Volatility and Hedging Errors. Jim Gatheral September, 25 1999. Background. Derivative portfolio bookrunners often complain that hedging at marketimplied volatilities is suboptimal relative to hedging at their best guess of future volatility but
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Volatility and Hedging Errors
Jim Gatheral
September, 25 1999
2 Paths with Volatility=20%
Spot
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Days
0
0
16
32
48
64
80
96
112
128
144
160
176
192
208
224
240
256
P&L vs Hedge Vol.
Whipsaw
60,000,000
40,000,000
20,000,000
Hedge Volatility

P&L
10%
15%
25%
40%
20%
30%
35%
(20,000,000)
(40,000,000)
(60,000,000)
Sine Wave
(80,000,000)
Trend
Range
3000
2900
2800
2700
Realised Volatility 12.17%
2600
2500
2400
2300
2200
2100
2000
3/3/91
4/22/91
6/11/91
7/31/91
9/19/91
11/8/91
12/28/91
2/16/92
4/6/92
5/26/92
Realised Volatility 12.45%
Range Scenario
Trend Scenario
Range Scenario
Trend Scenario
Volatility
Strike
where may itself be stochastic.
where the forward variance .
So, if we delta hedge using the BlackScholes (fixed) delta, the outcome of the hedging process is:
with a deterministic function of stock price and time.
We can extend the previous analysis to local volatility.
So, if we delta hedge using the marketimplied delta ,the outcome of the hedging process is:
Define
Then
If the claim being hedged is pathdependent then is also pathdependent. Otherwise all the can be determined at inception.
Writing the last equation out in full, for two local volatility surfaces we get
Then, the functional derivative
In practice, we can set by bucket hedging.
Note in particular that European options have all their sensitivity to local volatility in one bucket  at strike and expiration. Then by buying and selling European options, we can cancel the riskneutral expectation of gamma over the life of the option being hedged  a static hedge. This is not the same as cancelling gamma pathbypath.
If you do this, you still need to choose a delta to hedge the remaining risk. In practice, whether fixed, swimming or marketimplied delta is chosen, the parameters used to compute these are reestimated daily from a new implied volatility surface.
Dumas, Fleming and Whaley point out that the local volatility surface is very unstable over time so again, it’s not obvious which delta is optimal.