As it is impossible to know in advance what the future volatility of a security will be, the implied volatility is often used to calculate deltas. Delta hedging using this estimate causes the position to have equity market risk and, hence, it becomes path dependent (although the average or expected profit remains unchanged). Figure 72 shows that the profits from delta hedging are no longer independent of the direction in which the underlying moves. The fact that there is a difference between the correct delta (calculated using the remaining volatility to be realised over the life of the option) and the delta calculated using the implied volatility means returns are dependent on the direction of equity markets.
Figure 72. Profit from Cheap Options Is Not Constant if Volatility Is Not Known
-1 0 1 2 3 4 5 90% 95% 100% 105% 110% Straddle T=0 Straddle T=1 Small profit
Delta hedging a cheap option with delta calculated from implied volatility (as volatility is unknown) is always profitable, but profits are spot dependent
Large profit
Source: Santander Investment Bolsa.
If implied volatility = realised volatility, profits are path independent
If the implied volatility is equal to the realised volatility, then the estimated delta calculated from the implied will be equal to the actual delta (calculated from the realised). In this case, profits from hedging will exactly match the theta cost for all paths, so it is path independent. Figure 73. Profit (or Loss) from Continuously Delta Hedging Unknown Volatility
-50 -40 -30 -20 -10 0 10 20 30 40 50 -10 -8 -6 -4 -2 0 2 4 6 8 10
Realised vol - implied vol (%) P&L (%)
Average profit Profit +/- 1σ
When realised volatility = implied volatility, delta from implied volatility is correct hence profit is always 0
Source: Santander Investment Bolsa. If volatility is
unknown, the correct delta cannot be calculated
With continuous hedging, buying a cheap option is always profitable
If there is a difference between the actual delta and estimated delta, there is market risk but not enough to make a cheap option unprofitable (or an expensive option profitable). This is because in each infinitesimally small amount of time a cheap option will always reveal a profit from delta hedging (net of theta), although the magnitude of this profit is uncertain. The greater the difference between implied and realised, the greater the market risk and the larger the potential variation in profit.
DISCRETELYDELTAHEDGINGWITHKNOWNVOLATILITY
While assuming continuous delta hedging is mathematically convenient, it is impossible in practice. Issues such as the cost of trading and minimum trading size (even if this is one share) make continuous trading impossible, as do fundamental reasons, such as trading hours (if you cannot trade 24 hours then it is impossible to trade overnight and prices can jump between the close of one day and start of another) and weekends.
Discrete hedging errors can be reduced by increasing the frequency of hedging
The more frequent the discrete hedging, the less variation in the returns. If 24-hour trading were possible, then with an infinite frequency of hedging with known volatility the returns converge to the same case as continuous hedging with known volatility (ie, Black-Scholes). In order to show how the frequency of hedging can affect the payout of delta hedging, we shall examine hedging for every 5% and 10% move in spot.
Figure 74. Profit from Discrete Delta Hedging with Different Frequencies
-1 0 1 2 3 4 5 90% 95% 100% 105% 110%
Initial delta hedged straddle Straddle rehedged after +5% move
Changing hedging frequency has changed the profit made
Source: Santander Investment Bolsa.
Hedging every 5% move in spot
If an investor delta hedges every 5% move in spot, then an identical profit is earned if the underlying rises 10% as if the underlying rises 5% and then returns to its starting point. This shows that the hedging frequency should ideally be frequent enough to capture the major turning points of an underlying.
Discrete delta hedging adds noise to returns
Hedging every 10% move in spot
If the investor is hedged for every 10% move in the underlying, then no profit will be earned if the underlying rises 5% and then returns to its starting point. However, if the underlying rises 10%, a far larger profit will be earned than if the position was hedged every 5%. This shows that in trending markets it is more profitable to let positions run than to re-hedge them frequently.
Hedging error is independent of average profitability of trade
As the volatility of the underlying is known, there is no error due to the calculation of delta. As the only variation introduced is essentially ‘noise’, the size of this noise, or variation, is independent from the average profitability (or difference between realised vol and implied vol) of the trade.
Figure 75. Profit (or Loss) from Discrete Delta Hedging Known Volatility
-50 -40 -30 -20 -10 0 10 20 30 40 50 -10 -5 0 5 10
Realised vol - implied vol (%) P&L (%)
Average profit Profit +/- 1σ Profit +/- 1σ with 4x frequency
Hedging a 4x frequency halves the noise from discrete delta hedging
Source: Santander Investment Bolsa.
With discrete hedging, cheap options can lose money
With continuous delta hedging (with known or unknown volatility) it is impossible to lose money on a cheap option (an option whose implied volatility is less than the realised volatility over its life). However, as the error from discrete hedging is independent from the profitability of the trade, it is possible to lose money on a cheap option (and make money on an expensive option).
Hedging error is halved if frequency of hedging increased by factor of four
The size of the hedging error can be reduced by increasing the frequency of hedging. An approximation (shown below) is that if the frequency of hedging is increased by a factor of four, the hedging error term halves. This rule of thumb breaks down for very high-frequency hedging, as no frequency of hedging can eliminate the noise from non-24x7 trading (it will always have noise, due to the movement in share prices from one day’s close to the next day’s open).
N
vega
L P4
&π
σ
σ
≈
×
×
where N is the number of times position is hedged in a year Noise fromdiscrete delta hedging is independent of how cheap the option is