn from a standard normal distribution with mean of zero and standard deviation of 1.
In Figure 5.4 we hedge weekly, and in Figure 5.5 we hedge daily. The results are summarized in Table 5.1.
TABLE 5.1 Summary Statistics for the Hedging Frequency Experiment
Hedge Frequency P/L σP/L Kurtosis Skew
Daily −107.19 2,615.2 4.05 0.25
Weekly −140.36 5,714.6 3.66 0.16
Frequency
0 5 10 15 20 25 30 35
Percentage of Observations –18,000 –16,000 –14,000 –12,000 –10,000 –8,000 –6,000 –4,000 –2,000 2,000
0 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 More
P/L
FIGURE 5.5 The P/L Distribution for $1,000 Vega of Initially At-the-Money Op- tions When Hedged Once a Day
From this crude experiment we can draw some tentative conclusions:
r The average P/L is roughly zero.
r The distribution is roughly normal looking.
r The dispersion is inversely proportional to the hedging frequency;
more specifically, it is closely approximated by N–1/2 when N is the number of hedges (see Figure 5.6).
We said earlier that “realized volatility over their lifetime is in fact 30 percent.” This wasn’t exactly true. The process that generated the
0 2,000 4,000 6,000 8,000 10,000 12,000
0 50 100 150 200 250 300
Number of Hedges
Standard Deviation of P/L
FIGURE 5.6 The Distribution of the Hedging Error as a Function of the Number of Rebalances
underlying pricedidhave a volatility of 30 percent. But we were only ob- serving the process at discrete intervals. This introduces sampling error.
From Chapter 2, equation (2.10), we can see that that the sampling error is given by
σmeasured≈σ± σ
√2N (5.5)
So the volatility of the P/L as a function of the initial option value will be approximated by
σP/L ≈vega√σ
2N (5.6)
This argument isn’t rigorous and isn’t actually correct either. The actual relationship is given by
σP/L≈ π
4vega σ
√N (5.7)
(Kamal and Derman 1999).
This result can be used to derive a useful rule of thumb for the standard deviation of a straddle position. An approximation for the value of an at- the-money option (put or call) is given by
C ≈ 1
√2πSσ√
T (5.8)
(Brenner and Subrahmanyam, 1994). So the vega of a straddle (a put and a call) is approximately
vega≈ 2
√2πS√
T (5.9)
If we think of the straddle as a single volatility bet and hedge only once, equation 5.7 gives us
σP/L ≈ 1
√2S√
Tσ (5.10)
Also note that in this case only the actual realized volatility is relevant. It only matters where we are at expiry, not what the option market is implying after we have done our trade.
The important point is that a trader can be correct about his assess- ment of realized volatility and still not make money. This is just the risk we take for not hedging infinitely often and running up an infinite amount of trading cost. This variability is unfortunate but we can always reduce it
by hedging more frequently. Note that this is when the underlying follows a diffusive process. If there are jumps, we can’t alleviate the issue by hedging more often. In the real world there are simply some risks we can’t hedge away. An equity trader is trading drift but he is also exposed to volatility risk and path dependency. Being long a stock that moves smoothly from 100 to 110 in one-dollar increments is very different to being long a stock that moves from 100 to 110 but dropped to 50 before shooting up. In each case the final price does not tell the full story of the risks involved in the trade. We seek to trade volatility, but we also have volatility sampling risk (intimately related to the volatility of volatility) and path dependency.
VOLATILITY DEPENDENCY
This analysis has assumed that we were hedging our options at the true volatility. In practice this is unknown and we need to choose a hedging volatility. There are two obvious candidates,σforecastandσimplied. Which is better and what consequences are attached to choosing a volatility that will almost certainly not be the same as the realized volatility?
A simple set of examples shows that the results of this choice are again very path dependent. Consider the two underlying price paths shown in Figure 5.7. These have exactly the same annualized volatility (23.44 per- cent) but in one case the price drifts higher and in the other case it does not.
Assume we buy $1,000 vega of the one-month calls with a strike price of 100, at an implied volatility of 18.44 percent, and hedge at the end of each day. Also assume that we are able to perfectly forecast the realized
92 94 96 98 100 102 104 106 108 110 112
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Time
Price
FIGURE 5.7 Two Price Paths with Identical Volatility
TABLE 5.2 Results for Hedging the Same Option Portfolio under Different Volatilities and Different Paths
Path Hedge at Implied Hedge at Realized
Drifting $9,821 $10,516
Nondrifting $6,765 $3,078
volatility. (Note that here there will be no sampling-induced errors in the P/L because the price path has been exactly constructed rather than gener- ated by a specified random process.) Table 5.2 shows the P/L possibilities when we hedge along each path with either the implied volatility or the forecast volatility.
So obviously the choice matters a great deal, and equally obviously the situation is very path dependent. When the underlying is drifting and we are long gamma, our hedges are going to be losers. We will be selling into a rising market. So we want to hedge less often. If we use a higher volatility, we see a lower gamma so we hedge less. Conversely, in a choppy, trend- less market our hedges will tend to be winners, so we want to hedge more often. If we use a lower volatility we can gain more gamma. The situation is obviously opposite if we are short options. Traders refer to this hedging trick as “letting their deltas run” if they are long in a trending market, or
“hedging defensively” if they are short gamma in a trending market. This is summarized in Table 5.3.
Next we more generally examine the problem of choosing a hedging volatility. We assume we sell an option at an implied volatility,σi, and we then hedge at the realized volatility,σr. Let’s examine how the P/L of this position evolves through time as we hedge it. This is really just the same ar- gument we used in Chapter 1, but now we need to be very careful whether we are evaluating quantities at the implied or realized volatilities. Again our derivation will be informal. Readers interested in mathematical details should consult Carr (1999), Henrard (2003), or Ahmad and Wilmott (2005).
TABLE 5.3 The Direction One Should Bias Volatility for Different Option Positions as a Function of Market Direction
Position Market Hedging Volatility Bias
Short gamma Trending Low
Short gamma Range bound High
Long gamma Trending High
Long gamma Range bound Low
Expression 1.2 gives the value of the hedged portfolio after the first time step. Note that we need to evaluate the option atσibecause we are interested in the portfolio value as it is marked to market, butneeds to be evaluated at the realized volatility because that is how we have chosen to hedge. This is the P/L that hits our accounts at the end of each day.
C(St+1, σi)−C(St, σi)−(σr)(St+1−St)+r(C(σi)−(σr)St) (5.11) However, we also know that we can evaluate the portfolio at the real- ized volatility,σr, and here the option will be valued “correctly” (by defini- tion) so that
C(St+1, σr)−C(St, σr)−(σr)(St+1−St)+r[C(σr)−(σr)S]=0 (5.12) So over one time step our marked-to-market profit is
dC(σi)−dC(σr)+r[C(σr)−(σr)S]dt−r[C(σi)−(σr)S]dt (5.13) Now using the BSM expression (1.5), we could also write this one-step profit as
1 2
σi2−σr2
S2(σi)dt+[(σi)−(σr)][(µ−r)Sdt+σSd X] (5.14)
r Expression (5.13) tells us that we will make money ifσI> σr(leaving aside the issues associated with discrete rebalancing).
r Equation (5.14) tells us that this profit does not arrive smoothly. Equa- tion (5.14) contains a random variable.
r The way the profit is realized depends on the drift term,µ.
Figure 5.8 shows five possible paths for the P/L evolution as a function of time for an option hedged at realized volatility. This was for the case of 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 expiry. Drift, rates, and dividend yields were zero.
Traders should be familiar with this situation. These P/L swings essen- tially are due to other people (the market) marking options against our position. This isn’t meant to be taken as a conspiracy theory. We traded the options because they were mispriced. They still are. We end up car- rying the wrong amount of stock against our position. We are hedged in our mind but not according to the market. Anyone who has ever tried to explain this situation to their managers will know that it can be a difficult conversation.
–2,000.00 0.00 2,000.00 4,000.00 6,000.00 8,000.00 10,000.00 12,000.00
0 50 100 150 200 250 300
Time (Days)
P/L
FIGURE 5.8 Profit as a Function of Time for a Short Position, Hedged at the Re- alized Volatility
Next we look at what happens when we hedge at the implied volatil- ity. We go through the same analysis as before, but now all the relevant variables are evaluated at the implied volatility. The one time step mark-to- market profit is given by
dC(σi)−(σi)dS−r(C(σi)−(σi)S)dt= 1 2
σi2−σr2
S2(σi)dt (5.15) r Equation 5.15 contains no stochastic terms. The profit arrives deter-
ministically.
r There are no issues with the options supposedly being marked against us. Whatever they are marked at, we hedge accordingly.
r This is a viable hedging method even if we don’t know how to forecast volatility very well! If we sell options because our only opinion is that implied volatility is too high, we can still make money.
If we take the present value of the one-step profit and then sum over all time steps, we arrive at the total P/L for the position:
1 2
σi2−σr2 exp(−rt)S2(σi)dt (5.16)
We can now see the path-dependence effect that we observed in the earlier example. Even if our option is initially struck at the money, the drift in the underlying will take it away from there until the option’s gamma is small, and hence the potential profit in equation (5.16) is small. This makes
sense: We are putting on a volatility bet. We only get paid if we assume some volatility risk and if an option is sufficiently far from the at-the-money point such that we have no volatility exposure left. This is why traders want to end up near the strike at expiration. The gamma is largest at that point in time and space. If you are correct in your volatility assessment you make the most money there. This is also an argument for trading strangles or strips instead of straddles. We want to have a volatility position over a wide enough range of the underlying.
Path dependency is also a reason why different strikes should indeed trade at different implied volatilities. The path the underlying takes has different effects on options of different strikes because their gamma pro- file will be very different. Expression (5.16) tells us that the fair implied volatility (the volatility that makes the P/L zero) is gamma dependent. This concept is explored by Dupire (2006) in his work on “breakeven volatility skews.”
Note that this applies to both long and short gamma positions. This may come as a surprise. Most traders know that it is good for a short po- sition to gently settle near a strike at expiry. However, it is also good for a long position to finish at the strike. The difference is that practically hedg- ing the long position becomes next to impossible at very short timescales and the only time we can hedge easily is when realized volatility is low (i.e., we gently settle near our long strike), which means our volatility forecast is incorrect. We need to also note that expiring at a strike greatly increases the feedback effects of the dynamic hedging of the market makers. If they are long gamma, they will all be buying the underlying below the strike and selling it above the strike, thus compressing the realized volatility. This will be bad for the option holders, but it is the low volatility that is bad, not being near the strike. Being near the strike can be the cause of the low volatility.
Figure 5.9 shows five possible paths for the P/L evolution as a function of time for an option hedged at implied volatility. This was for the case of 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 implied volatility until expiry.
Drift, rates, and dividend yields were zero.
Figure 5.9 shows that hedging at the implied volatility gives a smoother P/L, but the end result is far more variable than when we hedge at realized volatility. Most traders find this a far easier situation to deal with.
Ahmad and Wilmott (2005) found expressions for the expected profit (as an integral) and the standard deviation (as a double integral) of an op- tion hedged at the implied volatility. The interested reader can refer to these, but it is probably more useful to gain a feel for this situation by running simulations (these can also be extended to cover different stochas- tic processes, an implied volatility smile, and a position consisting of more
–5,000 0 5,000 10,000 15,000 20,000 25,000
0 50 100 150 200 250 300
Time (Days)
P/L
FIGURE 5.9 Profit as a Function of Time for a Short Position Hedged at Implied Volatility
than one option type). Figures 5.10 and 5.11 shows the results of several such simulations. We can clearly see some features:
r Out-of-the-money option positions have far more variance than at-the- money options.
r We have less variance if we err on the high side when choosing a hedg- ing volatility.
0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Hedging Volatility
Standard Deviation
FIGURE 5.10 Standard Deviation for a Short At-the-Money Position Hedged at Various Volatilities
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Hedging Volatility
Standard Deviation
FIGURE 5.11 Standard Deviation for a Short 20 Delta Call Position Hedged at Various Volatilities
Of course we can hedge at any volatility we like. We are not restricted to implied volatility or the forecast volatility. The general case was exam- ined by Carr (1999) and Henrard (2003), but again, running simulations is probably the easiest way to get a good picture of the situation.
SUMMARY
One of the frustrating things about using options to trade the spread be- tween implied volatility and realized volatility is that we can be correct in this prediction and still lose money. Traders need to be aware of why and how this can happen. Some of the reasons for this disconnect between our prediction and our results can be addressed, but we also need to under- stand in order to learn. We need to learn from real errors, not try to draw lessons from what is essentially just bad luck.
r Hedged option positions are very path dependent.
r Correctly predicting realized volatility is no guarantee of being profitable.
r Some variance in P/L can be eliminated by hedging more frequently.
r Hedging at realized volatility makes our final P/L more certain but very noisy.
r Hedging at implied volatility smoothes the P/L but makes the final amount more uncertain.
100
C H A P T E R 6