- wangyangwangyang
**Posts:**9**Joined:**

Hi volatility gurus,The problem at hand is a plain option written on the following synthetic Volatility Target Index:I_t = I_{t-1} * (Exposure_t * Risky Index_t / Risky Index_{t-1} + (1 - Exposure_t) * Cash Component_t / Cash Component_{t-1})where,Exposure_t is adjusted daily by comparing the historical volatility (of Risky Index) over a certain range (say, the past 30 days) with some predefined Volatility Target, for example, Exposure_t = Min(Exposure_Cap, (Target Volatility/Historical Volatility_t)).The basic idea is attempting to confine the synthetic index volatility level within a certain range. I am currently trying to find a suitable stochastic vol model to properly price and hedge this product. My first impression is that it is going to be very sensitive to the volatility model assumed for pricing and therefore I would like to apply the well-known Bergomi model (directly on forward variances) to simulate this product. Does anybody have any opinion or suggestion as to which model to use? I would be happy to discuss. And happy new year to you all!

I am usually a bit sceptical about this vol targetting indices, but that's just personal. so are you trying to price an option on this index (I_t) or just the forwards? what's the underlying? how the instrinsic forwards look like for a fixed weights? - like a steep one (costly) or a flat one (cheap)? and how frequently you rebalance, how much it varies in between (say 0 to 100% ??) and how long dated trades you will write on this? I do not think there is any standard method to price these, just trying to figure out the prob

- wangyangwangyang
**Posts:**9**Joined:**

Thanks very much for your comments, prodiptag.I am pricing a European call option on this synthetic index. The underlying is a major stock index, say SP500. The life span is around 1.5 years, so I would expect the foward curve to be relatively flat. All this looks fine to me.My concern mostly comes from rebalancing the portfolio weights (happens everyday and could go from 20% to 140%) according to the variation of the realized volatility. In this sense, it is an option contingent on the realized vol of the underlying and I wonder if it is still appropriate to simulate the index using a local-vol-type-of model. I have run some simple test cases in Monte Carlo using both local vol and Bergomi type of models and already found some not-so-small discrepancies in pricing between the two types of models.I would greatly welcome any opinion on the model risks associated with local vol/stochastic vol/Bergomi for this speicific product.

why can't you price this using Black Scholes given that the volatility is constant ? edit: I can see why as the vol targetting means your exposure to the underlying varies so affecting your cost of carry. What are the standard ways of pricing these products ?

Last edited by daveangel on January 3rd, 2011, 11:00 pm, edited 1 time in total.

knowledge comes, wisdom lingers

In September's Risk Magazine there was a two page research report sponsored by RBS, where they compared B-S with target vol of 15% on similar srtucture on S&P and implied volatility to be inserted into B-S resulting from Monte Carlo simulation of the dynamic trading using local volatility, stochastic volatility and local volatility with jumps. The differences were maximum 2 p.p. of implied vol and dependant on option's moneyness. I think that there is high chance, that BID/ASK on such structures is wider than that.

ok, thinking aloud, I think your major exposure is to smile term structure + underlying vol correlations. So any model that can handle this should be fine. I think if you can have a decent price back for variance swaps (rather vol swap) then the other thing, i.e. vol/underlying correl becomes more important. So I would be happy with a stochvol term structure model. You say you get significantly different values using LV and Bergomi, do they had same implied autocorrels and vol/underlying correls?Also this would be perhaps doable only for the shorter end, else the heavy computation might be problematic for risk runs, so I think one way is to split the price in to, say a 3m horizon with a propoer MC, and then price the rest term using the worst possible weights, given today's market values

- wangyangwangyang
**Posts:**9**Joined:**

Thank you guys for your inputs. Pimpel, I have located the paper you referred to and it seems pretty helpful to the problem I'm looking at. Prodiptag, I basically agree with you on that the major exposure stems from smile term structure + underlying vol correlation. I shall test further to investigate where the discrepancy between LV and Bergomi creeps in.

Voltargets for non-dividend paying stocks/funds could be priced with just B&S (as hinted at by several others). Another thing that quite commonly is done is doing a voltarget where the actual underlying isn't the index itself but instead the rolling first future (in such amounts that the forward is flat). You can then just give no remuneration on the non-risky asset and you're left with something with fixed vol and that is flat forward, which is often even cheaper.The length of rebalancing periods as well as min/max allocations and divs can make that you still want to price this with MC rather that in a simple B&S setting.

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