# Beginner question on Black Scholes

Would you please confirm whether my understanding is correct please? (Sorry a lot of questions...)

1) BS is derived based on the assumption that during an infinitesimal time, we can replicate the payoff of the option by holding Delta amount of underlying stocks

2) Since the portfolio is risk free in that infinitesimally small period, its value should grow at risk free rate during the period

3) Based on such differential equation during the infinitesimally short time, we deduce the formula of option price (by integration)

4) If we were to carry out such replication, at infinite frequency, does it really mean we will have a risk less portfolio with 0 p&l? (Except for the risk free rate)

5) if yes, then where does the gamma pnl of the option come from? (Which says that whichever way the stock goes, option seller loses money equal to 0.5*gamma*change^2) Is there somehow an opposite effect embedded in the delta hedging side of the portfolio that would create a compensating loss?

6) If yes, can you explain this effect intuitivelly and mathematically please? It bothers me because I cannot understand how a dynamic hedging scheme can relate to the gamma pnl (that no matter which direction the stock goes, the option loses money and the hedge wins money)

Thank you very much

1) I think that..gamma being the second derivative of the option, would have a positive impact to the price of the option when stock moves, regardless of the hedging frequency. So the fact that positions are not hedged continuously is not the cause of gamma pnl. (The gamma does have an effect to the correct delta to hedge with when we are not hedging continuously, but that seems a different matter from the theoretical gamma pnl)

2) Indeed, the opposite effect of the gamma pnl is coming from Theta. So my statement about "hedging side offsetting the gamma" is wrong.

Now, I'm thinking how does the gamma and theta effect arise? Intuitively, it makes perfect sense as the more volatile the stock moves, the more probability the option is in the money, so volatility is always good. After all, a long option position is capped on the down side and "infinite" on the up side. As for theta, it makes sense because time decay reduces opportunity for such swings.

But in the context of black Scholes, how does it relate? BS states that the portfolio is hedged instantaneously. My only guess is that the gamma is coming from the transition between one instant to another instant, which is not hedged. (Delta only hedges the first derivative)

But again, what bothers me is that, if there is still a risk in the delta-hedged portfolio, how can we apply the "Risk neutral" argument to develop the BS PDE?

Is it because the portfolio is not really risk neutral (volatility is still in the equation), it is only stripped of the risk preference (drift), which is enough for us to apply a change of measure, since volatility is invariant under change of measure?

• (1) (2) (3) (4) are true in the limit as $\Delta t\rightarrow 0$. On the other hand the concept of Gamma P&L (5) is for finite $\Delta t$, the hedge is not perfect because you do not update it continuously, but only every $\Delta t$ and this of course is real life. – Alex C Jun 25 '18 at 4:34
• And the offsetting effect of the gamma p/l is time decay. The black scholes equation effectively says expected gamma p/l equals time decay. – dm63 Jun 25 '18 at 10:21