Why Discrete Price Jumps Limit Delta Hedging

Summary

The document addresses why an at-the-money call’s familiar delta of about one half does not match the full payoff change in a large, single-step move. Delta is a local sensitivity: in the theoretical continuous-time setting, it approximates how an option’s value responds to a sufficiently small change in the underlying. The explanation relies on small time steps and assumes the underlying has no jump over the interval.

In practice, hedges are adjusted at discrete times, so they capture only an approximation of the option’s changing exposure between adjustments. The example considers a call at the strike that expires after one period, with the underlying making a sizable move; over that interval the payoff change can differ sharply from the initial delta prediction. This illustrates a limitation of applying a local hedge ratio to a large discrete move, rather than invalidating delta’s role for small changes. No quantitative error analysis or alternative hedging method is given.

Key ideas

• Delta describes a local option-value sensitivity, not necessarily the payoff change across a large move.
• Continuous-time delta hedging assumes infinitesimal adjustments and, in this explanation, no jumps.
• Practical hedging takes place at discrete intervals, making delta an approximation.
• A large move over one interval can leave a discrete hedge far from the intended offset.

Tags

# Calculating deltas of call options?


# Calculating deltas of call options?












From a continuous standpoint, I understand why an ATM call has delta = 0.5 and for ITM call, the delta approaches 1 since each move in the underlying corresponds to same unit of value change in call option.

Howwever, if we take a discrete case where call option expires in 1 period. If strike price is 100 and the underlying is at 100. If the underlying moves to 105 in 1 period, then dS = 5 and the change in call option is also 5? So then this ratio of $\frac{dc}{dS} = 1$ not 0.5?

## Answer by Richi Wa (score 3)

https://quant.stackexchange.com/a/9895

The theoretical idea of Delta hedging is placed in a setting of infinitesimally small time steps. In such small time steps and if no jumps occur (eg. in the diffusion case) the underlying can not move that much and Delta makes sense (at least theoretically). In continuous time things work out.

As in practice (or in simulations) the Delta hedge can only be done in small discrete time steps (not continuously) and thus only captures a part of the change in value, it is merely an approximation.

In your case the time step is discrete and the move is too large. Thus the Delta Hedge does not work at all.

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