pricing of max(0, TotalReturnIndex-PriceIndex)

cfornarola · October 7th, 2008, 8:33 am

Hi, my question is on the pricing of a call option on the spread at maturity between a total return version of an index ( for example Dow Jones EUTO STOXX Return Index) and its price version ( for example Dow Jones EUTO STOXX Price Index).Do I need a model that takes into account the stochasticity of dividends? If it is not needed is this pricing equal to the pricing of a dividend swap?Thanks for your help, C.

horacioaliaga · October 7th, 2008, 4:56 pm

I think the historical difference might be some decreasing (almost) deterministic function.In that case, the realized volatility of the difference should be pretty small.The correlation might be pretty high too.A naive approach would involve taking a Yield Curve, forecasted dividends and implied volatilities on options to build up these two models.If you consider low correlations, you might get a price that is to high.

Zefle · October 8th, 2008, 2:26 pm

Hi cfornarola,I think that there is no optionality in your payoff as soon as you suppose that dividends are positive. The only probleme is the cost of replication of the Total Return. For the DJ EuroStoxx Return Index, in the computation of the index, the calculator assumes a level of tax which may differ from the one you will face. So if you forget the tax adjustment, the price of this payoff is the present value of dividends which can be computed with a dividend swapZefle

cfornarola · October 10th, 2008, 8:26 am

Thx Zefle, it is clear now.If I consider max(0,TotalReturnIndex-PriceIndex) then pricing is deterministic and it's equivalent to the pricing of a dividend swap. By the way i fyou consider a generalized payoff max(0, TotalReturn-PriceIndex-Strike) and Strike<>0 then a stochastic model for dividends is needed.Thanks for your help!C.

daveangel · October 10th, 2008, 11:41 am

QuoteOriginally posted by: cfornarolaThx Zefle, it is clear now.If I consider max(0,TotalReturnIndex-PriceIndex) then pricing is deterministic and it's equivalent to the pricing of a dividend swap. By the way i fyou consider a generalized payoff max(0, TotalReturn-PriceIndex-Strike) and Strike<>0 then a stochastic model for dividends is needed.Thanks for your help!C.Why does introducing a constant into the expression change the problem ?

Zefle · October 10th, 2008, 2:56 pm

hi daveangel,(TotalReturnIndex - PriceIndex)_t starts at 0 and is increasing with time (dividends are positive), so it will always be above 0, so the max has no value. However, it could be above or below the Strike.It is exactly the same idea as for standard call on a stock, you may not need to have a model to price max(0; S_T) since S_t is positive, but it could be a good idea to put stochasticity on S if you want to price max(0, S_T - Strike)Z.

daveangel · October 10th, 2008, 3:27 pm

I thought the difference TotalRt - Index retun was deterministic ?

Zefle · October 10th, 2008, 3:41 pm

What I meant was that you do not need a stochastic model to price max(0, TotalRt - Index retun) even if the dividend is stochastic