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Constraints and Practical Costs in Zero-Cost Portfolio Optimization

Article Quant Q&A · Author: Nick
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Summary

The document asks how to add an initial-cost limit to an integer linear portfolio optimization problem. Portfolio positions can be positive for purchases or negative for short sales, and the proposed constraint allows initial cost to be capped at a positive, zero, or negative amount. The practical motivation is whether proceeds from selling contracts can fund purchases within the same portfolio.

The cited discussion of zero-cost option strategies cautions that zero initial premium does not eliminate collateral, processing, valuation, or opportunity costs. The response recommends looking into sequential quadratic programming and suggests that the formulation needs clearer treatment of transaction timing, asset values through time, and buying versus selling. It does not establish that the proposed integer program is feasible or executable in a market, nor does it provide a worked optimization method; those questions require more detailed constraints and realistic trading costs.

Key ideas

  • An initial-cost cap can be formulated to allow positive, zero, or negative portfolio costs.
  • Short-sale proceeds may fund purchases in a portfolio, subject to market and financing constraints.
  • A zero-premium options structure can still require collateral and incur other economic costs.
  • Sequential quadratic programming is suggested as a relevant optimization approach.
  • Transaction timing and evolving asset values should be specified more clearly in the objective and constraints.

Tags

OptionsPortfolio constructionRisk management
Full text
# Portfolio with zero or negative initial cost


# 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.

## Answer by user20928 (score 1)

https://quant.stackexchange.com/a/26328

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.