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# Theta Question

Discussion in 'Options' started by trader_XX, May 21, 2006.

Hey guys, i'm a newb trying to understand the basics of black-scholes/the greeks.

My question involves theta. According to black-scholes, if I sell a call option and then I dynamic hedge it putting out my own capital, I should be earning a return of e^(rt)-1 on the capital that i've used.

I'm assuming that this return is somehow reflected by a decrease in the value of the call option i.e. my liabilities have decreased by the amount that I should be gaining. I'm assuming this because even if the underlying doesn't move, I should still make the same return and therefore the resk of my portfolio is exactly the same except for the call option that i'm long, so my call option must have decreased in value.

The problem that I'm seeing with this is that the equations in the text book for Theta don't seem to reinforce this reasoning. I would assume that there should be a -S*delta*r*(e(rt)-1) in the equation for theta. In the books:

Theta = -So*N'(d1)*sigma/2sqrt(T) - r*k*e^(-rt)*N(d2)

Where is my reasoning flawed and where in the black scholes equations is this fact represented?

Also, we know from black scholes that the underlying will inherently have a time value associated with it (i.e. the probability of the option finishing in the money is decreased with time). Doesn't this lead to a greater than r expected return on the trader's capital?

For example, imagine the trader writes the call, and delta hedges it. The stock stays constant, but the time value of the option keeps decreasing as time passes. At an point hte trader can buy a call for cheaper, close out his position and have earned greater than the market return (because the option is losing time value which will cause it to decay faster than it should decay to give the trader the expected market return r).

If anyone has any insight please let mek now.

Thanks