Interpreting a Delta-Hedged Call's Value at Expiration
Summary
The document clarifies a call option example in which a long call is hedged by shorting a quantity of the underlying stock. At expiration, the call is worth its intrinsic value, while the short stock position is valued using the ending share price. That ending value is not the hedge's profit by itself: the calculation must also account for the cash received when the shares were initially sold short.
The example separates the short position's change in value from its ending market value, showing how the initial short-sale proceeds affect the combined result. Before expiration, the option's value must include remaining time value rather than just intrinsic value. The discussion is a simplified illustration; it notes that borrowing costs and returns on cash can also affect realized profit, and it does not develop a full dynamic hedging or transaction-cost model.
Key ideas
- A long call can be delta-hedged by shorting shares of the underlying.
- At expiration, the short hedge's market value uses the ending stock price.
- To calculate the portfolio result, include the initial cash received from shorting the shares.
- Before expiration, an option valuation must account for remaining time value.
- Borrowing costs and returns on short-sale proceeds can change the realized result.
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# Formula 4.10 in Volatility Trading by Sinclair # Formula 4.10 in Volatility Trading by Sinclair I'm reading "Volatility Trading" by Sinclair and am confused about formula 4.10. I hope someone of you can enlighten me :) What he's saying there is that he purchases a call option and wants to hedge it with a quantity `h`: ``` Value of portfolio = C - h * S ``` So the value of that portfolio is the call option minus the hedge of the current stock price. He offers some examples. E.g. in formula 4.2 he purchases an ATM call at 100 and then the stock rises to 101. The value of the portfolio is ``` C - h * S = Max(101-100,0) - hS = 1 - h * 101 ``` I don't understand that second term. Is it the cost for the hedge? But then why is it not the cost of the initial stock price, but the current one? And he states earlier he sells a hedge, not purchases one. I'm confused. ## Answer by AlRacoon (score 0, accepted) https://quant.stackexchange.com/a/78558 As in Pleb's comment, this is the value of the portfolio at expiration. The equation is just measuring the moneyness of the option. At any time prior to expiration, this would have to be some option model price to account for any time value that would remain in the option. As for the second term that is the value of the hedge at expiration. the negative sign reflects that he has sold the delta (h) of the option as a hedge. If one is long a call, one is long the delta of the underlying. To hedge the delta risk of the option, one would short delta units of the stock. At the expiration of the option, the value of the hedge would be - h * the ending stock price. You are correct in that he is neglecting the cash he received for shorting the stock. In the example, the loss on the hedge would have been h*(101-100), but the value would be -h*101. Let's say the ATM delta is 0.5. At inception, the investor would be long 1 call (strike = 100) and short 0.5 shares (one cannot technically be short a fraction of a share). At expiration, the Call value would be 1. The short stock hedge value would be -(0.5)*101. But he received (0.5)*100 in cash when he shorted the stock at inception. The overall value of the portfolio would be: $$Value = 1 - (0.5)(101-100) = 0.5$$ In real life, cost to borrow the underlying to short, and the return on the cash invested would also contribute to the overall value or profit.
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