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For someone who has a delta hedged options position, the $\Gamma:= \frac{\partial^2V}{\partial S^2}$ roughly quantifies the amount of money made or lost if $$\frac{1}{\Delta t}\frac{(\Delta S)^2}{S^2} \neq \sigma^2$$ where $\sigma$ is the number that characterizes the process: $$ dS_t = S_t(\mu dt + \sigma dW_t) $$ If we consider the quantity on the left-hand side to be the "realized-variance" of our trajectory, gamma can be interpreted as the Greek that measures the impact of the difference between realized-variance and implied-variance/model-variance on our option price, assuming our portfolio is delta hedged. Thus, in my opinion, $\Gamma$ can (in a very precise sense) be considered the "realized volatility greek".

I have the following question then:

  1. What, fundamentally is $\nu $? I understand it is defined as $\frac{\partial C}{\partial \sigma}$, so that it quantifies changes in option prices given our inputted variance/volatility. However, many consider it to be the sensitivity of option price to "implied volatility". This makes little sense to me - implied volatility is computed using option prices in the first place, so it makes little sense to have a greek like this, if changes in implied volatility are a posteriori computed from changes in option prices.
  2. How does $\theta$ figure into the calculations of delta-hedged PnL?

Thanks in advance for the assitance.

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    $\begingroup$ You should have a look at Lorenzo Bergomi's book "stochastic volatility modeling". Think of the vega as the sensitity of your (possibly exotic) option to vanilla options, the latter being quoted in terms of implied volatility rather than in price. $\endgroup$ – Antoine Conze Dec 3 '20 at 14:44
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Gamma is the change in price due to realized volatility--in other words due to the fact that the underlying has moved. For a Delta hedged position, this will be a measure of how much your PnL was off due to a move in the underlying. As options prices that have any time left are not linear with respect to the move in the underlying, your delta will change as the underlying changes and hence your delta hedge will be off as the underlying moves. Gamma will approximate how much your hedge will be off.

Vega is the change in the price of volatility or the change in the implied volatility of options prices. This will be a measure of the PnL strictly due to the fact that the price of volatility has changed (the underlying has not changed in value).

Theta will measure the amount of PnL due to the passage of time. If you are long options, if nothing else has moved (implied vol or the underlying), the option will lose value as time passes (except really far out of the money options). Theta typically measures how much the option value will lose from the passage of a day for long options holders (or a gain for option writers).

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many consider it to be the sensitivity of option price to "implied volatility"

This may just be a semantic issue- it's the sensitivity to volatility, regardless of how that volatility is computed. Since the volatility in the black-scholes model is a forward-looking volatility, there's no observable way to measure it other than implying it from market prices. But suppose the market is wrong and is overpricing options? Meaning that the actual forward volatility is lower? Then vega tells you how sensitive the option value is to changes in that volatility factor.

Or, put another way, if you look at changes in market prices (and volatilities) from one day to another, vega will tell you how much of the change in market price was due to the change in volatility (ignoring any second-order and higher effects due to vol skew, etc.)

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    $\begingroup$ Have you ever written code to compute implied vol? It is usually done iteratively, starting from an initial guess. At each step you adjust the current vol up or down to try to match the market price of the option. How much should you adjust the Vol? One good method is to adjust vol by the current pricing error divided by the current Vega. It works because "Vega tells you how sensitive the option value is to changes in that volatility parameter". That's what Vega is. $\endgroup$ – noob2 Dec 3 '20 at 2:08
  • $\begingroup$ @noob2 So Vega's usefulness comes from Newton's method? $\endgroup$ – rubikscube09 Dec 3 '20 at 4:14
  • $\begingroup$ To a programmer, and to Isaac Newton, the answer is yes ;) $\endgroup$ – noob2 Dec 3 '20 at 13:38
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    $\begingroup$ Well, that's not the only way Vega is useful, but yes, it can help you find implied vol in fewer iterations. But it's by no means the only or even the most valuable use of Vega. $\endgroup$ – D Stanley Dec 3 '20 at 14:13
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You need to think of the option price as a function of 3 variables $C=C(t,S,\sigma)$. For a listed option the price is known from the market, and $\sigma$ is the only unknown.

  • Write down the Taylor series expansion in those 3 variables, including second order for spot and the spot/vol cross term. Include the theta!
  • At times $t_1, t_2$ you will see 2 option prices in the market and 2 spot prices, so you have sets $(t_1, S_1, C_1),(t_2, S_2, C_2)$.
  • You can convert these option prices to implied vols, and calculate the option Greeks
  • You can input the Greeks and the data changes $S_2-S_1,\sigma_2-\sigma_1$ and see how much of the price move from $C_1$ to $C_2$ you can explain. That should clarify for you what the implied volatility (and theta) is for.

The real utility of this process comes when you are modelling more than one option in a portfolio. The time and spot move are shared by all option Taylor series, and the various implied volatility moves become what you study to understand (and control) the total moves in portfolio value.

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OK,

many consider it to be the sensitivity of option price to "implied volatility". This makes little sense to me - implied volatility is computed using option prices in the first place, so it makes little sense to have a greek like this, if changes in implied volatility are a posteriori computed from changes in option prices.

Let's consider a simpler instrument. Suppose you have some corporate bonds in multiple currencies. Assume the FX rates and bond prices are observable. It is good to have a P&L explain that tells you how much of your P&L came from FX rate changes, how much from passage of time (pull to par etc), how much from the change in the risk-free interest rates in various currencies, how much from the various credits, how much from the idiosyncratic moves of indovodual bonds...

Now suppose you have an option whose price is observable in the market (not marked to model). Before Black-Scholes, traders did not have a good way to explain why option prices changed or what their views were. Today, we can get the implied vol from option price and, for example, we can see the book's sensitivity to impled vol; we can decide whether we are comfortable with this sensitivity, or we want to change it (by unwinding some positions or adding some hedges). We can explain the change in the option's price by attributing it to the passage of time, the changes in the price of the underlying and the interest rates, and the changes in the implied vol. Although the quotes often are the impled vols, still the implied vols are not as directly observable like the time, the interest rates, and the underlying price. Just think of it as the part that leaves no unexplained P&L.

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