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Delta Hedging, Fair Option Prices, and Dealer Markups

Article Quant Q&A · Author: Victor
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Summary

The discussion explains why continuously or frequently delta hedging a fairly priced option does not, by itself, create expected profit. For a short call, the hedge offsets directional exposure while leaving the trader exposed to the relationship between realized volatility and the option’s implied volatility. Under the stated Black–Scholes assumptions, with volatility unchanged and gamma hedged consistently, hedge losses can offset the premium received, producing zero profit on average.

The second answer describes how a dealer can earn a markup by selling an option above its theoretical value and hedging the position, provided hedging costs stay below that markup. Dealers may also manage risk across a portfolio, offsetting positions or reselling inventory before hedging fully. The explanation is conceptual: it does not quantify discrete hedging costs, transaction costs, model error, volatility changes, or the risks of holding an unhedged position. A lower option price later does not alone establish a locked-in profit, since future hedge costs and remaining risks still matter.

Key ideas

  • Delta hedging offsets directional exposure but does not guarantee profit.
  • Under the stated fair-pricing assumptions, frequent delta hedging can leave expected profit near zero.
  • A dealer can seek profit when an option’s sale markup exceeds the cost of hedging.
  • Portfolio offsets and inventory management can reduce how much a dealer must hedge externally.
  • Changes in volatility, discrete hedging, and trading costs can affect realized profit.

Tags

OptionsDerivatives pricingVolatilityMarket makingRisk management
Full text
# When should we delta hedge?


# When should we delta hedge?












Let's say I'm the seller of a European call option on a non-dividend paying stock. I pocket the premium $c_0$ of the call at $t=0$.

If I start to delta-hedge right away, this is equivalent to replicating the call and the cost of the strategy will converge to the price of the option $c_0$ if the option is priced fairly (according to the Black-Scholes formula) and if the delta-hedging is done sufficiently frequently.

So, it seems unless at $t=0$ the option is priced higher than its theoretical price, delta-hedging right away leads to an average null profit. Am I correct? So what are the precise scenarios when we should use delta-hedging?

For example, if at a time $t$, the call price $c_t$ is below the price at $t=0$ ($c_t < c_0$), the delta-hedging strategy will cost in average $c_t$. Does it mean I will lock an average positive profit?

## Answer by HowtoETF101 (score 9)

https://quant.stackexchange.com/a/45368

By delta hedging you are saying that you have a view on the path and the volatility of the option you are trading, but not on its direction; in your case, that being short delta.

From a theoretical perspective, all options are priced fairly and not delta hedging simply increase the variance of your payouts.

In your example, selling a call and delta hedging, then, at time T, if the volatility of the call stays constant and the option's gamma is "consistently" hedged, then it means that you will lose an amount on your delta hedge equal in value to the premium you received for your option, so PNL = 0.

## Answer by AlRacoon (score 4)

https://quant.stackexchange.com/a/45376

As @Alex C mentioned, this is how banks make money. They will sell/buy you an instrument with a markup and hedge their position. Over the life of the trade, they will adjust the hedge. If the costs of hedging over the life of the trade are less than the markup, they will profit. Of course the banks, as market makers, will have a portfolio of positions and they can efficiently hedge a number of trades as the risks of each individual trade will be offset by other positions on the book. Also, they will use this "inventory" of positions to buy and sell to clients such that they will not have to hedge the position over the entire life--only over their holding period.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.