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Correcting the Financing Sign in a Delta-Hedged Call P&L

Article Quant Q&A · Author: Easting
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Summary

The question examines the interest term in a derivation of the value change for a call option hedged by shorting delta shares. The reader reasons that buying the call requires borrowing its price, while short-sale proceeds from the shares are lent, implying net interest income of r times the share proceeds minus the call cost. This is compared with the opposite sign shown in the book's formula.

The accepted response confirms that the reader's sign is correct and says the derivation was corrected in the book's second edition. This resolves the specific financing-sign discrepancy, but the excerpt does not reproduce the revised formula or walk through the complete derivation. Readers applying the result should check the corrected edition and keep track of the cash account implied by their option and hedge positions.

Key ideas

  • A delta-hedged call position combines a long call with a short position in delta shares.
  • The question identifies a sign issue in the interest earned or paid on the position's net financing.
  • The accepted answer says the reader's sign is correct and the book corrected it in its second edition.
  • The excerpt does not provide the full revised derivation.

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OptionsDerivatives pricing
Full text
# Formula 1.2 in book "Volatility Trading" by Euan Sinclair


# Formula 1.2 in book "Volatility Trading" by Euan Sinclair












I am reading "Volatility Trading" by Euan Sinclair. In the derivation of BSM process in Chapter 1, formula 1.2 confused me.

It means that the value change of a hedged call position (1 Call $C_t$ and $\Delta$ short stocks).

The last term from author denotes the interest income, which is from the financing to build the position for the hedged option.

However, in my understanding, to build a position of a hedged call, you borrow money of $C(S_t)$ to buy the Call option, then short Delta stocks and lending the proceedings of $\Delta S_t$ which makes you receive $r(\Delta S_t-C)$ interest income instead of $r(C-\Delta S_t)$.

Any one can explain me which one should be correct?

## Answer by Hans-Peter Schrei (score 3)

https://quant.stackexchange.com/a/75396

Your understanding is correct, this has been corrected in the second edition on Page 3:

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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