# Vanna - any practical uses for risk or pnl attribution purposes?

What is the practical use for Vanna in trading?

How can it be used for a PnL attribution?

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If you can guess the puzzle after she turns the letters, you make a profit :P –  barrycarter Apr 6 '11 at 15:32

Pretty much irrelevant for vanilla markets but really cannot be ignored when pricing exotics such as barriers. Basically, if you do not hedge vega you are likely to sell lots of cheap exotics.

Webb discusses the practical relevance of vanna and vomma in "The Sensitivity of Vega" (Derivatives Strategy, November (1999), pp. 16 - 19).

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Thanks for the link, olaker. I was mostly interested in how could I incorporate the vanna into my pnl explanation code. The first order greeks like delta and vega are easy, gamma and vomma is a bit more complex, but doable too. I am at loss as to how to add the vanna (which I see as a cross-greek) into the whole picture... Ideas anyone? –  Andrew Apr 9 '11 at 8:11
And what do you think risk reversal is made of if not vanilla ? –  nicolas Mar 3 '12 at 8:50

Commonly used on FX option markets, see wikipedia

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Let us assume that you want to obtain the change in the price C of a plain vanilla call on a stock with price S varying with time t.

For trading, $\Delta$, $\Theta$ and $\Gamma$ matter, as in the following Taylor series expansion of C in terms of S and t:

$$dC\approx\Delta dS+\Theta dt+\frac{1}{2}\Gamma\left(dS\right)^{2}$$

Assuming a delta-neutral portfolio, gamma hedging consists of buying or selling further derivatives to achieve a gamma neutral portfolio, i.e. $\Gamma=0$. [...] Since [stocks and futures contracts] both have a constant $\Delta$ and thus $\Gamma=0$, [... they] can be used to make a gamma neutral portfolio delta neutral. [...] From [the] Black-Scholes formula it follows for a delta neutral portfolio consisting of stock options

$$rV=\Theta+\frac{1}{2}\sigma^{2}S^{2}\Gamma$$

with V consisting of the portfolio value [and r the continuous risk free interest rate]. $\Theta$ and $\Gamma$ depend on each other in a straightforward way. Consequently, $\Theta$ can be used instead of $\Gamma$ to gamma hedge a delta neutral portfolio.''

The preceding is an excerpt from : Franke, J. Haerdle, W.K., Hafner, C.M., Statistics of Financial markets - An Introduction'', Second Edition, Springer, 2008, pp. 104-107

The following is an excerpt from page 110 of the same source.

As for Vanna, the derivation of the Black Scholes formula yields:

$$Vanna=\left(\sqrt{\tau+\frac{1}{\sigma}}\right)\varphi\left(d1\right)$$

where $\varphi\left(\right)$ is the normal probability density function and $d1$ is the familiar value from the Black-Scholes equation:

$$d1=\frac{\ln\left(\frac{S}{K}\right)+\left(b+\frac{\sigma^{2}}{2}\right)\tau}{\sigma\sqrt{\tau}}$$

where, as is the custom,

$b$ is the continuous time equivalent of the dividend rate on the stock

$\sigma$ is the instantaneous volatility of the price of the stock

$K$ is the exercise price of the option

$\tau$ is the time to expiration of the option

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Ok, so what is the practical use for Vanna in trading? –  chrisaycock Jan 20 '12 at 2:02
how does that relate to the question ? –  nicolas Mar 3 '12 at 8:41
I have edited my answer in order to address that concern. –  Jean-Victor Côté Mar 4 '12 at 2:26

In trading, vanna relates to how much you are exposed to conditional, downside insurance.

It generally generates a good theta, as smile generally makes downside volatility higher than upside

In terms of PL attribution, beside this additional theta (which comes with 0 gamma !), you would be sensitive to smile change

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PL attribution is a sum of Greeks times [realized minus implied by the model]

Gamma attribution is Gamma times [realized vol minus implied vol (vol used to price)] Vanna attribution is Vanna times [realized asset/volatility covariance minus the asset/volatility covariance your model implies]

More or less, at least this should get you started

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can you expand you answer pls. tnx. –  Andrew Mar 9 '12 at 9:06
Option price is a function of risk factors, suppose we have just one risk factor, the spot price. –  mepuzza Mar 15 '12 at 16:28

Sorry, had to readd this Option price is a function of risk factors, suppose we have just one risk factor, the spot price. Then, provided that you delta hedge your position, the PL explanation will be the difference between Gamma times dS squared (which is what I call realized vol in my comment) and Theta times dt. Incidentally Theta times dt is equal to Gamma times sigma squared times spot squared times dt which is what I call implied vol in my comment. If realized vol is higher than implied vol you make (lose) money if you are long (short) the option and viceversa. Same considerations apply to a model with two risk factors i.e. spot and vol. In that case you have to look at the convexity of the price with respect to spot (gamma) to vol (volga) and cross convexity (vanna). Each convexity has an associated theta. Explanation will be Convexity times dfactor squared (e.g. gamma times dS squared) minus theta times dt (which is equal to gamma times implied dS squared). For vanna everything works the same, look at the Heston PDE and see what terms multiplies the cross derivative. That term times dt is the theta term corresponding to the vanna. The convexity term is just the cross derivative multiplied by dS times dVol

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No need to add a new answer; you can edit your original answer. Also, we allow $\LaTeX$ here, so feel free to put in stylistic math notation. –  chrisaycock Mar 15 '12 at 17:38