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Extension of stochastic volatility equity models 990.060.055Cliquet prices from SZHW and HHW with r x,r= −0.7 and the HestonSchöbel−Zhu−Hull−WhiteHeston−Hull−WhiteHeston0.060.055Cliquet prices from SZHW and HHW with r x,r= 0.7 and the HestonSchöbel−Zhu−Hull−WhiteHeston−Hull−WhiteHeston0.050.05Value of Cliquet at t=00.0450.040.0350.03Value of Cliquet at t=00.0450.040.0350.030.0250.0250.020.020.015−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.040.015−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04MinCouponMinCouponDownloaded by [Lech A. Grzelak] at 10:55 24 January 2012Figure 1. Pricing a cliquet product under the SZHW, the HHW and the Heston models. Both figures present the price of a globallyfloored cliquet as a function of MinCoupon given by (52) for T ¼ 3 years and M ¼ 36. The remaining parameters are as in table 4.Left: Pricing with x,r ¼ 0.7. Right: Pricing with x,r ¼ 0.7.Value of Diversification product at t = 01.31.251.21.151.11.051Diversification product prices from SZHW and HHW with r x,r= −0.7 and the HestonSchöbel−Zhu−Hull−WhiteHeston−Hull−WhiteHeston0.950 50% 100% 150% 200% 250%ωValue of Diversification product at t = 01.31.251.21.151.11.05Diversification product prices from SZHW and HHW with r x,r= 0.7 and the Heston1Schöbel−Zhu−Hull−WhiteHeston−Hull−WhiteHeston0.950 50% 100% 150% 200% 250%ωFigure 2. Pricing of a diversification hybrid product under different models. The simulations were performed with ¼ 10. Theremaining parameters are as in table 4. Left: Pricing with x,r ¼ 0.7. Right: Pricing with x,r ¼ 0.7.shows that the Heston model generates a significantlyhigher price, whereas the HHW and SZHW prices arerelatively close. The absolute difference between themodels increases with percentage !.4.4. Strategic investment hybrid (best-of-strategy)Suppose that an investor believes that if the price of anasset, S 1 t , goes up, then the equity markets under-performrelative to the interest rate yields, whereas if S 1 t goesdown, the equity markets over-perform relative to theinterest rate (Hunter and Picot 2005/2006). If the prices ofS 1 t are high, the market may expect an increase in inflationand hence in interest rates and low S 1 t prices could havethe opposite effect. In order to include such a feature in ahybrid product we define a contract in which an investoris allowed to buy a weighted performance coupondepending on the performance of another underlying.Such a product can be defined as follows:withðt 0 ¼ 0, T Þ¼E Q ðeV T ¼ max 0, ! L 0þð1L TR T0 r sds V T jF 0 Þ, ð56Þ!Þ S TS 0þ max 0, ð1 !Þ L 0L Tþ ! S TS 01 S 1T >S 1 01 S 1T 5S 1 0,where ! 0 is a weighting factor related to a percentage,and L T ¼ P Mi¼1 PðT, t iÞ with t 1 ¼ T is the T-value of theprojected liabilities for certain time t M , with!4100% !.Figure 3 shows the prices obtained from Monte Carlosimulation of the contract at time t 0 ¼ 0 for maturityT ¼ t 1 ¼ 3 and time horizon t M ¼ 12 with one year spacing.
100 L. A. Grzelak et al.10.9Strategic Investment SZHW, HHW with r x,r= −0.7 and the Heston modelSchöbel−Zhu−Hull−WhiteHeston−Hull−WhiteHeston10.9Strategic Investment SZHW, HHW with r = 0.7 and the Heston modelx,rSchöbel−Zhu−Hull−WhiteHeston−Hull−WhiteHestonValue of Strategic Investment at t = 00.80.70.60.5Value of Strategic Investment at t = 00.80.70.60.50.40.40% 50% 100% 150% 200% 250%ω0% 50% 100% 150% 200% 250%ωFigure 3. Discounted payoffs of the strategic investment hybrid priced with the SZHW, the HHW and the Heston models as afunction of !. The remaining parameters are as in table 4. Left: Pricing with x,r ¼ 0.7. Right: Pricing with x,r ¼ 0.7.Downloaded by [Lech A. Grzelak] at 10:55 24 January 2012Since we did not model the second underlying process, S 1 T ,we assume that S 1 T > S1 0 . We see that, for ! 2 [0%, 100%],the max over the sum of performances disappears andthe hybrid can be relatively easily priced, i.e. separatelyfor both underlyings (L 0 /L T and S T /S 0 ). The differencebetween the stochastic models becomes more pronouncedfor !40% since, then, the correlation plays a moreimportant role. The simulations performed for x,r ¼ 70% and x,r ¼ 70% show that the absolutedifference between the SZHW and HHW models becomessignificant for !4200%. The figure shows that, for small!, the prices of the SZHW and HHW models arerelatively close, whereas the Heston model gives lowerprices for !450%.5. ConclusionsIn this paper we have presented an extension of theScho¨bel–Zhu stochastic volatility model with a Hull–White interest rate process and evaluated it by means ofpricing structured hybrid derivative products.The aim was to define a hybrid stochastic process thatbelongs to the class of affine jump-diffusion models, asthis may lead to efficient calibration of the model. Wehave shown that the so-called Scho¨bel–Zhu–Hull–Whitemodel belongs to the category of affine jump-diffusionprocesses. No restrictions regarding the choice ofcorrelation structure between the different Wiener processesappearing need to be made.We also compared the model with the Heston–Hull–White hybrid model with an indirectly implied correlationbetween the equity and the interest rate. We found thatalthough the model is very attractive because of its squareroot volatility structure, it is unable to generate extremecorrelations.Due to the resulting semi-closed (for Scho¨bel–Zhu–Hull–White) and closed (Heston–Hull–White)characteristic functions we were able to calibrate themodels in an efficient way by means of the Fourier-cosineexpansion pricing technique, adapted to a stochasticinterest rate.It has been shown by numerical experiments fordifferent hybrid products that under the same plainvanilla prices the extended stochastic volatility modelsgive different prices than the Heston model.The present hybrid model cannot model a skew in theinterest rates, which will form part of our future work.AcknowledgementsThe authors would like to thank two anonymous refereesfor valuable suggestions that have significantly improvedthe paper. Moreover, the authors thank NataliaBorovykh and Roger Lord from RabobankInternational and Bin Chen and Fang Fang from theDelft University of Technology for helpful comments.ReferencesAlbanese, C. and Lawi, S., Laplace transforms for integrals ofMarkow processes. Working Paper, 2007. Available onlineat: http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.1599v1.pdf(accessed 2008).Andersen, L.B.G., Efficient simulation of the Heston stochasticvolatility model. J. Comput. Finance, 2007, 11, 1–42.Arnold, L., Stochastic Differential Equations, Theory andApplications, 1973 (Wiley: New York).Brigo, D. and Mercurio, F., Interest Rate Models—Theory andPractice: With Smile, Inflation and Credit, 2nd ed., 2007(Springer: Berlin).Broadie, M. and Kaya, O¨ ., Exact simulation of stochasticvolatility and other affine jump diffusion processes. Oper.Res., 2006, 54, 217–231.Carr, P.P. and Madan, D.B., Option valuation using the fastFourier transform. J. Comput. Finance, 1999, 2, 61–73.
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