## Upper Bounds for American Option Prices using Regression with Martingale Basis Functions (2004)

### BibTeX

@MISC{Firth04upperbounds,

author = {N. P. Firth},

title = {Upper Bounds for American Option Prices using Regression with Martingale Basis Functions},

year = {2004}

}

### OpenURL

### Abstract

High dimensional American options have no analytic solution and are di#cult to price numerically. Progress has been made in using Monte Carlo simulation to give both lower and upper bounds on the price. Building on an idea of Glasserman and Yu we investigate the utility of martingale basis functions in regression based approximation methods. Regression methods are known to give lower bounds easily, however upper bounds are usually computationally expensive.

### Citations

565 | Option pricing: a simplified approach
- Cox, Ross, et al.
- 1979
(Show Context)
Citation Context ...olutions have been found for American option prices 1 . Therefore much work has been done pricing American options numerically. Early examples include finite differences [12] and the binomial lattice =-=[20]. Gr-=-id based methods work well for single asset options, and have been extended to higher dimensions [8, 10]. However, these methods suffer from the ‘curse of dimensionality’, as they become exponenti... |

330 |
Carlo methods in financial engineering
- Glasserman, Monte
(Show Context)
Citation Context ...een the number of basis functions and number of paths required is investigated by Glasserman and Yu [28]. For a comparison of simulation approaches see Fu et al. [25] or the recent book by Glasserman =-=[27]. Glasserman-=- and Yu [29] investigate the relative merits of ‘regression now’ versus ‘regression later’. ‘Regression now’ involves using basis functions defined at the current time step and regressing ... |

291 | Valuing American Options by Simulation: A simple Least-Square Approach
- Longstaff, Schwarz
- 2001
(Show Context)
Citation Context ...ization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22], irregular grid approximations [5], and sparse grid methods [37]. Regression methods include =-=[18, 33, 43]-=-. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to its accuracy and simplicity. The paper explains the method, but an introduction to the method is also giv... |

158 |
Monte Carlo methods for security pricing
- Boyle, Broadie, et al.
- 2001
(Show Context)
Citation Context ...irst suggested in [7]. Progress in using simulation methods to price American style options was stimulated by Tilley [42]. This and other early methods [4, 6, 14, 15, 24] are reviewed in Boyle et al. =-=[9]-=-. Since then the stochastic mesh method [15] has been made more efficient [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include parameterization of the optimal exe... |

157 |
Options: A Monte Carlo approach
- Boyle
- 1977
(Show Context)
Citation Context ...options depending on more than three assets. Monte Carlo methods are better suited to high dimensional problems. Using Monte Carlo to price European style financial derivatives was first suggested in =-=[7]-=-. Progress in using simulation methods to price American style options was stimulated by Tilley [42]. This and other early methods [4, 6, 14, 15, 24] are reviewed in Boyle et al. [9]. Since then the s... |

104 |
Pricing American-style securities using simulation
- Broadie, Glasserman
- 1997
(Show Context)
Citation Context ...price European style financial derivatives was first suggested in [7]. Progress in using simulation methods to price American style options was stimulated by Tilley [42]. This and other early methods =-=[4, 6, 14, 15, 24]-=- are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include pa... |

93 | Numerical Valuation of High Dimensional Multivariate
- Barraquand, Martineau
- 1995
(Show Context)
Citation Context ...price European style financial derivatives was first suggested in [7]. Progress in using simulation methods to price American style options was stimulated by Tilley [42]. This and other early methods =-=[4, 6, 14, 15, 24]-=- are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include pa... |

91 | Regression methods for pricing complex americanstyle options
- Tsitsiklis, Roy
- 2000
(Show Context)
Citation Context ...ization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22], irregular grid approximations [5], and sparse grid methods [37]. Regression methods include =-=[18, 33, 43]-=-. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to its accuracy and simplicity. The paper explains the method, but an introduction to the method is also giv... |

88 | A stochastic mesh method for pricing highdimensional American options
- Broadie, Glasserman
- 2004
(Show Context)
Citation Context ...price European style financial derivatives was first suggested in [7]. Progress in using simulation methods to price American style options was stimulated by Tilley [42]. This and other early methods =-=[4, 6, 14, 15, 24]-=- are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include pa... |

83 |
Pricing American options: a duality approach
- Haugh, Kogan
- 2004
(Show Context)
Citation Context ...ession. However, they do not suggest basis functions and do not implement their method. Primal-Dual representations of the American option problem allow both an upper and lower bound to be calculated =-=[1, 32, 34, 38]-=-. However, the upper bound involves calculating an expectation, which has been done using another Monte Carlo simulation. This simulation on simulation is computationally expensive. Glasserman and Yu ... |

79 |
The Valuation of American Put Options
- Brennan, Schwartz
- 1977
(Show Context)
Citation Context ...Introduction. No closed form solutions have been found for American option prices 1 . Therefore much work has been done pricing American options numerically. Early examples include finite differences =-=[12] a-=-nd the binomial lattice [20]. Grid based methods work well for single asset options, and have been extended to higher dimensions [8, 10]. However, these methods suffer from the ‘curse of dimensional... |

76 |
Monte Carlo valuation of American options
- Rogers
(Show Context)
Citation Context ...ession. However, they do not suggest basis functions and do not implement their method. Primal-Dual representations of the American option problem allow both an upper and lower bound to be calculated =-=[1, 32, 34, 38]-=-. However, the upper bound involves calculating an expectation, which has been done using another Monte Carlo simulation. This simulation on simulation is computationally expensive. Glasserman and Yu ... |

74 | A primal-dual simulation algorithm for pricing multidimensional American options
- Andersen, Broadie
- 2004
(Show Context)
Citation Context ...ession. However, they do not suggest basis functions and do not implement their method. Primal-Dual representations of the American option problem allow both an upper and lower bound to be calculated =-=[1, 32, 34, 38]-=-. However, the upper bound involves calculating an expectation, which has been done using another Monte Carlo simulation. This simulation on simulation is computationally expensive. Glasserman and Yu ... |

65 |
Valuing American Options in a Path Simulation Model, Transactions of the Society of Actuaries
- Tilley
- 1993
(Show Context)
Citation Context ...nal problems. Using Monte Carlo to price European style financial derivatives was first suggested in [7]. Progress in using simulation methods to price American style options was stimulated by Tilley =-=[42]-=-. This and other early methods [4, 6, 14, 15, 24] are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient [2, 16] and has been modified to use low-dis... |

58 |
A Lattice Framework for Option Pricing with Two State Variables
- Boyle
- 1988
(Show Context)
Citation Context ...n options numerically. Early examples include finite differences [12] and the binomial lattice [20]. Grid based methods work well for single asset options, and have been extended to higher dimensions =-=[8, 10]. Ho-=-wever, these methods suffer from the ‘curse of dimensionality’, as they become exponentially more expensive as the dimension increases, and cannot be used for options depending on more than three ... |

58 |
J.F.Valuation of the early-exercise price for options using simulations and nonparametric regression
- Carriere
- 1996
(Show Context)
Citation Context ...ization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22], irregular grid approximations [5], and sparse grid methods [37]. Regression methods include =-=[18, 33, 43]-=-. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to its accuracy and simplicity. The paper explains the method, but an introduction to the method is also giv... |

58 |
Options on the maximum or the minimum of several assets
- Johnson
- 1987
(Show Context)
Citation Context ...l options can be priced easily analytically and potentially used as martingale basis functions. European options on the maximum or minimum of two asset are priced in [40], and the results extended in =-=[31]-=-. A well known example is an option on the geometric average of a basket of stocks [27]. Some martingales under geometric Brownian motion are given in [26]. 6. Low Biased Estimators. The LSMC estimato... |

56 |
Numerical Evaluation of Multivariate Contingent Claims,” The Review of Financial Studies
- Boyle, Evnine, et al.
- 1989
(Show Context)
Citation Context ...n options numerically. Early examples include finite differences [12] and the binomial lattice [20]. Grid based methods work well for single asset options, and have been extended to higher dimensions =-=[8, 10]. Ho-=-wever, these methods suffer from the ‘curse of dimensionality’, as they become exponentially more expensive as the dimension increases, and cannot be used for options depending on more than three ... |

45 | Alternative characterizations of American put options
- Carr, Jarrow, et al.
- 1992
(Show Context)
Citation Context ...ura International plc. for funding this research. The author would also like to thank Dr. W. T. Shaw, Dr. B. Hambly, A. Dickinson, and particularly Dr. R. A. Stalker Firth. 1 analytical work includes =-=[13, 17, 35]-=-sUpper Bounds for American Option Prices The rest of this paper is structured as follows: We formulate the American option pricing problem, following [29], as an optimal stopping problem whose solutio... |

31 |
Sensitivity Analysis for Monte Carlo Simulation of Option Pricing
- Fu, Hu
- 1995
(Show Context)
Citation Context |

30 |
The pricing of the American option
- Myneni
(Show Context)
Citation Context ...ura International plc. for funding this research. The author would also like to thank Dr. W. T. Shaw, Dr. B. Hambly, A. Dickinson, and particularly Dr. R. A. Stalker Firth. 1 analytical work includes =-=[13, 17, 35]-=-sUpper Bounds for American Option Prices The rest of this paper is structured as follows: We formulate the American option pricing problem, following [29], as an optimal stopping problem whose solutio... |

29 | Pricing American Options: A Comparison of Monte Carlo Simulation Approaches
- Fu, Laprise, et al.
- 2001
(Show Context)
Citation Context ...ven in [19, 43]. The relationship between the number of basis functions and number of paths required is investigated by Glasserman and Yu [28]. For a comparison of simulation approaches see Fu et al. =-=[25] or the rece-=-nt book by Glasserman [27]. Glasserman and Yu [29] investigate the relative merits of ‘regression now’ versus ‘regression later’. ‘Regression now’ involves using basis functions defined at... |

24 |
A quantization tree method for pricing and hedging multidimensional American options
- Bally, Pagès, et al.
- 2005
(Show Context)
Citation Context ... more efficient [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm =-=[3]-=-, wavelets [21, 22], irregular grid approximations [5], and sparse grid methods [37]. Regression methods include [18, 33, 43]. The regression method of Longstaff and Schwartz [33] has proved particula... |

23 |
The valuation of American options on Multiple Assets’. Mathematical Finance 7(3):241–286
- Broadie, Detemple
- 1997
(Show Context)
Citation Context ...ura International plc. for funding this research. The author would also like to thank Dr. W. T. Shaw, Dr. B. Hambly, A. Dickinson, and particularly Dr. R. A. Stalker Firth. 1 analytical work includes =-=[13, 17, 35]-=-sUpper Bounds for American Option Prices The rest of this paper is structured as follows: We formulate the American option pricing problem, following [29], as an optimal stopping problem whose solutio... |

23 | Monte Carlo valuation of American options through computation of the optimal exercise frontier - Ibanez, Zapatero - 2004 |

21 |
Martingale approach to pricing perpetual American option
- Gerber, Shiu
- 1994
(Show Context)
Citation Context ... priced in [40], and the results extended in [31]. A well known example is an option on the geometric average of a basket of stocks [27]. Some martingales under geometric Brownian motion are given in =-=[26]-=-. 6. Low Biased Estimators. The LSMC estimator has undetermined bias because it both generates a sub-optimal exercise strategy and uses future information (as described in [27]). If we fix a sub-optim... |

15 | Pricing American Options by Simulation Using a Stochastic Mesh with Optimized Weights. Probabilistic Constrained Optimization: Methodology and Applications
- Broadie, Glasserman, et al.
- 2000
(Show Context)
Citation Context ... style options was stimulated by Tilley [42]. This and other early methods [4, 6, 14, 15, 24] are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient =-=[2, 16]-=- and has been modified to use low-discrepancy sequences [11]. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22],... |

15 |
An analysis of a least squares regression algorithm for American option pricing
- Clément, Lamberton, et al.
- 2002
(Show Context)
Citation Context ...The paper explains the method, but an introduction to the method is also given in Tavella [41]. Further analysis can be found in Stentoft [39]. Convergence results for regression methods are given in =-=[19, 43]-=-. The relationship between the number of basis functions and number of paths required is investigated by Glasserman and Yu [28]. For a comparison of simulation approaches see Fu et al. [25] or the rec... |

14 | Efficiency improvements for pricing American options with a stochastic mesh
- Avramidis, Hyden
- 1999
(Show Context)
Citation Context ... style options was stimulated by Tilley [42]. This and other early methods [4, 6, 14, 15, 24] are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient =-=[2, 16]-=- and has been modified to use low-discrepancy sequences [11]. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22],... |

13 | Number of paths versus number of basis functions in american option pricing - Glasserman, Yu |

12 |
Convergence of the Least-Squares Monte Carlo Approach to American Option Valuation
- Stentoft
- 2002
(Show Context)
Citation Context ...ved particularly popular, due to its accuracy and simplicity. The paper explains the method, but an introduction to the method is also given in Tavella [41]. Further analysis can be found in Stentoft =-=[39]-=-. Convergence results for regression methods are given in [19, 43]. The relationship between the number of basis functions and number of paths required is investigated by Glasserman and Yu [28]. For a... |

11 |
Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben
- Reisinger
- 2004
(Show Context)
Citation Context .... Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22], irregular grid approximations [5], and sparse grid methods =-=[37]-=-. Regression methods include [18, 33, 43]. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to its accuracy and simplicity. The paper explains the method, but ... |

6 |
Simulation Estimators of Optimal Early Exercise
- Bossaerts
- 1989
(Show Context)
Citation Context |

5 |
Pricing American Options by Simulation: Regression Now or Regression
- Glasserman, Yu
- 2005
(Show Context)
Citation Context ...functions and number of paths required is investigated by Glasserman and Yu [28]. For a comparison of simulation approaches see Fu et al. [25] or the recent book by Glasserman [27]. Glasserman and Yu =-=[29] investigate-=- the relative merits of ‘regression now’ versus ‘regression later’. ‘Regression now’ involves using basis functions defined at the current time step and regressing discounted option values... |

4 |
Options on the minimum or the maximum of two risky assets: analysis and applications
- Stultz
- 1982
(Show Context)
Citation Context ... options. Various multi-dimensional options can be priced easily analytically and potentially used as martingale basis functions. European options on the maximum or minimum of two asset are priced in =-=[40]-=-, and the results extended in [31]. A well known example is an option on the geometric average of a basket of stocks [27]. Some martingales under geometric Brownian motion are given in [26]. 6. Low Bi... |

4 |
Quantitative Methods in Derivatives Pricing: An Introduction to Computational Finance
- Tavella
- 2002
(Show Context)
Citation Context ...on method of Longstaff and Schwartz [33] has proved particularly popular, due to its accuracy and simplicity. The paper explains the method, but an introduction to the method is also given in Tavella =-=[41]-=-. Further analysis can be found in Stentoft [39]. Convergence results for regression methods are given in [19, 43]. The relationship between the number of basis functions and number of paths required ... |

3 |
Pricing American–style options using low discrepancy mesh methods
- Boyle, Kolkiewicz, et al.
- 2000
(Show Context)
Citation Context ...rly methods [4, 6, 14, 15, 24] are reviewed in Boyle et al. [9]. Since then the stochastic mesh method [15] has been made more efficient [2, 16] and has been modified to use low-discrepancy sequences =-=[11]-=-. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22], irregular grid approximations [5], and sparse grid methods ... |

2 | An irregular grid method for solving high-dimensional problems in finance
- Berridge, Schumacher
- 2002
(Show Context)
Citation Context ...ow-discrepancy sequences [11]. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets [21, 22], irregular grid approximations =-=[5]-=-, and sparse grid methods [37]. Regression methods include [18, 33, 43]. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to its accuracy and simplicity. The p... |

2 |
Wavelet methods in PDE valuation of financial derivatives
- Dempster, Richards
- 2000
(Show Context)
Citation Context ...t [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets =-=[21, 22]-=-, irregular grid approximations [5], and sparse grid methods [37]. Regression methods include [18, 33, 43]. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to... |

2 |
Solution of PDEs by wavelet methods
- Dempster, Eswaran
- 2001
(Show Context)
Citation Context ...t [2, 16] and has been modified to use low-discrepancy sequences [11]. Other methods include parameterization of the optimal exercise boundary [6, 24, 30], a quantization tree algorithm [3], wavelets =-=[21, 22]-=-, irregular grid approximations [5], and sparse grid methods [37]. Regression methods include [18, 33, 43]. The regression method of Longstaff and Schwartz [33] has proved particularly popular, due to... |

1 |
Why use QuantLib? URL http://www.maths.ox.ac.uk/∼firth/research/quantlib.pdf
- Firth
(Show Context)
Citation Context ...pact of their work if both their papers and software implementations are available on the internet. For a survey of open–source derivatives pricing libraries and the benefits of adopting QuantLib se=-=e [23]. Fo-=-r instructions on how to download, install, run and contribute see the project website [36]. 8.2. Regression Precision. Under certain conditions on the residuals, ‘regression later’ gives an estim... |

1 | Monte Carlo methods for multiple exercise problems - Meinshausen, Hambly - 2004 |