Optimizing Dynamic Hedging Strategies for Short Options

Volatility Review: Russells Weekly Rundown

• VIX Options: A moderate volume week.
  • M: 266K
  • T: 400K
  • W: 572K
  • Th: 614K
  • Total 7.45m (5.73m Calls, 1.72m Puts)
• Crude Oil: OIV/OVX - 46, almost touched 80 back in mid-February.
• Gold: GVZ 19.32 - Well off recent highs.

Volatility Voicemail: One giant listener question.

Question from Jeremy - Hi Guys, This show is a great resource and has taught me a lot throughout the years. I was listening to the most recent show on March 11th "Uncovered Covered Calls and IRA Options Debate" and I wanted to write in regarding the discussion on dynamic hedging of a short option position. First, let me quickly introduce myself to provide some context.

My name is Jeremy and I am in the final year of my PhD in computer engineering with a MS in applied mathematics and statistics and a BS in math. I have spent the past 4+ years researching for the NASA Ames Research Center. My PhD research focuses on the modeling, simulation and analysis of stochastic processes and the application of optimization algorithms to the air traffic control system. Here is a link to my LinkedIn to verify my academic record and list of publications.

In addition to my PhD research, I am working on starting a fund with two other academics. The fund specializes in delta neutral trading and has developed two tools to execute volatility premium harvesting strategies. I provide a short description of the tools below. In particular, the second tool is directly related to the dynamic hedging of a short option position and the discussion on the previous show.

1) Our stochastic volatility tool estimates volatility using high-frequency data, such that the microstructure noise is directly accounted for. The tool uses a Bayesian MCMC algorithm to provide estimates of the volatility where estimates are consistent across all sampling time scales. Moreover, the tool provides samples of arbitrarily large number of volatility paths though time.

2) Our stochastic optimal control tool to hedge a short option position (or a combination of short option positions). After the option is sold, the risk can be controlled by effecient delta hedging. This can be done by calculating the optimal price boundaries to trigger a hedge in the underlying asset. Moreover, these boundaries are informed by the dynamics of the volatility as calculated from our stochastic volatility tool.

The strategy that the Greasy Meatball provided on the show to hedge the short option at the close of every trading day is a good start. The problem with the strategy is that this approach is a sub-optimal timing mechanism. Many days you will hedge at the close when you don't need to, leading to increased trasaction costs. Instead, this problem should be formulated as an optimization problem that we can solve to provide the best strategy.

A little background on the academic literature. In an ideal world under the Black Scholes model with no transaction costs the options can be hedged perfectly. In the presence of transaction costs, a continuous hedging strategy is prohibitively expensive. Hence, it is impossible to perfectly replicate the option in this setting when there are transaction costs and, as a result, trading in an option involves an essential element of risk.

How to optimize with respect to this risk and uncertainty has been a topic of research throughout the economic and control literature. There have been many different strategies throughout the years. Briefly I will present one strategy which minimizes the standard deviation of hedging error when compared to many, if not all, of the other strategies.

The attached paper was published in 2009 in the Journal of Economics Dynamics and Control by a Professor of Statistics at Stanford. As can be found in the paper, the problem of option hedging in the presence of proportional transaction costs can be formulated as a singular stochastic control problem. The authors approach is based on minimization of a Black–Scholes-type measure of pathwise risk, defined in terms of a market delta, subject to an upper bound on the hedging cost. The approach can also be applied to solve the problem of maximizing the investors utility at the terminal time of the option contract.

The main idea of the attached paper is to calculate price boundaries that trigger a hedge in the stock. Moreover, the number of shares that should be held when we reach the boundary is also calculated. Attached to this email is a screen shot from the paper that illustrates these price boundaries. Without getting into too many details, the top figure illustrates the price boundaries for an option with "1 unit of time" left (think 1 month). And the bottom figure illustrates the price boundaries for an option with "0.25 units of time" left (think 0.25 months). As can be seen in the figure, the shape of these boundaries changes with respect to how much time is left to expiration.

This email is most likely too long so I will stop my explanation here, but I would love to continue this discussion with you guys. If you are interested in learning more about the stochastic optimal control or the stochastic volatility tools me and my fund have developed please ask. In particular, I can provide more detail on our stochastic control and how it is an enhancement of the control that is developed in the attached paper. The key difference being that their tool assumes a constant volatility over the time horizon, whereas our tool allows for the volatility to change as a function of time.

I hope to hear your thoughts and get any feedback you may have! Best, Jeremy

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