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## Optimal robust mean-variance hedging in incomplete financial markets

Venue: | Journal of Mathematical Sciences |

Citations: | 3 - 1 self |

### Citations

1002 | The Pricing of Options on Assets with Stochastic Volatilities - JC, White - 1987 |

957 |
Robust Statistics: The Approach Based on Influence Functions
- Hampel, Ronchetti, et al.
- 1986
(Show Context)
Citation Context ...nd term are common in robust statistic theory. In contact to optimal B-robustness (see Section 2), here we develop the approach, known in robust statistics as optimal V -robustness, see Hampel et al. =-=[12]-=-. Note that our approach allows incorporating current information on the underlying model, and hence is adaptive. Namely, passing from time value t to t + τ, τ > 0, when more information about market ... |

240 |
Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,”
- Scott
- 1987
(Show Context)
Citation Context ...länder and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and Scott =-=[39]-=-, where the stock price volatility is an random process, is a popular model of incomplete market, where the mean-variance hedging approach can be used (see, e.g., Laurent and Pham [18], Biagini et al.... |

168 |
Statistics of random processes
- Liptser, Shiryaev
- 2001
(Show Context)
Citation Context ... t(n) < · · · < ] → 0, as n → 0, calculate the realization of yield dXs, and then calculate the sum Xs ∑n−1 Sn(t) = j=0 |R t (n) j+1 − R (n)| t j 2 . It is well-known (see, e.g., Lipster and Shiryaev =-=[23]-=-) that Sn(t) P → ∫t 0 σ 2 s ds as n → ∞. Since σ 2 t (ω) = f(Yt) is a continuous process we get where F(t, ω) = ∫t 0 σ 2 F(t + ∆, ω) − F(t, ω) t (ω) = lim , ∆↓0 ∆ σ 2 s(ω)ds. Hence, the realization (y... |

164 | Pricing and hedging derivative securities in markets with uncertain volatilities
- AVELLANEDA, LEVY, et al.
- 1995
(Show Context)
Citation Context ...ion rate in contrast to our consideration, where the model misspecification is due to the volatility parameter. The consideration of misspecified asset price models was initiated by Avellaneda et al. =-=[1]-=-, Avellaneda and Paras [2]. Various authors in different settings attacked the robustness problem. The method used in Section 3 was suggested by Toronjadze [41] for asset price process modelled by the... |

143 |
Martingale Methods in Financial Models,
- MUSIELA, RUTKOWSKI
- 1997
(Show Context)
Citation Context ...eveloped in Hampel et al. [12] and in Rieder [34]; the theory of the trend parameter estimates for diffusion process with small noise is developed in Kutoyants [17]; the book of Musiela and Rutkowsky =-=[29]-=- is devoted to the mathematical theory of finance and finally, the general theory of martingales can be found in Jacod and Shiryaev [15].OPTIMAL ROBUST MEAN-VARIANCE HEDGING 9 2. Optimal B-Robust Est... |

125 |
Mean–Variance Analysis in Portfolio Choice and Capital Markets,
- MARKOWITZ
- 1987
(Show Context)
Citation Context ...-measurable square-integrable random variable (r.v.), x is an initial investment, Θ is a class of admissible trading strategies, T is an investment horizon. The mean-variance formulation by Markowitz =-=[26]-=-, provides a foundation for a single period portfolio selection (see, also Merton [27]). In recent paper of Li 1991 Mathematics Subject Classification. 60G22, 62F35, 91B28, 62F35, 62M05, 62M09. Key wo... |

87 | An Analytical Derivation of the Efficient Portfolio Frontier.
- Merton
- 1972
(Show Context)
Citation Context ... a class of admissible trading strategies, T is an investment horizon. The mean-variance formulation by Markowitz [26], provides a foundation for a single period portfolio selection (see, also Merton =-=[27]-=-). In recent paper of Li 1991 Mathematics Subject Classification. 60G22, 62F35, 91B28, 62F35, 62M05, 62M09. Key words and phrases. Stochastic volatility, small diffusion, robust parameter estimate, op... |

78 |
Option hedging and implied volatilities in a stochastic volatility model,
- Renault, Touzi
- 1996
(Show Context)
Citation Context ... is a vector of unknown parameters, and ε, 0 < ε ≪ 1, is a small number. Assume that the system (1.2) has an unique strong solution. This model is analogous to the model proposed by Renault and Touzi =-=[32]-=- (RT hereafter). The principal difference is the presence of small parameter ε in our model, which due to the assumption that the volatility of randomly fluctuated volatility process is small (see, al... |

74 |
Mean-variance hedging in continuous time.
- Duffie, Richardson
- 1991
(Show Context)
Citation Context ...n. Therefore, the mean-variance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson =-=[9]-=-, Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [1... |

72 |
Limit theorems for stochastic processes, Grundlehren der mathematischen Wissenschaften
- Jacod, Shiryaev
- 1987
(Show Context)
Citation Context ... developed in Kutoyants [17]; the book of Musiela and Rutkowsky [29] is devoted to the mathematical theory of finance and finally, the general theory of martingales can be found in Jacod and Shiryaev =-=[15]-=-.OPTIMAL ROBUST MEAN-VARIANCE HEDGING 9 2. Optimal B-Robust Estimates 2.1. Basic model. CULAN estimates. The basic model of observations is described by the SDE dYs = a(s, Y ; α) ds + ε dws, Y0 = 0, ... |

67 |
Approximation Pricing and the Variance-Optimal Martingale Measure,
- Schweizer
- 1996
(Show Context)
Citation Context ...iance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], =-=[38]-=-, Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatilit... |

65 | Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation.
- Li, Ng
- 2000
(Show Context)
Citation Context ...2F35, 91B28, 62F35, 62M05, 62M09. Key words and phrases. Stochastic volatility, small diffusion, robust parameter estimate, optimal mean-variance robust hedging. 12 N. LAZRIEVA, T. TORONJADZE and Ng =-=[22]-=- the concept of Markowitz’s mean-variance formulation for finding the optimal portfolio policy and determining the efficient frontier in analytical form has been extended to multiperiod portfolio sele... |

57 |
Mean-variance hedging and numeraire.
- Gourieroux, Laurent, et al.
- 1998
(Show Context)
Citation Context ...ade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. =-=[11]-=- (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and Scott [39], where the stock price volatility is an random process, is a popular model of in... |

57 |
Approximating random variables by stochastic integrals.
- Schweizer
- 1994
(Show Context)
Citation Context ...an-variance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], =-=[37]-=-, [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic vol... |

55 |
Mean–Variance Hedging for General Claims,
- SCHWEIZER
- 1992
(Show Context)
Citation Context ...the mean-variance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer =-=[36]-=-, [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochast... |

51 | Managing the volatility of risk of portfolios of derivative securities: the Lagrangian uncertain volatility model.
- AVELLANEDA, PARAS
- 1996
(Show Context)
Citation Context ...r consideration, where the model misspecification is due to the volatility parameter. The consideration of misspecified asset price models was initiated by Avellaneda et al. [1], Avellaneda and Paras =-=[2]-=-. Various authors in different settings attacked the robustness problem. The method used in Section 3 was suggested by Toronjadze [41] for asset price process modelled by the one-dimensional process. ... |

51 |
Identification of Dynamical Systems with Small Noise,
- Kutoyants
- 1994
(Show Context)
Citation Context ...4]; the theory of robust statistics is developed in Hampel et al. [12] and in Rieder [34]; the theory of the trend parameter estimates for diffusion process with small noise is developed in Kutoyants =-=[17]-=-; the book of Musiela and Rutkowsky [29] is devoted to the mathematical theory of finance and finally, the general theory of martingales can be found in Jacod and Shiryaev [15].OPTIMAL ROBUST MEAN-VA... |

36 | Mean-Variance Hedging for Continuous Processes: New Proofs and Examples,
- Pham, Rheinlander, et al.
- 1998
(Show Context)
Citation Context ...investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. =-=[31]-=-, Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and Scott [39], where the stock price volatility is an random process, ... |

34 |
Dynamic programming and mean-variance hedging.
- Laurent, Pham
- 1999
(Show Context)
Citation Context ...9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham =-=[18]-=-). A stochastic volatility model, proposed by Hull and White [13] and Scott [39], where the stock price volatility is an random process, is a popular model of incomplete market, where the mean-varianc... |

29 | Weighted norm inequalities and hedging in incomplete markets.
- Delbaen, Monat, et al.
- 1997
(Show Context)
Citation Context ...werful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. =-=[8]-=-, Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by... |

27 |
Föllmer-Schweizer decomposition and meanvariance hedging for general claims. The Annals of Probability
- Monat, Stricker
- 1995
(Show Context)
Citation Context ...h above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker =-=[28]-=-, Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and... |

22 |
Hedging of Non-redundant Contingent Claims, Contributions to Mathematical Economics
- Foellmer, Sondermann
- 1986
(Show Context)
Citation Context ...semimartingale X, the mean-variance approach suggests to use the quadratic criterion to measure the hedging error, i.e. to solve the mean-variance hedging problem introduced by Föllmer and Sondermann =-=[10]-=-: minimize E ( H − x − ∫T θtdXt ) 2 over all θ ∈ Θ, (1.1) 0 where contingent claim H is a FT-measurable square-integrable random variable (r.v.), x is an initial investment, Θ is a class of admissible... |

19 |
Statistical estimation. Asymptotic theory. Transl. from the Russian by Samuel Kotz.
- Ibragimov, Khas’minskij
- 1981
(Show Context)
Citation Context ...pinion this “double robust” strategy should be more attractive to protect the hedger against the possible errors. The general asymptotic theory of estimation can be found in Ibragimov and Khas’miskii =-=[14]-=-; the theory of robust statistics is developed in Hampel et al. [12] and in Rieder [34]; the theory of the trend parameter estimates for diffusion process with small noise is developed in Kutoyants [1... |

17 | Stochastic volatility, smile & asymptotics
- Sircar, Papanicolaou
- 1999
(Show Context)
Citation Context ...ipal difference is the presence of small parameter ε in our model, which due to the assumption that the volatility of randomly fluctuated volatility process is small (see, also Sircar and Papanicolau =-=[40]-=-). Thus assumption enables us to use the prices of trading options with short, nearest to the current time value maturities for volatility process filtration and parameter estimation purposes (see bel... |

15 | Mean-variance hedging for stochastic volatility models.Mathematical Finance
- Biagini, Guasoni, et al.
- 2000
(Show Context)
Citation Context ...ere ϕ = (ϕt)0≤t≤T is a R d -valued, F-predictable process with ∫T 0 ϕ ′ t d〈M〉tϕt < ∞. Let τ be F-stopping time. Denote 〈k ′ · M〉Tτ = 〈k ′ · M〉T − 〈k ′ · M〉τ. Proposition 3.1 (see also Biagini et al. =-=[3]-=-, LLaurent and Pham [18]). 1. Equation (3.13) is equivalent to equation ET(ϕ ′ · M ∗ ) ET(L) = ce 〈k′ ·M〉T , (3.15) where the R d -valued process M ∗ = (M ∗ t )0≤t≤T is given by the relation dM ∗ t = ... |

13 |
A semimartingale Bellman equation and the varianceoptimal martingale measure
- Mania, Tevzadze
- 2000
(Show Context)
Citation Context ...latility is an random process, is a popular model of incomplete market, where the mean-variance hedging approach can be used (see, e.g., Laurent and Pham [18], Biagini et al. [13], Mania and Tevzadze =-=[24]-=-, Pham et al. [31]). Consider the stochastic volatility model described by the following system of SDE dXt = Xt dRt, X0 > 0, dRt = µt(Rt, Yt) dt + σ.dw R t , R0 = 0, σ 2 t = f(Yt), dYt = a(t, Yt; α) d... |

11 |
On L 2 -Projections on a Space of Stochastic Integrals
- RHEINLÄNDER, SCHWEIZER
- 1997
(Show Context)
Citation Context ... decomposition of r.v. H (see, e.g., Pham et al. [31]) putting ξ H t = ∂v(t, Xt, Yt) ∂x , L H T ∫ = ε 0 T ∂v ∂y (t, Xt, Yt) dw σ t , and calculate ψ H t , LT and V ∗ t using (4.13) and (4.14) of RSch =-=[33]-=-. Thus one get the explicit solution of the mean-variance hedging problem. Finally, here is the short summary of approach: a) Incorporate the robust procedure in statistical analysis of volatility pro... |

11 |
A Stochastic Control Approach to Risk Management under Restricted
- Runggaldier, Zaccaria
- 2000
(Show Context)
Citation Context ...nformation about market prices are available,OPTIMAL ROBUST MEAN-VARIANCE HEDGING 7 becomes smaller, and hence the estimation procedure becomes more precise. In the paper of Runggaldier and Zaccaria =-=[35]-=- the adaptive approach to risk management under general uncertainty (restricted information) was developed. As it is mentioned in this paper there exist a series of investigations dealt with various t... |

7 | Dynamic spanning: are options an appropriate instrument - Bajeux-Besnainou, Rochet - 1996 |

6 |
Controlled Diffusion Processes. Translated from the Russian by
- Krylov
- 1980
(Show Context)
Citation Context ... This region shrinks to the estimate α ∗,0 t , as ε → 0. Now if the coefficient a(t, y; α) in (1.2) is such that the solution Y ε t (α) of SDE (1.2) is continuous w.r.t parameter α (see, e.g., Krylov =-=[16]-=-), then the confidence region of parameter α is mapped to the confidence interval for 0 = Y ), Further, by the function f, the latter Y ε ∗ t (α), which shrinks to Yt t (α∗,0 t interval is mapped to t... |

5 |
Diffusion coefficient estimation and asset pricing when risk premia and sensitivities are time varying
- Chesney, Elliott, et al.
- 1993
(Show Context)
Citation Context ...s(ω)ds. Hence, the realization (yt)0≤t≤T of the process Y can be found by the formula ), 0 ≤ t ≤ T. yt = f −1 (σ2 t More sofisticated methods using the same idea can be found, e.g., in Chesney et al. =-=[5]-=-, Pastorello [30]. We can use the reconstructed sample path (yt), 0 ≤ t ≤ T. to estimate the unknown parameter α in the drift coefficient of diffusion process Y . The second, market price adjusted pro... |

5 | A semimartingale backward equation related to the p-optimal martingale measure and the lower price of a contingent claim
- Mania, Santacroce, et al.
- 2002
(Show Context)
Citation Context ...) process: Yt ε j+1 − Yt ε j = a(tε j, Yt ε j ; α)(tε j+1 − t ε j) + ε(w σ t ε j+1 − wσ t ε j ). Using the data {yt ε j } one can construct the MLE αε t of parameter α, see, e.g., Chitashvili et al. =-=[25]-=-, [26], Lazrieva and Toronjadze [19]. Remember the scheme of construction of MLE. Rewrite the previous AR(1) process, using obvious simple notation, in form Then ∂ ∂y P {Yj+1 ≤ y | Yj} = Yj+1 − Yj = a... |

4 | Robust asymptotic statistics. Springer Series in Statistics - Rieder - 1994 |

2 |
Toronjadze, Asymptotic theory of M-estimators in general statistical models
- Chitachvili, Lazrieva, et al.
(Show Context)
Citation Context ... equation L ψ,ε t (X; α) = 0, where L ψ,ε t (X; α) is defined by (2.14), ψ ∈ Ψ0. The asymptotic theory of M-estimates for general statistical models with filtration is developed in Chitashvili et al. =-=[7]-=-. Namely, the problem of existence and global behaviour of solutions is studied. In particular, the conditions of regularity and ergodicity type are established, under which M-estimates have a CULAN p... |

1 |
Torondzhadze, Asymptotic properties of an M-estimate in a general statistical experiment scheme. (Russian) Statistics and control of random processes (Russian) (Preila
- Lazrieva, A
- 1987
(Show Context)
Citation Context ...j, Yt ε j ; α)(tε j+1 − t ε j) + ε(w σ t ε j+1 − wσ t ε j ). Using the data {yt ε j } one can construct the MLE αε t of parameter α, see, e.g., Chitashvili et al. [25], [26], Lazrieva and Toronjadze =-=[19]-=-. Remember the scheme of construction of MLE. Rewrite the previous AR(1) process, using obvious simple notation, in form Then ∂ ∂y P {Yj+1 ≤ y | Yj} = Yj+1 − Yj = a(tj, Yj; α)∆ + ε∆w σ j . ( 1 √ exp −... |

1 |
Robust estimators in statistical models with filtration. Shrinking neighbourhoods
- Lazrieva, Toronjadze
- 1994
(Show Context)
Citation Context ...resents the Huber gross error model (as it explain in Remark 2.2). The model of type (1.4) of contamination of measures for statistical models with filtration was suggested by Lazrieva and Toronjadze =-=[20]-=-, [21]. In Section 2 we study the problem of construction of robust estimates for contamination model (1.4). In subsection 2.1 we give a description of the basic model and definition of consistent uni... |

1 |
Robust estimators in statistical models associated with semimartingales
- Lazrieva, Toronjadze
- 1998
(Show Context)
Citation Context ...s the Huber gross error model (as it explain in Remark 2.2). The model of type (1.4) of contamination of measures for statistical models with filtration was suggested by Lazrieva and Toronjadze [20], =-=[21]-=-. In Section 2 we study the problem of construction of robust estimates for contamination model (1.4). In subsection 2.1 we give a description of the basic model and definition of consistent uniformly... |

1 |
Diffusion coefficient estimation and asset pricing when risk premia and sensitivities are time varying: A comment
- Pastorello
- 1996
(Show Context)
Citation Context ...he realization (yt)0≤t≤T of the process Y can be found by the formula ), 0 ≤ t ≤ T. yt = f −1 (σ2 t More sofisticated methods using the same idea can be found, e.g., in Chesney et al. [5], Pastorello =-=[30]-=-. We can use the reconstructed sample path (yt), 0 ≤ t ≤ T. to estimate the unknown parameter α in the drift coefficient of diffusion process Y . The second, market price adjusted procedure of reconst... |

1 | Optimal mean-variance robust hedging under asset price model misspecification
- Toronjadze
(Show Context)
Citation Context ...odels was initiated by Avellaneda et al. [1], Avellaneda and Paras [2]. Various authors in different settings attacked the robustness problem. The method used in Section 3 was suggested by Toronjadze =-=[41]-=- for asset price process modelled by the one-dimensional process. As it will be shown in Remark 3.2 below, in simplest case when asset price process is a martingale w.r.t initial measure P, and it is ... |