# stochastic dividend modeling

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- 1. Stochastic Dividend ModelingFor Derivatives Pricing and Risk ManagementGlobal Derivatives Trading & Risk Management Conference 2011Paris, Thursday April 14th, 2011Hans Buehler, Head of Equities QR EMEA, JP Morgan. QUANTITATIVE RESEARCH
- 2. QUANTITATIVE RESEARCH Part I Vanilla Dividend Market Part II General structure of stock price models with dividends Part III Affine Dividends Part IV Modeling Stochastic Dividends Part V CalibrationPresentation will be under http://www.math.tu-berlin.de/~buehler/ 2
- 3. Part IVanilla Dividend Market
- 4. QUANTITATIVE RESEARCHVanilla Dividend MarketDividend Futures Dividend future settles at the sum of dividends paid over a period T1 to T2 for all members of an index such as STOXX50E. T2 i T1 i Standard maturities settle in December, so we have Dec 13, Dec 14 etc trading. 4
- 5. QUANTITATIVE RESEARCHVanilla Dividend MarketVanilla Options Refers to dividends over a period T1 to T2. Listed options cover Dec X to Dec Y. Payoff straight forward T2 i T1 K i note that dividends are not accrued. Note in particular that a Dec 13 option does not overlap with a Dec 14 option ... makes the pricing problem somewhat easier than for example pricing options on variance.Market Active OTC market in EMEA EUREX is pushing to establish a listed market for STOXX50E At the moment much less volume than in the OTC market 5
- 6. QUANTITATIVE RESEARCHVanilla Dividend MarketQuoting The first task at hand is now to provide a Quoting mechanism for options on dividends this does not intend to model dividends; just to map market $ prices into a more general implied volatility measure. For our further discussion let t* be t* :=max{T1,t} and T2 i T1 i Fut Past Fut i t* i , Past i T i , EFut : E t [Fut] 2 T t* 1 Imply volatility from the market. The simplest quoting method is as usual Black & Scholes: Et T2 i T1 i K : BSEFut, K Past; T2 t , s div Adjust strike by BS forward equal to past dividends. expected future dividends 6
- 7. QUANTITATIVE RESEARCHVanilla Dividend MarketQuoting ... term structure looks a bit funny though. Ugly kink Graph shows ATM prices for option son div for the period T1=1 and T2=2 at various valuation times. 7
- 8. QUANTITATIVE RESEARCH Vanilla Dividend Market Quoting Basic issue is that dividends are an average so using straight Black & Scholes doesnt get the decay right. Alternative is to use an average option pricer for simplicity, use the classic approximation x 11 x i 1 x sWs ds sWs ds s x Y s x 1 Wx s 2 x i 0 x 0 e i 0 E[ i ] e x x e 0 e 3 3 6x and define the option price using BS formula as Imply a different volatility from the T2 i K market. Et i T1 : BS EFut, K Past; (T1 t* ) (T2 t* ) / 3, s div Basically the average pricing translates to a new scaling in time. 8
- 9. QUANTITATIVE RESEARCHVanilla Dividend MarketQuoting ... gives much better theta: Average option method yields decent theta, 9
- 10. QUANTITATIVE RESEARCHVanilla Dividend MarketQuoting... market implied vols by strike: 10
- 11. QUANTITATIVE RESEARCHVanilla Dividend MarketQuoting Using plain BS gives rise to questionably theta, in particular around T1 using an average approximation leads to much better results. After that, market quotes can be interpolated with any implied volatility model. Using SABR to interpolate impliedAt that level no link to the actual stock price volatilities let us focus on that now. Dec 12 Dec 13 Dec 14 a0 25% 25% 31% r -0.85 -0.84 -9.59 n 102% 47% 28% d t a t t dWt da t a tn rdWt 1 r 2 dWt 11
- 12. Part IIThe Structure of Dividend Paying Stocks
- 13. QUANTITATIVE RESEARCHThe Structure of Dividend Paying StocksAssumptions on Dividends We assume that the ex-div dates 0 0 (it is straight forward to incorporate simple credit risk *1,2+ but well skip that here ) 13
- 14. QUANTITATIVE RESEARCHThe Structure of Dividend Paying StocksStock Price Dynamics In the absence of friction cost, the stock price under risk- neutral dy

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