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Delta-Hedged Option P&L, Volatility Risk, and Market-Maker Spreads

Article Quant Q&A · Author: Ouissem
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Summary

The document asks how traders and banks can earn money while selling options and maintaining delta hedges. Its answers distinguish the costs and economics of hedging from the market maker’s sources of revenue. Maintaining a hedge exposes the seller to costs tied to realized underlying volatility, as well as bid-ask spreads and market impact. Market makers may charge option prices that reflect implied volatility above their estimate of actual volatility to cover costs and seek a return.

A second explanation emphasizes the bid-ask spread: customers trade at the market maker’s bid or ask, and the price difference contributes to profit. Delta hedging helps reduce inventory risk, but market makers typically manage net exposure across their whole book rather than hedge each option separately. Offsetting positions can therefore reduce trading needs. The discussion is qualitative; it does not quantify hedge losses, spread revenue, or the conditions under which a hedge is perfect.

Key ideas

  • Delta hedging can incur costs related to realized volatility, bid-ask spreads, and market impact.
  • Market makers may price options to compensate for hedge costs and earn a return.
  • The bid-ask spread is a direct source of market-maker revenue.
  • Hedging reduces inventory exposure, while offsetting positions across a book can reduce required trades.
  • The explanations are qualitative and do not establish that a perfect hedge guarantees a particular profit.

Tags

OptionsVolatilityExecutionRisk management
Full text
# Pnl on delta hedged option


# Pnl on delta hedged option












When we sell an option and we hedge it using Delta, we replicate the option payoff until maturity according to its Delta. If we replicate the option perfectly and with high frequency, we should be able to pay just the payoff at the end if it is exercised, otherwise we end with zero PnL at maturity.

How traders and banks makes their PnL if the hedge is perfect ?

## Answer by AlRacoon (score 1)

https://quant.stackexchange.com/a/78016

The phenomenon you describe is the cost of maintaining the delta hedge due to the actual volatility of the underlying (other costs include bid-ask spread, market impact etc.) To compensate for the costs of delta hedging and to make some money to make a market in the option, the market maker charges more for the option than the option value. Hence the "implied volatility" (the volatility implied by the price of the option, all other inputs constant) tendency to being greater than the actual volatility.

Further, the market maker hedges their net delta of their full book, rather than each and every option. Should there be offsetting deltas from being long/short and puts vs calls, etc., the market maker does not have to trade the delta of each individual option, and therefore reduces their overall delta hedging costs.

## Answer by nbbo2 (score 0)

https://quant.stackexchange.com/a/78014

At all times the market maker posts 2 prices: the bid and the ask (which is higher than the bid). When someone sells to the marketmaker, they receive the bid price, when someone wants to buy they have to pay the ask price. So the market maker earns the spread between these two prices, just like a street vendor sells you an apple for a higher price than he paid at the wholesale fruit market - that is his profit.

The hedging makes it possible to eliminate the "inventory risk" i.e. the danger of price changes between the time the marketmaker buys from X and the time he sells to Y. But the profit comes (mostly) from the price difference.

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.