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Abstract:
This chapter provides a comprehensive explanation of hedge fund replication. This chapter first reviews the characteristics of hedge fund returns. Then, the emergence of hedge fund replication products is discussed. Hedge fund replication methods are classified into three categories: rule-based, factor-based, and distribution replicating approaches. These approaches attempt to capture different aspects of hedge fund returns. This chapter explains the three methods.

hedge funds, alternative investments, hedge fund clones

Abstract:
There are now available wide variety of swap products which exchange Libors with different currencies and tenors. Furthermore, the collateralization is becoming more and more popular due to the increased attention to the counter party credit risk. These developments require clear distinction among different type of Libors and the discounting rates. In this brief note, we will explain the method to construct the multiple swap curves consistently with all the relevant swaps with and without a collateral agreement.

Libor, swap, tenor, yield curve, collateral, overnight index swap, cross currency, basis spread

Abstract:
This paper provides a new method to construct a dynamic optimal portfolio for asset management in a complete market. The method generates a target payoff distribution by the cheapest dynamic trading strategy. It is regarded as an extension of Dybvig (1988a) to continuous-time framework and dynamic portfolio optimization where the dynamic trading strategy is derived analytically by applying Malliavin calculus. As a practical example, the method is applied to hedge fund replication, which extends Kat and Palaro (2005) and Papageorgiou, Remillard and Hocquard (2008) to multiple trading assets with both long and short positions.

Hedge Fund Replication, Asset Management, Dynamic Portfolio Optimization, Malliavin Calculus, Martingale Method

Abstract:
This paper studies the probability distribution and option pricing for drawdown in a stochastic volatility environment. Their analytical approximation formulas are derived by the application of a singular perturbation method (Fouque et al. [7]). The mathematical validity of the approximation is also proven. Then, numerical examples show that the instantaneous correlation between the asset value and the volatility state crucially affects the probability distribution and option prices for drawdown.

drawdown, stochastic volatility, singular perturbation

Abstract:
An asymptotic expansion scheme in finance initiated by Kunitomo and Takahashi [15] and Yoshida[68] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. [46], [47] and [53] provide explicit formulas of conditional expectations necessary for the asymptotic expansion up to the third order. In general, the crucial step in practical applications of the expansion is calculation of conditional expectations for a certain kind of Wiener functionals. This paper presents two methods for computing the conditional expectations that are powerful especially for high order expansions: The first one, an extension of the method introduced by the preceding papers presents a general scheme for computation of the conditional expectations and show the formulas useful for expansions up to the fourth order explicitly. The second one develops a new calculation algorithm for computing the coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate their effectiveness, the paper gives numerical examples of the approximation for λ-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.

asymptotic expansion, stochastic volatility, λ-SABR model, Libor market model, Malliavin calculus, cross currency model, foreign exchange rate option (Forex Option)

Abstract:
This paper shows pricing and hedging efficiency of a three factor stochastic mean reversion Gaussian model of commodity prices using oil and copper futures and forward contracts. The model is estimated using NYMEX WTI (light sweet crude oil) and LME Copper futures prices and is shown to fit the data well. Furthermore, it shows how to hedge based on a three-factor model and confirms that using three different futures contracts to hedge long-term contract outperforms the traditional parallel hedge based on a single futures position by time series data and simulation. It also finds that the three factor model outperforms its two-factor version in replication of actual term structures and that stochastic mean reversion models outperform constant mean reversion models in Out of Sample hedges.

Abstract:
This short note proposes an approximation method of pricing barrier options under stochastic volatility environment by applying an asymptotic expansion approach combined with a static hedging method. In particular, through numerical examples it shows that the fifth-order normal approximation of an asymptotic expansion scheme (Shiraya-Takahashi-Toda, Takahashi-Takehara-Toda) with a modification of a static hedging method by Fink provides good approximations under the lambda-SABR model.

barrier option, knock-out option, static hedge, asymptotic expansion, stochastic volatility, lambda-SABR model

Abstract:
This paper proposes a new hedging scheme of European derivatives under uncertain volatility environments, in which a weighted variance swap called the polynomial variance swap is added to the Black-Scholes delta hedging for managing exposure to volatility risk. In general, under these environments one cannot hedge the derivatives completely by using dynamic trading of only an underlying asset owing to volatility risk. Then, for hedging uncertain volatility risk, we design the polynomial variance, which can be dependent on the level of the underlying asset price. It is shown that the polynomial variance swap is not perfect, but more efficient as a hedging tool for the volatility exposure than the standard variance swap. In addition, our hedging scheme has a preferable property that any information on the volatility process of the underlying asset price is unnecessary. To demonstrate robustness of our scheme, we implement Monte Carlo simulation tests with three different settings, and compare the hedging performance of our scheme with that of standard dynamic hedging schemes such as the minimum-variance hedging. As a result, it is found that our scheme outperforms the others in all test cases. Moreover, it is noteworthy that the scheme proposed in this paper continues to be robust against model risks.

European Derivatives, Black-Scholes Delta Hedging, Uncertain Volatility Risk, Polynomial Variance Swap

Abstract:
This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for the valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates.

Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained.

An asymptotic expansion approach provides a closed-form approximation formula for their values, which can be calculated in a moment and thus can be used for calibration or for an explicit approximation of Greeks of options. Moreover, this scheme develops Fourier transform method with an asymptotic expansion as well as with closed-form characteristic functions obtainable in parts of a model, extending the method proposed by Takehara and Takahashi [2008] to be applicable to a general class of models.

It also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of the closed-form characteristic functions. Finally, a series of numerical examples shows the effectiveness of our scheme.

Currency option, libor market model, stochastic volatility, asymptotic expansion, Monte Carlo simulation

Abstract:
This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula in Malliavin calculus is effectively applied in an asymptotic expansion approach. First, the paper derives an expansion formula for generalized Wiener functionals. After it is applied to pricing path-dependent derivatives with discrete monitoring, the paper presents an analytic (approximation) formula for valuation of discrete barrier options under stochastic volatility environment. To our knowledge, this paper is the first one that shows an analytical formula for pricing discrete barrier options with stochastic volatility models.

discrete barrier option, barrier option, knock-out option, Malliavin calculus, stochastic volatility, asymptotic expansion, Malliavin weight

Abstract:
This paper proposes a new approximation formula for average options on commodities under stochastic volatility environment. In particular, it derives a formula under two stochastic volatility models such as Heston and λ-SABR models including the SABR model as a special case by using an asymptotic expansion method.

To our knowledge, this paper is the first one that shows an analytic (approximation) formula under stochastic volatility models for valuation of average options structured for commodity contracts. Then, it confirms its sufficient accuracy through numerical examples. It also implements calibration to the WTI futures option market that is one of the most liquid commodity markets. Using the parameters obtained by calibration, it compares model-based prices with those of traded average options in NYMEX.

Abstract:
This paper reviews an asymptotic expansion approach to numerical problems for pricing financial assets and securities.

Abstract:
This paper proposes a new approximation method of pricing barrier and average options under stochastic volatility environment by applying an asymptotic expansion approach. In particular, a high-order expansion scheme for general multi-dimensional diffusion processes is effectively applied. Moreover, the paper combines a static hedging method with the asymptotic expansion method for pricing barrier options. Finally, numerical examples show that the fourth or fifth-order asymptotic expansion scheme provides sufficiently accurate approximations under the lambda-SABR and SABR models.

barrier option, average option, knock-out option, stochastic volatility, static hedge, asymptotic expansion,lambda-SABR model, SABR model

Abstract:
This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston[1993], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.

asymptotic expansion, currency options, libor market model, Malliavin calculus, stochastic volatility

Abstract:
This paper develops a Fourier transform method with an asymptotic expansion approach for option pricing. The method is applied to European currency options with a libor market model of interest rates and jump diffusion stochastic volatility models of spot exchange rates. In particular, we derive closed form approximation formulas of the characteristic functions of log prices of the underlying assets and the prices of currency options based on a third order asymptotic expansion scheme; we use a jump diffusion model with a mean-reverting stochastic variance process such as in Heston[1993]/Bates[1996] and log normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

Currency option, libor market model, stochastic volatility, asymptotic expansion, Fourier transform

Abstract:
This paper studies the approximation accuracy of a singular perturbation method for option pricing up to the second order under a stochastic volatility model. First, numerical experiments confirm that the first order approximation provides sufficiently accurate option prices in a fast mean-reversion volatility case. On the other hand, it creates relatively large errors in a non-fast mean-reversion volatility environment. Then, the second order approximation formula is derived and the improvement of the approximation is investigated.

option pricing, stochastic volatility, partial differential equation, singular perturbation, approximation accuracy

Abstract:
This paper studies portfolio selection and performance analysis of hedge funds located or invested in Asia-Pacific. It investigates the characteristics of the funds' returns and recommends optimization methods to create a 'Fund-of-Funds'. The returns of the hedge funds are then decomposed into asset class factors. Finally, portfolio optimizations and performance analyses are integrated to show how these methods are utilized in practice.