How Interest Carry Relates to Black–Scholes Option Theta

Summary

The document asks whether standard Black–Scholes–Merton (BSM) theta includes the cost of financing an option premium. It quotes a discussion from Taleb’s Dynamic Hedging that distinguishes theta from the trader’s financing carry: the option’s present value reflects interest rates, while a self-financing trader’s cash account accrues or pays interest. With consistent financing assumptions, the option’s price change and funding cash flows offset, so the convention described excludes premium financing costs from theta.

The document raises the issue but does not provide an answer or derive the BSM formula. The general distinction is between the model’s sensitivity to the passage of time, holding other inputs fixed, and the total profit or loss of a funded position, which also depends on how the premium is financed. Exact treatment depends on the pricing convention and whether theta is reported with or without carry adjustments; the quoted passage alone does not settle every convention used by practitioners.

Key ideas

• Theta conventions may distinguish option time decay from the financing cost of holding the premium.
• BSM pricing reflects interest rates through discounting and the underlying’s carry assumptions.
• A self-financing position accounts for interest earned or paid in its cash balance.
• Realized position P&L can differ from quoted theta when financing cash flows are included.

Tags

# breaking down different theta costs


# breaking down different theta costs












Reading through Taleb's Dynamic Hedging, when I came across this part:

> Theta, Interest Carry, and Self-Financing Strategies Traders eliminate the interest costs of holding the premium to compute the theta because the carry of an option should be neutral to a trader who funds himself. In other words, if the trader incurs carry costs, the price of the theta will be increased by the interest paid on the premium, making it totally neutral. Thus, if interest rates are 20%, theta will be lower by 20% of the total premium (the option will have a lower price because of the present-value effect) but the carry costs of holding the option will offset these savings. It is assumed that the trader has borrowed the money to buy the option and that he would pay the difference in higher interest. Many traders erroneously factor the premium costs in the theta computation. As a convention, in this book, theta includes no premium costs. In the derivation of an option value through a self-financing strategy, the seller of the option is supposed to buy an interest-yielding instrument. This is equivalent to the option trader funding himself from his firm by paying for negative balances and earning interest on positive ones.

I'm a bit confused. Does the standard BSM formula for theta have the interest costs for holding the premium or not? If so, what exactly is being removed? Would appreciate some general explanation of what he's talking about, thanks.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.