# Portfolio with zero or negative initial cost

Let's say I have formulated an integer linear programming (ILP) problem with the objective function $$F(X)=V(T,X)-C(t,X),$$ where $V(T,X)$ is the payoff of portfolio, and $C(t,X)$ is the initial cost of portfolio, $0<t<T$ is the calendar time, then I have setup a system of constraints and found the optimum solution $X=(x_1, x_2, \ldots, x_n)$, where $x_i$ is the number of units of an $i$-th asset in the portfolio, with $x_i>0$ for buying, $x_i<0$ for short selling.

Now I'd like to extend the system of constraints and add new constraint on the initial cost $C(t,X)$. Let's say $C(t,X)\le c$, where $c$ can be either a positive number or zero, or even negative number. I think that theoretically I can find the optimum solution $X$ with the constraint $C(t,X)\le c$.

In the study (Bartoňová M., 2012) was demonstrated the usage of zero-cost option’s strategy in hedging of sales. But on the page 125 the author conclude:

And are they really zero-cost? As for initial fee, than yes. It is necessary to take into consideration that there is necessary general agreement with bank for option trading. It must be covered by collateral. There are also costs of contract processing, expert's opinions for assets evaluation, opportunity costs influencing of pledge, also of call option sale... Any zero-cost options are not really zero.

My question: Can I assume than an ivestor can use the money received from the sale of some contracts to buy of other contracts in the portfolio? Can I realize the optimal porfolio with the zero or negative initial cost on a market?

Update.

Peter Carr and Dilip Madan. Towards a theory of volatility trading. In R. Jarrow, editor, Volatility , pages 417-427. Risk Publications, 1998.

• The question is not clear to me. If the desired payoff is only a function of the price the underlying will take at a single future date $T$ (we call that a European option), then it is always possible to write it as a continuous sum of call and put options. This is a model free result (see Carr & Madan decomposition). In that case $M_f$ would be the forward price $F (0,T)$. – Quantuple May 29 '16 at 12:40
• @Quantuple, thanks you for the comment. I have updated my topic. I have the trouble with transformation economical task to mathematical problem. I'm searching models/papers but I feel a lack of keywords in my mind ((( – Nick May 29 '16 at 17:24

## 1 Answer

Interesting question.

To answer it directly, try searching for the term 'Sequential Quadratic Programming'. This should lead you to relevant references.

More details, if I am reading your question correctly you are hoping to minimize the loss of a strategy involving sequential transactions (buying or selling) of options. I think your question would be more clear if you also indexed the the asset value by time (the equation does look like you are assuming the asset price is fixed at some future date throughout the duration of the strategy period) and if you do not impose symmetry with respect to the time you do a transaction involving an option. I also think it would be easier to express this problem as a minimization problem and to make it clear whether you are buying or selling the option at a given time period. This should help clarify your objective function.

• I would like to create a portfolio (buying or selling) of options at the one market session. Then I going to manage the portfolio and close and open some positions. Why are you recommend the term 'Sequential Quadratic Programming'? Should be objective function non-liner? – Nick Jun 1 '16 at 0:39
• Sequential, because not all of the portfolio managing decisions occur at the same point in time. Quadratic, because you expressed interest in formulating the problem as a non-linear one and this should expose you to problems with non-linear objective functions and how they are formalized mathematically. Program, to complete the phrase. In graduate school we studied Quadratic Programming as a technique to solve optimization problems with non-linear objective functions subject to linear constraints. – user20928 Jun 3 '16 at 22:56