Assume I want to price a call on a CPPI based on an index as underlying. There are two ways to derive CPPI risk-neutral returns for pricing the option:1. transform the index returns in the risk neutral measure and then run the CPPI with these returns2. run the CPPI with the index risk/return in normal measure then transform the CPPI returns in the risk-neutral measure Which one should be the correct way of transforming returns.

Hi UncleAlba,Since the underlyings of the CPPI are traded assets, the CPPI is a traded asset and both approaches should yield the same risk-free measure.Provided, of course the market is complete :-)Hans

Thanks Hans,Indeed the results looks very different. One reason that I identified is that the volatility of the CPPI returns are dependent on the drift of the underlying index. So if you would transform the index in risk-neutral measure and then derive CPPI, the volatility of the CPPI returns is different compared to deriving CPPI returns in index observed measure. And this yield different option prices. Any comments?

Hey UncleAlba,I'd be tempted to say that Hans is in a way wrong, as the underlying in CPPI's is not really a traded asset in the way mathematical finance defines it. The liquidity is often discretionary (see hedge funds or even mutual funds), and their NAV is the result of a set of investment rules aplied by the fund manager. These conditions establish the trading timing and the stake to put at risk by investing in the index. Hence the underlying NAV can influence the CPPI, as you well know that technically, a change in the 10th decimal place of a gap close to rebalance can cause a rebalance, hence modify definitely the path, which is the crucial issue.So even if you are trying to price a call on a 100% initial equity weight CPPI on S&P, that does not seem obvious to me that the 2 pricing strategies. I'd say that if you want to price something on a very liquid asset, use the typical assumptions on the risk neutral transformations for the underlying the run the CPPI, as the assumption on completeness and all the others hypothesis hold quite well for a "normal" underlying, which the CPPI is all but not (assumption 1). Plus the CPPI is not a primary market product, so it is not liquid in the way that CPPI writers have to abide by their underlying redemption/subscription rules - for an option on CPPI, the gamma swings so much that the position is sometimes a bit tricky to manage.Well, I don't know if you solved you issue, but that is what I think !Good luck

Hi UncleAlba,I can imagine that the two methods give not the same results since in method one you change the underlying. You just calculate the value of a call on two different underlyings. CPPI is not my expertise but it sounds to me that if you first apply any kind of mathematical transformation to the index risk/return and after that you run the CPPI, you will not have the same asset proportion than in the other way. Those 2 underlyings have (as you told yourself before) a different volatility. From the way you define your underlying in your firs message I deduce you should first run the CPPI with the index risk/return in normal measure and then transform the CPPI returns in the risk-neutral measure.

Uncle Alba,One more thing about "risk neutral valuation" (not "riskk free valuation" as someone write before).I want to make clear that the point here was not to check if your underlying fit the conditions for a risk neutral valuation(I assume you did this before starting your calculations).P.

I opted for the first version (first transform the returns in risk-neutral and then run the CPPI). In a paper by Bertrand and Pringent they derive the terminal value of the strategy in closed form ( cases when returns are normally distributed and a fixed multiplier). Based on these results the call price could be calculated in closed form. The option values are as expected.With the second strategy (generating CPPI returns and then transforming with risk-neutral world) I have not found a closed form for comparison.UA

If you have a rule to define your asset allocation this rule define your underlying.It “seems” obvious that you should define your underlying (running CPPI) before using a mathematical “trick” (like risk neutral valuation or any other) to solve the price of your option. In the other way, if you transform the returns in the risk neutral measure and then run the CPPI with these returns you will probably have another asset allocation.The option price that you will obtain by this way is the price of an option on something different. Was this the option you needed to price ? or did I misunderstood something ?If the answer to both questions is negative then I have to disagree with the method consisting in running CPPI at the end.If the answer to last question is positive then I need help.P.

If you have a rule to define your asset allocation this rule define your underlying.It “seems” obvious that you should define your underlying (running CPPI) before using a mathematical “trick” (like risk neutral valuation or any other) to solve the price of your option. In the other way, if you transform the returns in the risk neutral measure and then run the CPPI with these returns you will probably have another asset allocation.The option price that you will obtain by this way is the price of an option on something different. Was this the option you needed to price ? or did I misunderstood something ?If the answer to both questions is negative then I have to disagree with the method consisting in running CPPI at the end.If the answer to last question is positive then I need help.P.

The expected value of CPPI payoff is the same if the risk neutrality is taken in the underlying returns or after CPPI returns. It is equal with the forward value of the initial investment value. If we take the risk-neutrality at the first or second step the results are the same.This holds in the case when we have the assumptions of the constant interest rate, fixed multiplier, continuous rebalancing etc.At the end by creating the CPPI returns we are not increasing the sources of uncertainity (fixed multiplier) so it should not matter where the risk neutrality is taken.Although, I begin this thread having the same remarks/doubt as P.