International Journal of Theoretical and Applied Finance, 2010
Asian options incorporate the average stock price in the terminal payoff. Examination of the Asia... more Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.
By using Sturrn's Comparison theorem on the quasi-differential equation we generate a sequen... more By using Sturrn's Comparison theorem on the quasi-differential equation we generate a sequence of functions which has the property that its members separate the mutually exclusive behaviour of oscillation and non-oscillation of the given equation. It is shown that this ‘separation property’ is an increasing function of the order of members in the sequence generated i.e. Fk+l is a ‘sharper’ and ‘better’ separating function than Fk whenever is a suitably constructed sequence for the differential equation. Using several examples we establish necessary and sufficient conditions for this trend to hold.
We revisit the classical Merton portfolio selection model from the perspective of in-
tegrability... more We revisit the classical Merton portfolio selection model from the perspective of in- tegrability analysis. Applying a nonlocal transformation the nonlinear partial dierential equation for the two asset model is mapped to a linear option valuation equation with a consumption dependent source term - an identical result to that obtained by Cox and Huang using measure theory and stochastic integrals. The nonlinear two asset equation is then an- alyzed using Lie symmetry group theory. We show that the linearisation is directly related to the structure of the generalized symmetries.
Numerous studies have presented strong empirical evidence that certain financial
assets may exhib... more Numerous studies have presented strong empirical evidence that certain financial assets may exhibit mean reversion, stochastic volatility or jumps. This evidence discredits the classical continuous time paradigm with continuous sample price paths inherent in the use of governing the asset price process solely by a Brownian motion processes. This paper explores the valuation of European options when the underlying asset follows a mean reverting log-normal process with stochastic volatility and Jumps. A closed form representation of the characteristic function of the process is derived for the computation of European option prices via the Fast Fourier Transform. A numerical example comparing this technique to a Monte Carlo simulation is demonstrated to highlight the efficiency of the Fast Fourier Transform approach.
In the standard approach to fund valuation, it is often assumed that markets are perfectly liquid... more In the standard approach to fund valuation, it is often assumed that markets are perfectly liquid and hence assets have unique prices. In practice, however, as has been widely documented, this is not the case. Asset values are impacted by deterioration of market liquidity (market depth). However, the work by Acerbi and Scandolo (Quantitative Finance, 2008, 8(7), 681) highlighted, in addition, the key role that the liquidity policies of the fund can have on the fund value. Funds with the identical positions but differing liquidity policies have different values. In this paper, we describe and analyze, the relationship between the fund value and the liquidity policy (constraints that can potentially be called upon, and that are imposed externally on the fund manager). These constraints, for example, can be a requirement to be able to generate an amount of cash (due to possible adverse redemption), a minimum weight requirement per asset, or a market risk constraint such as a maximum expected shortfall (ES). In this paper we employ a Power-Law Marginal Supply-Demand Curves (MSDCs) to model market depth, together with various types of liquidity policies. These liquidity restrictions affect, to a varying degree, the liquidation process of illiquid and high volatility assets. We find that as the liquidity policy becomes stricter with the portfolio ES constraint, the liquidity costs increase significantly.
International Journal of Theoretical and Applied Finance, 2010
Asian options incorporate the average stock price in the terminal payoff. Examination of the Asia... more Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.
By using Sturrn's Comparison theorem on the quasi-differential equation we generate a sequen... more By using Sturrn's Comparison theorem on the quasi-differential equation we generate a sequence of functions which has the property that its members separate the mutually exclusive behaviour of oscillation and non-oscillation of the given equation. It is shown that this ‘separation property’ is an increasing function of the order of members in the sequence generated i.e. Fk+l is a ‘sharper’ and ‘better’ separating function than Fk whenever is a suitably constructed sequence for the differential equation. Using several examples we establish necessary and sufficient conditions for this trend to hold.
We revisit the classical Merton portfolio selection model from the perspective of in-
tegrability... more We revisit the classical Merton portfolio selection model from the perspective of in- tegrability analysis. Applying a nonlocal transformation the nonlinear partial dierential equation for the two asset model is mapped to a linear option valuation equation with a consumption dependent source term - an identical result to that obtained by Cox and Huang using measure theory and stochastic integrals. The nonlinear two asset equation is then an- alyzed using Lie symmetry group theory. We show that the linearisation is directly related to the structure of the generalized symmetries.
Numerous studies have presented strong empirical evidence that certain financial
assets may exhib... more Numerous studies have presented strong empirical evidence that certain financial assets may exhibit mean reversion, stochastic volatility or jumps. This evidence discredits the classical continuous time paradigm with continuous sample price paths inherent in the use of governing the asset price process solely by a Brownian motion processes. This paper explores the valuation of European options when the underlying asset follows a mean reverting log-normal process with stochastic volatility and Jumps. A closed form representation of the characteristic function of the process is derived for the computation of European option prices via the Fast Fourier Transform. A numerical example comparing this technique to a Monte Carlo simulation is demonstrated to highlight the efficiency of the Fast Fourier Transform approach.
In the standard approach to fund valuation, it is often assumed that markets are perfectly liquid... more In the standard approach to fund valuation, it is often assumed that markets are perfectly liquid and hence assets have unique prices. In practice, however, as has been widely documented, this is not the case. Asset values are impacted by deterioration of market liquidity (market depth). However, the work by Acerbi and Scandolo (Quantitative Finance, 2008, 8(7), 681) highlighted, in addition, the key role that the liquidity policies of the fund can have on the fund value. Funds with the identical positions but differing liquidity policies have different values. In this paper, we describe and analyze, the relationship between the fund value and the liquidity policy (constraints that can potentially be called upon, and that are imposed externally on the fund manager). These constraints, for example, can be a requirement to be able to generate an amount of cash (due to possible adverse redemption), a minimum weight requirement per asset, or a market risk constraint such as a maximum expected shortfall (ES). In this paper we employ a Power-Law Marginal Supply-Demand Curves (MSDCs) to model market depth, together with various types of liquidity policies. These liquidity restrictions affect, to a varying degree, the liquidation process of illiquid and high volatility assets. We find that as the liquidity policy becomes stricter with the portfolio ES constraint, the liquidity costs increase significantly.
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tegrability analysis. Applying a nonlocal transformation the nonlinear partial dierential
equation for the two asset model is mapped to a linear option valuation equation with a
consumption dependent source term - an identical result to that obtained by Cox and Huang
using measure theory and stochastic integrals. The nonlinear two asset equation is then an-
alyzed using Lie symmetry group theory. We show that the linearisation is directly related
to the structure of the generalized symmetries.
assets may exhibit mean reversion, stochastic volatility or jumps. This evidence
discredits the classical continuous time paradigm with continuous sample price
paths inherent in the use of governing the asset price process solely by a Brownian
motion processes. This paper explores the valuation of European options when
the underlying asset follows a mean reverting log-normal process with stochastic
volatility and Jumps. A closed form representation of the characteristic function
of the process is derived for the computation of European option prices via the
Fast Fourier Transform. A numerical example comparing this technique to a
Monte Carlo simulation is demonstrated to highlight the efficiency of the Fast
Fourier Transform approach.
tegrability analysis. Applying a nonlocal transformation the nonlinear partial dierential
equation for the two asset model is mapped to a linear option valuation equation with a
consumption dependent source term - an identical result to that obtained by Cox and Huang
using measure theory and stochastic integrals. The nonlinear two asset equation is then an-
alyzed using Lie symmetry group theory. We show that the linearisation is directly related
to the structure of the generalized symmetries.
assets may exhibit mean reversion, stochastic volatility or jumps. This evidence
discredits the classical continuous time paradigm with continuous sample price
paths inherent in the use of governing the asset price process solely by a Brownian
motion processes. This paper explores the valuation of European options when
the underlying asset follows a mean reverting log-normal process with stochastic
volatility and Jumps. A closed form representation of the characteristic function
of the process is derived for the computation of European option prices via the
Fast Fourier Transform. A numerical example comparing this technique to a
Monte Carlo simulation is demonstrated to highlight the efficiency of the Fast
Fourier Transform approach.