Delta hedging frequency for plain vanilla European options under trading costs

Question

I am looking for methods to select points in time when delta hedging plain vanilla European options under trading costs.

It is easy to come up with ad hoc ideas such as

1. Time-based: for example at fixed time intervals.
2. Price-based: hedge when the price has moved a more than a given percentage.
3. Delta-based: hedge when the option delta has reached a given threshold.

How is the delta hedging frequency choosen in practice? Could you point out more sophisticated methods in the literature? Would Taleb's book on dynamic delta hedging be a good investment?

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Edit: What I have found so far:

• Option Pricing and Replication with Transaction Costs, Leland, Journal of Finance, 1985.
  He disscusses rehedging at fixed regular intervals.
• Hedging of Option Portfolios and Options on Several Assets with Transaction Costs and Nonlinear Partial Differential Equations, V. Zakamouline, 2008.
  Extends Leland's approach to other options, especially path-dependent options.

Answer 1

First when transaction costs are involved the trader has to make a tradeoff between return and risk. Continuous rebalancing/hedging could lead to infinite transaction costs but provides (in theory) a perfect hedge. Discrete hedging enables to minimize transaction cost but leads to hedging errors and more risk. To find a price one must introduce an optimization criteria (e.g. a utility function)

As already mentioned it is hard to name "the one approach" that is done in practice. I will thus refer you to some texts to get you started on the topic. Banks etc. will develop their own methods using results from research/academia. Good methods equal good money - so people will not readily part with their secrets. Knowing the theory and the reasoning behing some of the approaches will, however, enable you to make your own condlusions.

Some interesting papers/results on the topic:

• You have already found: Option pricing and replication with transaction costs by Leland It is one of the first papers an the topic and a fairly easy read for the hedge is done in a B&S setting.
• Unfortunately the results derived by the authors do not always hold. Thus you should read the following paper On Leland’s Option Hedging Strategy with Transaction Costs for a more comprehensive understanding of the strategy suggested by Leland.
• Another approach is presented by Clewlow and Hodges in Optimal Delta-Hedging under transaction costs.
• Option pricing with transaction costs and a nonlinear Black-Scholes equation

For a good overview I also suggest the following Phd-Thesis - e.g. check the Bibliography.

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Some general remarks: A method must not be sophisticated to work well. There are different types of hedging approaches. Some traders might use algorithms other might trust their gut.

You should also be aware that the hedge itself strongly depends on the type of product and on the assumptions you made on the model/market. A hedge of a plain vanilla call will differ depending on the model (e.g. B&S vs. Heston) An American option is hedged differently than a European one. The model assumptions on the other hand depend on the type of product you want to price/hedge.

Also there is often a big gap between what is written in books and done in practice. The productive solutions are frequently a mixture of different approaches presented in different sources.

Some quant stack exchange topics that you might find interesting/relevant:

• Pre-trade evaluation and risk assessment of option trading strategies (in market practice) (this question does not explicitly deal with delta hedging but is interesting/relevant from a "practical" perspective)
• When does delta hedging result in more risk?

Answer 2

You can formulate the problem of finding the optimal hedging frequency under transaction costs to maximize the Sharpe ratio of the delta-hedging strategy. The risk is the P&L volatility and the reward is smaller transaction costs. Analytical solutions and similar analysis can also be applied to spot- and delta-based hedging strategies.

Details in http://ssrn.com/abstract=1865998 When You Hedge Discretely: Optimization of Sharpe Ratio for Delta-Hedging Strategy under Discrete Hedging and Transaction Costs

Answer 3

At the end of the day, any hedging strategy is a strategy, which means it comes with basic measures of performance like pnl, variance, sharpe etc... You can either look at the metrics purely for the hedging part (as a delta-1 strategy) or at the metrics for the combined portfolio (option + hedge).

Now for each type of utility function (total pnl or total variance say but you could choose others) that you choose to decide between those strategies you can formulate a mathematical model of stochastic control (with the hedges acting as the control) and try to get the optimal hedge either analytically or numerically. Alternatively you can be more empirical and use a simulation approach to fine tune the parameters that work best in your practical situation depending on what is at your disposal (in terms of capital, alpha signals or other aspects that can allow you to trade more efficiently).

Sorry for not being super specific but that's really a quite general question for which there is not a single best answer imho.