The martingale method of shortfall risk minimization in a discrete time market

2012 ◽  
Vol 39 (4) ◽  
pp. 413-424
Author(s):  
Marek Andrzej Kociński
Keyword(s):  
Discrete Time ◽  
Martingale Method ◽  
Risk Minimization ◽  
Shortfall Risk

Shortfall risk through Fenchel duality

2018 ◽  
Vol 05 (02) ◽  
pp. 1850019 ◽  
Author(s):  
Zhenyu Cui ◽  
Jun Deng
Keyword(s):  
Minimization Problem ◽  
Optimization Problem ◽  
Risk Measures ◽  
Fenchel Duality ◽  
Risk Minimization ◽  
Decomposition Approach ◽  
Shortfall Risk ◽  
Stochastic Optimization Problem ◽  
Duality Approach ◽  
Randomized Test

In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first translate the associated stochastic optimization problem to an equivalent static optimization problem, and then obtain the explicit structure of the optimal randomized test for both complete and incomplete markets. For the incomplete market case, to the best of our knowledge, we obtain for the first time the explicit randomized test, while previous literature only established the existence through the supermartingale optional decomposition approach. We also solve the shortfall risk minimization problem for an insider through the enlargement of filtrations approach.

SHORTFALL RISK MINIMIZATION UNDER FIXED TRANSACTION COSTS

2018 ◽  
Vol 21 (05) ◽  
pp. 1850034
Author(s):  
NIV NAYMAN
Keyword(s):  
Analytical Solution ◽  
Transaction Costs ◽  
Dynamic Programming Algorithm ◽  
Control Policy ◽  
Programming Algorithm ◽  
Main Question ◽  
Risk Minimization ◽  
Shortfall Risk ◽  
Markov Decision ◽  
Augmented State

In this work, we deal with market frictions which are given by fixed transaction costs independent of the volume of the trade. The main question that we study is the minimization of shortfall risk in the Black–Scholes (BS) model under constraints on the initial capital. This problem does not have an analytical solution and so numerical schemes come into the picture. The Cox–Ross–Rubinstein (CRR) binomial models are an efficient tool for approximating the BS model. In this paper, we study in detail the CRR models with fixed transaction costs. In particular, we construct an augmented state-action space forming a Markov decision process (MDP) and provide a proof for the existence of optimal control/policy. We further suggest a dynamic programming algorithm for calculating the optimal hedging strategy and its corresponding shortfall risk. In the absence of transaction costs, there is an analytical solution in both CRR and BS models, and so we use them for testing our algorithm and its convergence. Moreover, we point out various insights provided by our numerical results, for example, regarding the change in the investor’s activity in the presence of friction.

scholarly journals Risk minimization for game options in markets imposing minimal transaction costs

2016 ◽  
Vol 48 (3) ◽  
pp. 926-946 ◽  
Author(s):  
Yan Dolinsky ◽  
Yuri Kifer
Keyword(s):  
Transaction Costs ◽  
Continuous Time ◽  
Risk Minimization ◽  
Proportional Transaction Costs ◽  
Shortfall Risk ◽  
Partial Hedging ◽  
Black Scholes ◽  
Fixed Transaction Cost ◽  
Game Options ◽  
Binomial Models

Abstract We study partial hedging for game options in markets with transaction costs bounded from below. More precisely, we assume that the investor's transaction costs for each trade are the maximum between proportional transaction costs and a fixed transaction cost. We prove that in the continuous-time Black‒Scholes (BS) model, there exists a trading strategy which minimizes the shortfall risk. Furthermore, we use binomial models in order to provide numerical schemes for the calculation of the shortfall risk and the corresponding optimal portfolio in the BS model.

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