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Worst-case Value-at-Risk of nonlinear portfolios
- Management Science
, 2013
"... Abstract Portfolio optimization problems involving Value-at-Risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are compounded when the portfolio cont ..."
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Abstract Portfolio optimization problems involving Value-at-Risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR over all return distributions of the derivative underliers with given first-and second-order moments. The derivative returns are modelled as convex piecewise linear or-by using a delta-gamma approximation-as (possibly non-convex) quadratic functions of the returns of the derivative underliers. These models lead to new Worst-Case Polyhedral VaR (WPVaR) and Worst-Case Quadratic VaR (WQVaR) approximations, respectively. WPVaR serves as a VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that-unlike VaR that may discourage diversification-WPVaR and WQVaR are in fact coherent risk measures. We also reveal connections to robust portfolio optimization.
Robust Optimization of Currency Portfolios∗
, 2009
"... We study a currency investment strategy, where we maximize the re-turn on a portfolio of foreign currencies relative to any appreciation of the corresponding foreign exchange rates. Given the uncertainty in the estimation of the future currency values, we employ robust optimization techniques to max ..."
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We study a currency investment strategy, where we maximize the re-turn on a portfolio of foreign currencies relative to any appreciation of the corresponding foreign exchange rates. Given the uncertainty in the estimation of the future currency values, we employ robust optimization techniques to maximize the return on the portfolio for the worst-case for-eign exchange rate scenario. Currency portfolios differ from stock only portfolios in that a triangular relationship exists among foreign exchange rates to avoid arbitrage. Although the inclusion of such a constraint in the model would lead to a nonconvex problem, we show that by choosing ap-propriate uncertainty sets for the exchange and the cross exchange rates, we obtain a convex model that can be solved efficiently. Alongside robust optimization, an additional guarantee is explored by investing in currency options to cover the eventuality that foreign exchange rates materialize outside the specified uncertainty sets. We present numerical results that show the relationship between the size of the uncertainty sets and the dis-tribution of the investment among currencies and options, and the overall performance of the model in a series of backtesting experiments. Key words: robust optimization, portfolio optimization, currency hedg-ing, second-order cone programming 1
Robust International Portfolio Management
, 2010
"... We present an international portfolio optimization model where we take into account the two different sources of return of an international asset: the local returns denominated in the local currency, and the returns on the foreign exchange rates. The explicit consideration of the returns on exchange ..."
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We present an international portfolio optimization model where we take into account the two different sources of return of an international asset: the local returns denominated in the local currency, and the returns on the foreign exchange rates. The explicit consideration of the returns on exchange rates introduces non-linearities in the model, both in the objec-tive function (return maximization) and in the triangulation requirement of the foreign exchange rates. The uncertainty associated with both types of returns is incorporated directly in the model by the use of robust op-timization techniques. We show that, by using appropriate assumptions regarding the formulation of the uncertainty sets, the proposed model has a semidefinite programming formulation and can be solved efficiently. While robust optimization provides a guaranteed minimum return inside the uncertainty set considered, we also discuss an extension of our formu-lation with additional guarantees through trading in quanto options for the foreign assets and in equity options for the domestic assets.
COMISEF WORKING PAPERS SERIES Worst-Case Value-at-Risk of Non-Linear Portfolios
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Minimizing Downside Risk in Axioma Portfolio with Options
"... The market downturn in 2008 is considered by many economists to be the worst financial crisis since the Great Depression of the 1930s. During this time period, the Dow Jones Industrial Average fell 33.8%, the S&P 500 index fell 38.6%, and the NASDAQ fell 40.5%. Options are often used by portfoli ..."
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The market downturn in 2008 is considered by many economists to be the worst financial crisis since the Great Depression of the 1930s. During this time period, the Dow Jones Industrial Average fell 33.8%, the S&P 500 index fell 38.6%, and the NASDAQ fell 40.5%. Options are often used by portfolio managers to insure a portfolio against adverse market movements as in
Modeling a Risk-Based Criterion for a Portfolio with Options∗
, 2013
"... The presence of options in a portfolio fundamentally alters the portfolio’s risk and return profiles when compared to an all equity portfolio. In this paper, we advocate modeling a risk-based criterion for optioned portfolio selection and rebalancing problems. The criterion is inspired by Chicago Me ..."
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The presence of options in a portfolio fundamentally alters the portfolio’s risk and return profiles when compared to an all equity portfolio. In this paper, we advocate modeling a risk-based criterion for optioned portfolio selection and rebalancing problems. The criterion is inspired by Chicago Mercantile Exchange’s risk-based margining system which sets the collateralization requirements on margin accounts. The margin criterion computes the losses expected at the portfolio level using expected stock price and volatility variations, and is itself an optimization problem. Our contribution is to remodel the criterion as a quadratic programming subproblem of the main portfolio optimization problem using option Greeks. We also extend the margin subproblem to a continuous domain. The quadratic programming problems thus designed can be solved numerically or in closed-form with high efficiency, greatly facilitating the main portfolio selection problem. We present two extended practical examples of the application of our approach to obtain optimal portfolios with options. These examples include a study of liquidity effects (bid/ask spreads and limited order sizes) and sensitivity to changing market conditions. Our analysis shows that the approach advocated here is more stable and more efficient than discrete approaches to portfolio selection. 1
