Why do we need the self-financing assumption in risk-neutral pricing?

A portfolio is self-financing if the purchase of a new asset must be financed by the sale of an old one.

\begin{align*} x_t(1+R) + y_tS_t = x_{t+1} + y_{t+1}S_t \end{align*}

This says that, at each time $$t$$, the market value of the "old" portfolio $$(x_t, y_t)$$ (which was created at $$t-1$$) equals the purchase value of the new portfolio $$(x_t+1, y_t+1)$$, which is formed at $$t$$ (and held until $$t+1$$).

Why do we need a particular portfolio (self-financing/replicating portfolio) in risk-neutral world?

3 Answers

In practice, the self-financing condition can be regarded as an economic consequence of market competition. Take the perspective of an investment bank trading in hedgeable derivatives. If the hedging strategy is not self-financing, then it must be either:

• Generating cash outflows for the bank. It is therefore uneconomical for the bank to trade this product; supply will decrease, pushing up the market price of the derivative until its value is sufficient to allow for the construction of a self-financing strategy.
• Generating cash inflows to the bank. The bank is therefore extracting a profit surplus from trading the derivative. Competing banks can offer lower prices, bounded from below by the price which allows for a self-financing strategy.

In short, in a competitive market, supply-demand dynamics should push the derivative price towards its equilibrium, which is the price that enables the bank to perfectly replicate its payoff without any additional cash outflow or inflow.

I am not sure (or at least I do not remember) whether there is an intrinsic, purely mathematical reason which requires the self-financing assumption.

You don’t just need self-financing in a risk-neutral world but it’s a much more fundamental principle. If you look at a portfolio that is not self-financing, i.e. you can inject or withdrawal funds at any time, you can hedge any derivative easily. If you can always add the amount of money you need, then hedging becomes trivial. Thus, one requires the self-financing property in the definitions of arbitrage portfolios and hedging strategies. These concepts are even more fundamental than going over to a risk-neutral world.

Look at an arbitrage: a self-financing portfolio which has zero initial cost but a positive probability of paying a positive payoff. In any state, it pays at least zero. If you drop the assumption of self-financing, you can think of a portfolio that buys and sells nothing (zero initial cost) and hence pays nothing but you inject 1 unit of the currency (numeraire) in the next period. Then, you always have a positive payoff but it does not really capture the idea of arbitrage and a free lunch.

Thus, self-financing is the key property of admissible portfolios.

It really simplifies your life when dealing with valuation. As stated already a non-self-financing portfolio either generate or absorb cashflow. Such cash flows would need to be taken into account when valuing a certain derivative based on replication.

So, in general, is way easier to just deal with a self-financing portfolio.

On the other hand, when you try to replicate a derivative, like a European Option, that may deliver a pay-off only at maturity, so no intermediate cash flow, it comes naturally to replicate such contingent claim with a self-financing strategy, matching its cashflow schedule.