Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates

Belomestny, Denis

doi:10.1007/s00780-010-0132-x

Cited by 62 publications

(58 citation statements)

References 26 publications

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“…, J and some q > a. The condition (7) bounds the probability of staying in the δ-vicinity of the exercise boundary {Y * j ≤ Z j } in the case of continuation and is similar to the so-called margin condition in Belomestny (2011). As we will see, (7) leads to faster convergence rates of the standard Andersen-Broadie dual algorithm.…”

Section: Discussionmentioning

confidence: 86%

Multilevel dual approach for pricing American style derivatives

2013

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In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.

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Section: Discussionmentioning

confidence: 86%

Multilevel dual approach for pricing American style derivatives

2013

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“…Andersen [2], Belomestny [4], Agarwal and Juneja [1]) that simulation methods based on even a crude approximation of the optimal exercise boundary produce good estimates for the true American option price. Thus, in Section 2.2, we use the exercise boundary approximation x 0 + √ εx 1 in our setting to price the finite maturity American put via Monte Carlo.…”

Section: Asymptotic Analysismentioning

confidence: 99%

American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

Agarwal

Juneja

Sircar

2015

Quantitative Finance

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American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate.

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“…As was pointed out by Belomestny (2009), sometimes much better rate of convergence results can be derived for the Monte Carlo estimate of (20) considered as estimate of the price V 0 of the option. Because in view of a good performance of the stopping time it is not important that the estimate of the continuation values are close to the continuation values, instead it is important that they lead to the same decision as the optimal stopping rule.…”

Section: Algorithms Based On Nonparametric Regressionmentioning

confidence: 99%

A Review on Regression-based Monte Carlo Methods for Pricing American Options

Köhler

2010

Recent Developments in Applied Probability and Statistics

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In this article we give a review of regression-based Monte Carlo methods for pricing American options. The methods require in a first step that the generally in continuous time formulated pricing problem is approximated by a problem in discrete time, i.e., the number of exercising times of the considered option is assumed to be finite. Then the problem can be formulated as an optimal stopping problem in discrete time, where the optimal stopping time can be expressed by the aid of so-called continuation values. These continuation values represent the price of the option given that the option is exercised after time t conditioned on the value of the price process at time t. The continuation values can be expressed as regression functions, and regression-based Monte Carlo methods apply regression estimates to data generated by the aid of artificial generated paths of the price process in order to approximate these conditional expectations. In this article we describe various methods and corresponding results for estimation of these regression functions. Pricing of American options as optimal stopping problemIn many financial contracts it is allowed to exercise the contract early before expiry. E.g., many exchange traded options are of American type and allow the holder any exercise date before expiry, mortgages have often embedded prepayment options such that the mortgage can be amortized or repayed, or life insurance contracts allow often for early surrender. In this article we are interested in pricing of options with early exercise features.It is well-known that in complete and arbitrage free markets the price of a derivative security can be represented as an expected value with respect to the so called martingale measure, see for instance Karatzas and Shreve (1998 There are various possibilities for the choice of the process (X t ) 0≤t≤T . The most simple examples are geometric Brownian motions, as for instance in the celebrated Black-Scholes setting. More general models include stochastic volatility models, jump-diffusion processes or general Levy processes. The model parameters are usually calibrated to observed time series data.The first step in addressing the numerical solution of (1) is to pass from continuous time to discrete time, which means in financial terms to approximate the American option by a so-called Bermudan option. The convergence of the discrete time approximations to the continuous time optimal stopping problem is considered in Lamberton and Pages (1990) for the Markovian case but also in the abstract setting of general stochastic processes.For simplicity we restrict ourselves directly to a discrete time scale and consider exclusively Bermudan options. In analogy to (1), the price of a Bermudan option is the value of the discrete time optimal stopping problemwhere X 0 , X 1 , . . . , X T is now a discrete time stochastic process, f t is the discounted payoff function, i.e., f t (x) = d 0,t g t (x), and T (0, . . . , T ) is the class of all {0, . . . , T }-valued stopping times. He...

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Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates

Abstract: Bermudan options, Nonparametric regression, Boundary condition, Suboptimal stopping rules, 62G08, 65C05, 60G40, G10, G12, G13,

Cited by 62 publications

References 26 publications

Multilevel dual approach for pricing American style derivatives

Multilevel dual approach for pricing American style derivatives

American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

A Review on Regression-based Monte Carlo Methods for Pricing American Options

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