Self Financing portfolio in continuous time

+1 vote
63 views

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)

by (280 points)
selected

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.