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=m∑i=1qier(T−Ti)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=m∑i=1qier(t−Ti)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=e−rTE[(ST−K)+]=e−rTE[(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+)−e−rTKN(d−) with d±=lnS0K+I0+(r±σ22)Tσ√T For this price formula at time 0, I can understand it. An then I tried to compute for an arbitrary time t Ct=e−r(T−t)E[(ST−K)+|Ft]=e−r(T−t)E[(GT−(K+I0))+|Ft] Again, the calculation routine of Black-Scholes should give Ct=GtN(d+)−e−r(T−t)(K+I0)N(d−) with d± should be d±=lnGtK+I0+(r±σ22)(T−t)σ√T−t But in the Musiela's book, he give the different result without detail proof. His result is Ct=StN(ˆd+)−e−r(T−t)(K+It)N(ˆd−) with ˆd±=lnStK+It+(r±σ22)(T−t)σ√T−t 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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