Volatility by Adam S. Iqbal

Author:Adam S. Iqbal
Language: eng
Format: epub
ISBN: 9781119501688
Publisher: Wiley
Published: 2018-10-02T00:00:00+00:00

FIGURE 4.2 The figure is identical to Figure 4.1, apart from between Friday and Monday spot is simulated lower to 1.3600 rather than higher.

Her PnL explain is therefore

(4.10)

The important point to note here is that by delta hedging, the trader's PnL is 9 thousand USD regardless of whether the call option expired ITM with spot following the path from Figure 4.1, or OTM with spot following the path from Figure 4.2. Had she not delta hedged, her PnL would have been +739 thousand USD if the option expired ITM at 1.38501 or USD had the option expired OTM.

The reader is correct to suspect that I chose the ending spot level of 1.3600 in Figure 4.2 so that the delta hedged case PnL would be 9 thousand USD, equal to that in the case where spot went higher to 1.3850. There are clearly many simulations in which the PnL would be significantly different to 9 thousand USD. However, I chose this example to illustrate that an important feature of delta hedging is to decrease PnL variance. We will see in the BSM model in Chapters 9 and 10 that delta hedging in continuous time is able to diminish the variance of the PnL to zero. In practice, delta hedging dramatically decreases PnL variance, but because implied volatility is itself random and since continuous hedging is not practical, delta hedging cannot decrease this variance all the way to zero.

We saw in Chapter 1 in the context of an ATMS option that delta hedging leaves the trader exposed to volatility only. This is consistent with the arguments in this subsection. PnL variance relating to an option is derived from at least two sources: spot moving and implied volatility changing. By delta hedging, one is able to remove the former and so the overall PnL variance diminishes.

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