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Self Financing portfolio in continuous time

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asked Jun 10 in BUS 4028F - Financial Economics by anonymous

How do you show a portfolio to be self financing in continuous time question 3 i)? Please provide a general description and then the question specific method of doing this. BUS4028F tutorial 6 2017 _SV_.pdf (0,1 MB)

1 Answer

+2 votes
answered Jun 12 by hssnic004 (280 points)
selected Jun 18 by Richard van Gysen
 
Best answer

In words: to show a portfolio is self-financing, you want to show that the (instantaneous) change in the value of the portfolio is equal to the change in the value of your cash/bank account and risky asset(s), i.e.:

$$ dV_t = \phi_t dS_t  + \psi_t dB_t$$

where \(  \phi_t \) is the quantity of the risky asset, and \( \psi_t \) is the quantity of cash/bank account. (NOTE: this is bookwork – cf. Alex's slides). This is exactly what you want to do in the tutorial question. 

For the tutorial question in particular, you can either start with \( dV_t \) and show that this equals \( \phi_t dS_t + \psi_t dB_t \), or you can do it in the opposite order (i.e. start with  \( \phi_t dS_t + \psi_t dB_t \) and go from there). 

Furthermore, you need to use some of the additional information given in the question. 

From there, it's mostly a matter of manipulation of the expressions (e.g. using Ito's lemma) and substitution to show that self-financing holds. As such,

i) If you start with \( dV_t \), then the key thing to note is that

$$ B_t E_t = e^{rt}e^{-rt}V_t = V_t $$

and so 

$$ dV_t = d(B_t E_t) $$

ii) if you start with \( \phi_t dS_t + \psi_t dB_t \), then the key thing to note is that

$$B_t D_t = e^{rt} e^{-rt} S_t = S_t $$

and so 

$$dS_t = d(B_t D_t) $$

Remember that all of the definitions of the terms in the above expressions are given in the question – use them as such. 

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