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Black–Scholes Delta Hedging and the Cost of Continuous Rebalancing

Article Quant Q&A · Author: noob-mathematician
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Summary

The document raises two questions about dynamic delta hedging of a short European call in the Black–Scholes framework. It asks why, when rebalancing is frequent and transaction costs are absent, the discounted cash left after hedging and settling the option should match the option’s initial Black–Scholes value. It also asks how a trading desk obtains the delta used at each hedge adjustment, including whether it is calculated numerically.

The text provides the setup and the questions but no answers, proof, references, or numerical evidence. Its scope is therefore introductory: it identifies the replication intuition behind delta hedging and the practical need to estimate or obtain option sensitivities. The claimed equality depends on the idealized Black–Scholes assumptions, including continuous or sufficiently frequent trading and no transaction costs; actual hedging can diverge because rebalancing is discrete and market conditions and costs are not ideal.

Key ideas

  • A short call can be hedged by adjusting the underlying position as its delta changes.
  • The document asks why discounted hedging cash should equal the initial Black–Scholes option value under ideal assumptions.
  • It also asks how desks calculate or obtain delta at each hedge adjustment.
  • The document offers no proof or implementation details, and the stated relationship assumes no transaction costs and frequent rebalancing.

Tags

OptionsDerivatives pricingRisk management
Full text
# Hedging costs and BS-price


# Hedging costs and BS-price












I'm looking at the chapter, "The Greek Letters" in Hull's book (Options and derivatives...) and in particular the paragraph "Dynamic Aspects of Delta Hedging". He demonstrates two examples of how dynamic hedging works (within the Black-Scholes framework), for a short position on a European call option with expiry 2 weeks.

In particular, after shorting a call option, given that delta (assume that it can be computed somehow) changes over time, the seller should buy or sell stocks (might need to borrow money from some cash account) in order to neutralise the change in the value of the option due to changes in the value of the stock (delta neutral). During this hedging lifecycle, some rebalancing/hedging costs accumulate (occurring from borrowing money to buy extra stocks while hedging at discrete times) till the expiry of the option where the seller closes his position with the buyer.

Question 1:

The author claims that, if there are no transaction costs and the rebalancing is frequent, then the Discounted accumulated hedging cost at expiry (which appears as the cash remaining "to return to the bank" after having delivered the payoff and sold the owned stocks) should be equal to the Black-scholes price of the call option at inception. Anyone can (mathematically) prove why is that or provide reference?

Question 2: (for sanity)

How does a trading desk/seller knows what the delta of the option is every time they hedge the option? Do they approximate numerically using Black-Scholes? If yes, which approximation scheme of the derivative is used?

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.