A model-free approach to delta hedging

2.3 A variant

When taking into account variants like the cost of carry for commodities options
(see, e.g., [27]), replace Equation (3) by

dΠtrend ⁼ dVtrend ^- ∆dStrend + q∆Strenddt

where qSdt is the amount required during a short time interval dt to finance the
holding. Combining the above equation with

Vtrend - r∏trend(0) (exp ft r(τ)dτj

The derivation of the initial conditions ∆(0) and Πtrend(0) remains unaltered.

3 Numerical simulations

3.1 Two examples of delta hedging

Take two derivative prices: one put (CFU9PY3500) and one call (CFU9CY3500).
The underlying asset is the CAC 40. Figures 1-(a), 1-(b) and 1-(c) display the
daily closing data. We focus on the 223 days before September 18^th, 2009.
Figures 2-(a) and 2-(b) (resp. 3-(a) and 3-(b)) present the stock prices and
the derivative prices during this period, as well as their corresponding trends.
Figure 3-(c) shows the daily evolution of the risk-free interest rate, which yields
the tracking ob jective. The control variable ∆ is plotted in Figure 3-(d).

3.2 Abrupt changes

3.2.1 Forecasts

We assume that an abrupt change, i.e., a jump, is preceded by “unusual” fluc-
tuations around the trend, and further develop techniques from [13], and from
[11, 12]. In Figure 4-(a), which displays forecasts of abrupt changes, the symbols
o indicate if the jump is upward or downward.

3.2.2 Dynamic hedging

Taking advantage of the above forecasts allows to avoid the risk-free tracking
strategy (5), which would imply too strong variations of ∆ and cause some type
of market illiquidity. The Figures 4-(b,c,d) show some preliminary attempts,
where other less “violent” open-loop tracking controls have been selected.

Remark 3.1 Numerous types of dynamic hedging have been suggested in the lit-
erature in the presence of jumps (see, e.g., [5, 22, 27] and the reference therein).
Remember [7] moreover the well known lack of robustness of the BSM setting
with jumps.

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