# 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?

• If you can guess the puzzle after she turns the letters, you make a profit :P
– user59
Commented Apr 6, 2011 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).

• 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? Commented Apr 9, 2011 at 8:11
• And what do you think risk reversal is made of if not vanilla ? Commented Mar 3, 2012 at 8:50

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

One application not mentioned in the answers thus far, but perhaps well-known by now:

The implied volatility where the Black-Scholes vanna of a vanilla option is zero (notation $$I_-$$) is approximately the volatility swap strike (assuming the smile is generated by a (rough) stochastic vol model). So, knowing the varswap and volswap strike means you can monitor the vol-of-vol quite easily and almost in real-time, which is a nice thing for vol trading purposes.

Also, the implied volatility where vanna is nonzero but where volga is zero (notation $$I_+$$), is approximately the volswap strike under the share measure, i.e. the implied volatility of a vanilla option under the share measure where vanna is zero.

Furthermore, the difference $$I_+ - I_-$$ is the expectation of the covariation between the simple return of underlying stock and realized volatility. Also useful if you have a view on covariation/correlation.

Commonly used on FX option markets, see wikipedia

PnL attribution is a sum of Greeks times [realized - implied by the model]

Gamma attribution is Gamma times [realized vol - 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

• can you expand you answer pls. tnx. Commented Mar 9, 2012 at 9:06
• Option price is a function of risk factors, suppose we have just one risk factor, the spot price. Commented Mar 15, 2012 at 16:28

long vanna means => vega increases as stock goes up and vega decreases as stock goes down. If we assume standard equity markets; when the stock goes up, the vol goes down then I will sell vega when stock goes up and buy vega when stock down.

So you are buying vol when it's expensive and selling vol when it's cheap. Overall loss due to convexity.

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

• 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. Commented Mar 15, 2012 at 17:38

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

• Ok, so what is the practical use for Vanna in trading? Commented Jan 20, 2012 at 2:02
• how does that relate to the question ? Commented Mar 3, 2012 at 8:41
• I have edited my answer in order to address that concern. Commented Mar 4, 2012 at 2:26