Friday, August 10, 2018

Black-Scholes formula with deterministic discrete dividend (Musiela approach)



For deterministic discrete dividend, there are two approach



  • Musiela approach, works when every dividend are paid at maturity of the option.

  • Hull approach, works when every dividend are paid immediately after ex-dividend date.


I spend 1 day to understand the Musiela approach, but I can not understand his formula. In his book "Martingal Method for Financial Modelling 2nd Edit" $3.2.2, his first approach firstly define quantity :




  • Timeline $0 < T_1




  • Value of all posterior-t dividend compounded to Maturity time : It=mi=1qier(TTi)1[0,Ti](t)

    Note that It decrease in time t and piecewise constant. At each time Ti, It drop down qi



  • Value of all anterior-t dividend compound to time t. Dt=mi=1qier(tTi)1[Ti,T](t)
    Here, Dt increase in time t. At each time Ti, Dt jump up qi

  • He define the capital gain process Gt=St+Dt


Note that DT=I0G0=S0GT=ST+DT=ST+I0


And all jump in price process St are separated to Dt, he can model Gt by the geometric brownian as usual, i.e under risk-neutral measure dGtGt=rdt+σdWt
Now, he can give the B&S formula for European Call option at time zero C0=erTE[(STK)+]=erTE[(GT(K+I0))+]
Since the modelled process is Gt, this price at time 0 is easily found by Black-Scholes calculation routine. C0=S0N(d+)erTKN(d)
with d±=lnS0K+I0+(r±σ22)TσT
For this price formula at time 0, I can understand it. An then I tried to compute for an arbitrary time t Ct=er(Tt)E[(STK)+|Ft]=er(Tt)E[(GT(K+I0))+|Ft]
Again, the calculation routine of Black-Scholes should give Ct=GtN(d+)er(Tt)(K+I0)N(d)
with d± should be d±=lnGtK+I0+(r±σ22)(Tt)σTt
But in the Musiela's book, he give the different result without detail proof. His result is Ct=StN(ˆd+)er(Tt)(K+It)N(ˆd)
with ˆd±=lnStK+It+(r±σ22)(Tt)σTt
So the annoying differences are



  • He have strike term as K+It, I have K+I0


  • He have random process as St, I have Gt


Can anyone help please. I've spent to much time without success.


PS : one more question. There are maybe something that I am missing. The fact that he use Dt to model the dividend, but in the result, he use It, that seems strange.




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