Delta Hedging Simulation (Equity)

The criteria for this simulation requires that I find a mispriced option contract due to my interpretation of what the implied volatility should be.  In order to forecast volatility I will use my experience with fluctuating premiums and directional moves to calculate an assumed future volatility.

I plan on buying the option and selling the underlying so for the model I have to find a stock, determine an expiration date, assume an interest rate, and define an adjustment interval to balance the fluctuating deltas.  Since this example is based on equity which is subject to stock-type settlement, the interest rate will be used to calculate only the carrying costs for the initial outlay.  If futures were used then the fluctuation of cash as a result of the MTM treatment would debit or credit variation costs calculated with the interest rate.

CME is a good candidate for this simulation because it historically has had rapid price movement due to political fundamentals relating to legislation which if passed would require OTC swaps to be cleared through a credible exchange.  The two leading candidates for this are CME and ICE.

Backtesting of this strategy can be found in the following links:
Link to CME 2010 Backtesting CALLS
Link to CME 2010 Backtesting PUTS

Stock CME Closed at 310.84 on 1/11/11
Strike 310
Expiration Mar-11 66 days from 1/11/11
Impl Vol 26.95%
Forecast Vol 31% 3/18/11
Delta 0.52 1/11/11
Interest Rate 8% Theoretical
Adj Interval Weekly

Buy 10 CME MAR 11 310 CALLS @ 14.60
Sell 520 shares CME @ 310.84

Risk profile of position

Week Stock Price Delta Tot Delta Pos Adj Tot Adj
0 310.84 0.52 0 0 0

The forecast volatility of 31% will stay constant throughout the simulation.  With a volatility of 31% the premium of the MAR 11 CALLS is 18.04.  At the current levels the volatility implies 15.47.

Delta Hedging

The point of delta hedging is to profit from undervalued or overvalued options while limiting risk of moves against the position in the short-run.  The purchase or sale of a mispriced option requires a hedge in the underlying contract in the amount of the option delta to remain delta neutral and then it is adjusted at regular intervals as the deltas fluctuate.  This ensures that the increase or decrease of the option position is offset exactly by the opposing position of the underlying and because of this there is no upward or downward bias in the underlying market.

An option that is mispriced has the wrong implied volatility due to incorrect market consensus assumptions (i.e. if implied volatility is too low then option premium will be too cheap).  Delta hedging is most profitable when there is a big difference in the future volatility and the implied volatility.  Plugging your forecast volatility assumption into the Black-Scholes Model you can find the premium at which the option should be trading.  If the forecast volatility assumption is accurate then the difference in premium calculated using the forecast and implied volatility will approximate the profit when implied volatility reaches the forecast volatility or at expiration (whichever comes first).

There are four components that make up the profit and loss from this strategy.

  1. Original Hedge – The net of the liquidated option position and the opposing position in the underlying.
  2. Adjustments – The adjustments made to the hedge to remain delta neutral.  The beauty of delta adjustments is that since it depends on the deltas it forces you to buy low and sell high.
  3. Option Carrying Costs – Depending on the interest rate plugged into the model and whether both components of the position are subject to stock-type settlement there will be a debit for the initial cash outlay.
  4. Variation Costs – If the underlying is subject to futures-type settlement then depending on the brokerage firm, the fluctuation of cash going in and out of the account will earn interest or incur an interest expense.

At expiration, when the options expire the position is closed in three ways:

  1. Letting OTM options expire worthless
  2. Selling ITM options (if you’re long the options) for their intrinsic value (parity) or exercising them to offset the underlying position
  3. Liquidating the underlying at market price

This explanation is theoretical and doesn’t take into consideration external factors such as liquidity, different interest rates of return to traders on variation costs, commission paid, and tax implications.