RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 103 (2), 2009, pp. 373–385 Matem´ atica Aplicada / Applied Mathematics The price of liquidity in constant leverage strategies Marcos Escobar, Andreas Kiechle, Luis Seco and Rudi Zagst Abstract. In this paper we develop a formula for the Liquidity Premium of constant leverage strategies (CLS). These financial products are path dependent options where the underlying typically is a hedge fund portfolio. We describe and explain the functionality of CLSs, showing a closed form expression for the price of a CLS on a hedge fund assuming a Geometric Brownian Motion, discrete rebalancing for the hedge fund investment as well as stochastic interest rates. The risk of default before the next rebalancing date leads to a liquidity premium for the CLS which increases with the volatility of the underlying hedge fund portfolio and the leverage of the strategy. An increasing rebalancing period first leads to a higher liquidity premium, however, as the rebalancing period is extended further the liquidity premium begins to shrink again. El precio de liquidez en estrategias con apalancamiento constante Resumen. La funcionalidad de las estrategias de apalancamiento constante (CLS) es investigada en este art´ ıculo. Estos productos financieros son opciones depedendientes del camino, donde los t´ ıpicos subyacentes son Hedge Funds. En particular se encuentra una f´ ormula cerrada para el precio de liquidez de este derivado en el contexto de procesos brownianos geom´ etricos con reajuste discreto de la cartera y tasa de inter´ es estoc´ astica. El riesgo de bancarrota antes de un reajuste conlleva a un precio de liquidez para el CLS, el cual es proporcional a la volatilidad del activo subyacente y al apalancamiento de la estrategia. Un incremento en el periodo entre reajustes implica un incremento inicial en el precio, sin embargo, el precio disminuye para largos periodos de reajuste. 1 Introduction Constant Leverage Strategies are options where the underlying typically is a hedge fund portfolio. In this context leverage is defined as the ratio of debt to equity. An investment strategy with constant leverage works as follows. In addition to his own money, which we will refer to as the equity, the investor takes a loan and invests the sum of the equity and the loan in a portfolio. As the portfolio value changes over time, the leverage of the portfolio changes as well. The aim of a constant leverage strategy is to keep the initial leverage constant, which implies that the financing of the portfolio has to be adjusted. In the case that the value of the underlying portfolio has increased more than the interest on the loan, the investor has to take an additional loan to uphold the leverage. In the other case, the investor pays back a fraction of the loan. In the end, the investor pays back the cumulative loan (including interest) and his payoff is the value of the portfolio minus the loan and accrued interest. A CLS is a product where the issuer of the option sells a constant leverage strategy, typically on a hedge fund portfolio, to an investor. The underlying portfolio is Presentado por / Submitted by Alejandro Balb´ as. Recibido / Received: February 10, 2009. Aceptado / Accepted: April 1, 2009. Palabras clave / Keywords: CPPI, hedge fund, liquidity premium, constant leverage strategy. Mathematics Subject Classifications: 91B30, 47N30. c  2009 Real Academia de Ciencias, Espa˜ na. 373