A new approach to optimizing or hedging a portfolio of nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can be applied also to the optimization of percentiles in contexts outside of nance.

/pdf/optimization-of-conditional-value-at-risk-338oy3ddeu.pdf

Optimization of conditional value-at-risk

This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.

/pdf/valuing-american-options-by-simulation-a-simple-least-131pl9xpzk.pdf

Valuing American Options by Simulation: A Simple Least-Squares Approach

This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniqes cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.

/pdf/valuing-american-options-by-simulation-a-simple-least-2a6d5l0pu9.pdf

Valuing American Options by Simulation: A Simple Least-Squares Approach - eScholarship

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The Act of Creation

Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

http://dsec.pku.edu.cn/~tieli/notes/numer_anal/MCQMC_Caflisch.pdf

Monte Carlo and quasi-Monte Carlo methods

http://www.cs.hunter.cuny.edu/~felisav/StochasticProcesses/Course_Material_files/BBG_jedc96.pdf

Monte Carlo methods for security pricing

This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.

/pdf/model-specification-and-risk-premia-evidence-from-futures-59xmgu3max.pdf

Model specification and risk premia: evidence from futures options

http://www.axelkind.com/teaching/financial_derivatives/papers/broadie_glasserman_1997_jedc.pdf

Pricing American-style securities using simulation

We develop lower and upper bounds on the prices of American call and put options written on a dividend-paying asset. We provide two option price approximations, one based on the lower bound (termed LBA) and one based on both bounds (termed LUBA). The LUBA approximation has an average accuracy comparable to a 1,000-step binomial tree with a computation speed comparable to a 50-step binomial tree. We introduce a modification of the binomial method (termed BBSR) that is very simple to implement and performs remarkable well. We also conduct a careful large-scale evaluation of many recent methods for computing American option prices. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods

The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results, and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. This paper suggests a method for the exact simulation of the stock price and variance under Hestons stochastic volatility model and other affine jump diffusion processes. The sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O(s-1/2) convergence rate, where s is the total computational budget. The convergence rate for the Euler discretization method is O(s-1/3) or slower, depending on the model coefficients and option payoff function.

/pdf/exact-simulation-of-stochastic-volatility-and-other-affine-2xz0o4ay8f.pdf

Mark Broadie

Papers

Monte Carlo methods for security pricing

Model specification and risk premia: evidence from futures options

Pricing American-style securities using simulation

American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods

Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes