## Wednesday, December 19, 2012

### Option delta off expected volatility (in Excel)

I think many of us have used the default delta values off implied volatility, and noticed that it can be off significantly when implied is way out of line against realized vol. This error makes the P&L of individual trades path dependent, and frequently hurt long volatility trades as delta becomes understated. A way reduce this error is to input realized vol for delta calculation, or 1 step better -> Expected Realized Vol.

The increased hedging accuracy would lower the volatility of individual trade returns, path dependance. It all goes toward achieving that positive EV:

EV = BSM(Realized Vol) - BSM(Implied) - transaction/hedging costs - implementation shortfalls
Where BSM() = Black Scholes Merton option valuation model.

Deriving delta

S -> Underlying value
K -> Strike price
r -> “Risk free” rate of return (often off 10year treasury)
V -> Expected realized volatility
T -> Option life with respect to year

We need to solve for d1 from the Black Scholes Merton model,

Then we can derive the “correct deltas”,
delta(Call) = N(d1)
delta(Put) = 1 – N(d1)
where N() = Position within the Standard Normal Distribution

Doing it in excel
So I used the latest SPY values for the example, and an arbitrary interest rate of 0.5% and realized vol of 12% (I didn't bother doing a forecast value since that's already covered in the earlier post).

Formulas:

We end up with