A displaced diffusion version of SABR model and it’s underlaying’s probability distribution
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Baronas, Sarunas1, Forfatter
Gydesen, Mads2, Vejleder
1Det Samfundsvidenskabelige Fakultet, Københavns Universitet, København, Danmark, diskurs:7001              
2Økonomisk Institut, Det Samfundsvidenskabelige Fakultet, Københavns Universitet, København, Danmark, diskurs:7014              
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Ukontrollerede emneord: The SABR model, displaced diffusion, probability density function
 Abstract: In this paper I examine the performance of different versions of the SABR model applied to interest rate derivatives, when forward rates tend towards zero. I analyze probability density functions of the three versions of initial model and discuss how they influence the payoffs of the derivatives.
The thesis is divided into two main parts: the theoretical and the applications. Theoretical part starts with introduction of some classical results from finance theory regarding arbitrage free pricing and martingale measures. Next section covers the interest rates concept and theoretical discussion of plain vanilla derivatives. After that we move to the pricing of more difficult derivatives – constant maturity swaps. Furthermore, the so-called stochastic alpha beta rho (SABR) model is introduced. From that follows a theoretical discussion of the displaced and shifted SABR versions.
Second part of the thesis is dedicated for empirical applications. First of all, I calibrate different models to the market data on different dates and obtain implied volatility smiles. Then I price the chosen interest rate derivatives and discuss how the expected payoff from the financial product changes, when we use the displaced diffusion SABR or shifted SABR models in comparison to the original SABR model presented by Hagan et al. (2002). I also obtain risk-neutral densities of an underlying asset and discuss the influence of the changes in distribution to the payoffs of the derivatives under review.
In this thesis I show that the displaced diffusion SABR model can actually produce similar volatility smile to the one generated by the original SABR model. Furthermore, shifting the price of an underlying and all strikes in the SABR model setting does not affect the shape of the implied volatility curve.
Moreover, it turns out that the shifted SABR model results in the different cumulative density function and the different probability density function in comparison to the ones obtained by the original SABR model. Graphical analysis shows that moving ATM forward from left to right flatters the probability distribution of an underlying swap rate. The effect of the different distributions is analyzed with the pricing examples of CMS derivatives. Both graphically and with calculated values I prove that shifting the SABR model results in different CMS products’ prices. In essence, this means that shifting the SABR model is acceptable for the pricing of plain vanilla derivatives, yet this method cannot be ‘blindly’ applied for more sophisticated derivatives.
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 Type: Speciale
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Sprog: English - eng
 Datoer: 2013-08-19
 Sider: -
 Publiceringsinfo: København : Københavns Universitet
 Indholdsfortegnelse: I Introduction 7
2 Essential finance theory 9
21 Financial framework 9
22 BlackScholesMerton result 11
24 Interest rate concepts 13
3 Interest rate derivatives 15
31 Interest rate swaps 15
32 Interest rate options 18
33 European Swaptions 19
34 Constant maturity swaps derivatives 21
341 Valuation of CMS swaps, caps and floors 23
4 The SABR model 29
41 The parameters of SABR 32
42 The choice of beta parameter 33
43 The displaced diffusion version of the SABR model and shifted SABR model 35
431 The displaced diffusion SABR model 35
432 The shifted SABR model 39
54 Implied probability distributions 40
5 The risk measures under the SABR model 44
 Note: -
 Type: Speciale
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