# Volatility arbitrage - how is the profit extracted?

Is there any paper that describes in detail how the profit is extracted in directional volatility bet (vol arb)? I mean in the case that I bet the realized volatility will be lower than currently implied vol, I take a short position in call and long in the underlier to get delta hedge. So now, how do I actually make profit on realized volatility? What if the actual volatility during the following period is lower, so my bet was correct, but the implied volatility stays the same for the whole period anyway? If under those circumstances I liquidate the position, wouldn't the profit be 0?

For some reason I struggle to wrap my mind around this, but on the other hand I can see how pure option strategy like short straddle work....

-

Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage:

You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging.

Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize profit. If the option ends up in the money, your hedge would still be an open position, but it will then be fully covered by option exercise.

With short position its the otherwise, you buy high and sell low.

In the end, you hope that your hedging lost/gained less/more money than you sold/bought the option for.

In practice you watch you portolio and its greeks daily and can see whether you are winning or losing. Let's assume you adjust the delta hedge at the end of the every day. I'm denoting market move as $\delta S$. The value of your option at the end of the day can be approximated as: $$O(t+1,S+\delta S) \approx O(t,S) + \Delta\,\delta S + \frac{1}{2} \gamma (\delta S)^2 + \theta$$

So, if the volume of your hedge at the beginning of the day was exactly $-\Delta$, the $\Delta \delta S$ change of the value is compensated and you are left with $$P(t+1,S+\delta S) \approx P(t,S) + \frac{1}{2} \gamma (\delta S)^2 + \theta$$

(now $P$ stands for the value of the whole portfolio of option + hedge)

The $\theta$ term is completely deterministic. You are guaranteed to lose/gain some value every day while being long/short the option. The term with $\gamma$ depends on your luck. Notice, that the $(\delta S)^2$ term is always positive. So the whole term has the same sign as gamma.

So, if you are e.g. gamma positive (and your theta is negative), you are losing theta term every day, and gaining gamma term depending on the market move. If the market moves will turn out to be generally higher on average, the gamma term will earn more over time than the theta terms will lose. But you can see the luck factor in there. The bigger the difference between real and implied volatility there is, the less luck is needed.

If the option is priced fair, the gamma term will be equal to theta term on average: $$\rm{E}\biggl(\frac{1}{2} \gamma (\delta S)^2\biggr) = -\theta$$

-
There is this additional subtlety, whether to compute the greeks and values according to implied volatility or according to your expected (true) volatility. In the former case, you portfolio value at the beginning is zero, and you gain money over time. In the latter case, your portfolio starts with positive value and keeps it stable. – airguru Apr 24 '14 at 8:18
Thank you, very good explanation. This is also helpful to see what's going on. It's a bit hard to see why it behaves just like that intuitively (like how would I explain that to a child or undergrad?), but I guess that's the price we have to pay for nonlinearity and just trust the math. :-) – Paya Apr 24 '14 at 12:24
I see the intent behind this answer but I don't see how it addresses the fact that implied volatility is not obligated to track or even correlate with the observed volatility. He even says on his question, if my underlying, iv and time value offset, but my observed vol is different, isn't my pnl still 0 - which is absolutely correct. The only way to capture profit on the kind of strategy the original poster describes is by not only having his book delta neutral but also by trading a print indexed volatility derivative. – evan_irl Apr 24 '14 at 13:11
As far as I understand, the whole trick is in the not-so-intuitive behavior of gamma and that it affects the value of option based on actual volatility (the 1/2*gamma*market_change^2 term). This article was also quite good to explain the mechanics. And as volcube points out, the role of continuous delta-hedging is just to lock-in profits from the actual vol, otherwise rally of 5 points followed by drop of 5 points would earn nothing. – Paya Apr 24 '14 at 13:55
Riaz and Wilmott wrote a very readable paper which highlights the path-dependency of volatility arbitrage via dynamic replication. In short, if your delta hedge is constructed at the true future volatility, your P&L will be erratic but your profit ultimately guaranteed (assuming correct future vol prediction). If you hedge at the implied vol, your P&L is smooth, but your final profit will be stochastic. – experquisite Apr 24 '14 at 16:13

What if the actual volatility during the following period is lower, so my bet was correct, but the implied volatility stays the same for the whole period anyway? If under those circumstances I liquidate the position, wouldn't the profit be 0?

I think I know where your confusion comes from.

1) this isn't arb - it is not a risk free strategy

arbitrage is the practice of taking advantage of a price difference between two or more markets

The key point being that there are two or more markets. In true arbitrage, you have to have simultaneous buying interest and selling interest with a positive netback. In your scenario you have a single market so you cannot have arbitrage (ignoring stat and other time arb). You'd just be buying and selling into the same market. In your strategy you need to hold the asset until the market agrees with your value, but during that time lots of things can happen (fundamentals change, margin call, market/credit risk in general)

2) implied volatility and observed volatility are very different things

IV can do whatever it wants, and is dictated by buyers and sellers. Observed volatilities often correlate with IV but they don't have to. It depends on the market and the outlook on the asset.

-

If you are really after volatility arbitrage, rather have an opinion on volatility, you can use VIX options and futures. This can help you manage your views on volatility far more concisely than by buying and selling individual securities and delta hedge them. Delta hedging has a lot of transaction cost, and time and effort involved.

If you are buy side and have large book then exploiting inefficiencies in individual securities may be okay. If you are a dealer then delta hedging is a necessity and part of the business.

Id you are EOD trader may be VIX ETF/ETNs may be good.

-