Funding Costs in Present Value and No-Arbitrage Portfolios

# When does funding cost of a portfolio enter into the portfolio's present value?


# When does funding cost of a portfolio enter into the portfolio's present value?












This question comes from some confusion when reading Hull's book and from the general concept of no-arbitrage/self-financing portfolios in stochastic finance books. I am not fully seeing the distinction made between the present value of a portfolio and the cash flows made in setting up the portfolio, and when to use which.

In some cases when doing a no-arbitrage proof, I have seen that the funding cost of a portfolio doesn't enter into the portfolio present value. For example from https://maths.ucd.ie/~vlasenko/MST30030/fm5_0.pdf is a typical proof for European put-call parity:

So here, in Portfolio A, for example, the trader buys a European call and a zero coupon bond worth K at maturity, but the Portfolio's value considers them to be 'positive'. This seems to me 'as if the call and zero-coupon bond were free'. So here it seems like the cash flows themselves are not taken into consideration.

From Hull's book, Chapter 5.14:

> We suppose that the speculator puts the present value of the futures price into a risk-free investment while simultaneously taking a long futures position. The proceeds of the risk-free investment are used to buy the asset on the delivery date. The asset is then immediately sold for its market price.

The present value of this portfolio/investment is written as (according to Hull): $-F_0e^{-rT} + E(S_T)e^{-kT}$

So here the trader is putting money into a zero coupon bond that will give $F_0$ at maturity and also entering into a futures contract. But the present value of the bond investment is negative, rather than positive, unlike the earlier example. And now, even though the futures contract has 0 value at initiation the present value is written as $E(S_T)e^{-kT}$.

I could see this portfolio value arising from calculating the present value of a portfolio which in the future will consist of 'exchanging an amount $F_0$ for $S_T$', but then I am still confused why the portfolio takes into account the funding cost ($F_0$) of buying the stock, whereas the put-call parity example ignores funding costs.

Another issue that adds to the confusion is when showing that an arbitrage exists, the present value of the portfolio includes the proceeds of the transaction. For example, when a european call is mispriced and the present call value is above the present stock price, an arbitrage can be done by shorting the call and longing the stock. So the present value (at time $t$) of the portfolio would be $-c_t + S_t + proceeds$, where $proceeds = c_t - S_t$. So the present value of the portfolio is 0. So here the portfolio consists of both the assets and the cash flow. I am unsure why the funding costs are relevant here but not in other examples.

All the examples I gave above make sense to me in their own way, but I am unsure in which situations I would include the funding costs or not, and what the rationale is for (not) doing so.

Any clarifications/help would be greatly appreciated! Thanks!