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I am reading page 126 (Chapter 8) of the book "Option Volatility and Pricing 2E" by Sheldon Natenberg and have two questions I seem to be stumped on. (The bulleted text below the charts in the screenshots are my own notes.)

  1. Why is it that in making discrete adjustments to remain delta neutral, the variance of our P&L is "smoothed out"? (as described in 1st photo)

  2. In calculating the total theoretical profit, how do the small profits from the unhedged portions (2nd photo) add up to the total options' theoretical value? Moreover, is my interpretation of how we are making $$$ through these mismatches correct? (3rd photo) I can see how, if after we sell the underlying at a higher price to offset positive delta exposure after an increase in the underlying, we then buy back the underlying at a lower price to reestablish our hedge, yields a profit... but am having trouble understanding which interpretation Natenberg is alluding too with the visual graph. Where is the profit coming from exactly / how are we so-called "capturing" value?

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  • $\begingroup$ What you call "smoothing out" is a form of diversification: if instead of making a 1000 USD bet you make 100 bets of 10 USD each you are less affected by random luck on your bet(s). Your outcome is closer to the statistically expected value, with less influence of randomness. $\endgroup$
    – nbbo2
    Jun 23, 2023 at 6:19

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  1. Imagine that you had an option position that you decided you were only going to hedge once, at creation, then let run until maturity. As the market moves and as time passes (and as IV changes), your net delta (option delta + futures delta) will change from 0. If you are long the option, you will be glad to see the market move far in any direction, as your net position has produced positive P/L. Because of your decision to not hedge frequently, you are now exposed to the market returning to it's original price. If you subscribe to the view that markets follow a GBM (or GBM-esque) process, then over the hedging interval you are exposed to market movements which are proportional to $σ \sqrt{dt}$. For a choice of hedging frequency $\frac{T}{ dt}$, your P/L over that period will be given by $\frac{1}{2}\Gamma S^2 (\sigma^2 _r z^2 - \sigma^2 _i)dt$ where z is a random variable (assumed to be standard normal). This gives a total P/L over the horizon, T, of: $$\sum_{i=0}^\frac{T}{dt} \frac{1}{2}\Gamma_iS^2_i(\sigma^2_r z_i^2 - \sigma^2_{implied})\, dt$$ Holding $\Gamma$ and $S$ constant, the final P/L has the same expected value, regardless of hedging frequency, as the $dt$ terms cancel. However the variance, and thus volatility of the P/L is affected by the hedging frequency: $$ V[x] = E[x^2] - E[x]^2 $$ Which simplifies for the above case to: $$ V[PnL] = \frac{$\Gamma^2 * T* dt}{4} (2\sigma_r^4)$$ for a volatility of: $$ \sqrt{V[PnL]} = \frac{$\Gamma * \sqrt{2T* dt}}{2} \sigma_r^2$$ Which, when defining $dt$ as $\frac{T}{N}$, becomes: $$ \frac{$\Gamma \, \sigma_r^2 \, T}{2} \sqrt{\frac{2}{N}}$$ So the volatility of your hedged P/L is inversely proportional with the square root of the frequency at which you hedge, approaching 0 as N $\to \infty $ More generally you can replace the $\sqrt{2}$ in the final formula with $\kappa -1$, where $\kappa$ is the raw kurtosis of the underlying returns (3 for a normal distribution). More generally I think you can skip all this math and just think that you will have $\frac{T}{dt}$ hedging periods where deltas accumulate, where those deltas will contain market risk proportional to $\sigma\sqrt{dt}$, leaving something proportional to $\frac{T}{\sqrt{dt}}$ in the end.

  2. If you hold the law of one price, and replication arguments, which state that the price of a derivative should equal it's replicating portfolio, you will see that the price of an option should be equal to the potential P/L from replicating it, this potential P/L in the continuous case is given by: $$ \frac{1}{2}\int_{0}^{T} \Gamma_t S_t^2 \sigma_r^2 E[dW^2] dt$$ This total P/L is the fair price of the option (under constraints like 0 costs). Which then means that the $\sigma_i$ should be equal to $\sigma_r$, which as per the answer in part 1, creates an expected PnL of 0, with 0 variance in the continuous hedging case. If the option is mispriced, and you buy cheap / sell expensive, then transacting and hedging delta allows you to capture the mispricing between implied and realised volatility, which expresses itself through $\frac{1}{2}\int_0^T \Gamma S^2 (\sigma_r^2 - \sigma_i^2) dt $, imagine that at each hedging frequency you are faced with that familiar parabola shape with respect to spot, where the intercepts of the parabola are $ = \sigma_i \sqrt{dt}$, and each return, r, a random variable, lands you somewhere on that parabola, if r = 0, you pay the max loss, the $\theta$, but if the standard deviation of r > $\sigma_i$ over your chosen hedging frequency then you will expect to make money over time (if buying, flip r and i when selling).

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