I am trying to understand how an delta neutral profile is generated. I sell a call for strike of 50$ and the delat of this call is 0.5. I buy 0.5*100 stocks to remain delta neutral. Now when the market moves up buy 2$ the new delta is 0.6 . So I buy 10 more stocks and the market now moves down by 2$ which makes me sell 10 stocks. Every time I buy at 52$ and sell at 50$ since the total portfolio value seems to decrease when the stock moves up and down. How does this work in the BSM model ?


1 Answer 1


If you sell a call option, in the language of vol trading you are short gamma and vega, and long theta.

So yes, as the underlying wiggles around, if you keep delta-hedged by trading it you will lose some money. This can alternatively be seen as gamma losses (coming from the second-order move in the underlying, which you have not hedged) or as coming from the hedging strategy.

On the other hand, you are compensated for this by the time decay of the option price, ie. its theta.

By the end of the option lifetime, if realized vol over the period (which is effectively what you pay out to hedge) was higher than the implied vol price you received for the option (which was the premium you received), you probably* lost money, and if it was lower then you made money.

*this isn't exactly true due to vol itself being stochastic... here is a great reference on the topic Wilmott, Ahmad: Which Free Lunch...


Responding to comment: The first section of the wikipedia page does a good job answering this... if we re-express BS equation as \begin{align} {\frac {\partial V} {\partial t}} + {\frac 1 2} \sigma^2 S^2 {\frac {\partial^2 V} {\partial S^2}} = rV - rS {\frac {\partial V} {\partial S}} \end{align} then you see that the term on the right (risk-free growth coming from a portfolio of the option and the short delta hedge) exactly cancels the term on the left which is the gamma plus the theta.

So we can hedge away delta, but we are still exposed to these two.

  • $\begingroup$ According to black scholes if Istart buying the stock when I sell the option , I should not lose any money. What am I not considering here? $\endgroup$
    – roller
    Commented Dec 26, 2020 at 20:12
  • $\begingroup$ First, BS assumes you hedge in infinitely small quantities without transaction costs after every move in the underlying, which hedges out your delta completely. Second, BS assumes vol is deterministic and known at the beginning. However, we would still get some losses from the gamma component due to GBM's "quadratic variation" (basically bigger infinitesimal jumps), it's just that these losses would exactly cancel out our theta profit. This is expressed by the BS equation's partial differentials $\endgroup$
    – StackG
    Commented Dec 26, 2020 at 23:25

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