On the role of F\"ollmer-Schweizer minimal martingale measure in Risk Sensitive control Asset Management

Kuroda and Nagai \cite{KN} state that the factor process in the Risk Sensitive control Asset Management (RSCAM) is stable under the F\"ollmer-Schweizer minimal martingale measure . Fleming and Sheu \cite{FS} and more recently F\"ollmer and Schweizer \cite{FoS} have observed that the role of the minimal martingale measure in this portfolio optimization is yet to be established. In this article we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal martingale measure. We achieve this by using a "trick" of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability measure. We obtain some startling insights. Firstly, if short-selling is not permitted and if the factor process evolves under the minimal martingale measure then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then even if the market allows short selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal martingale measure .Comment: A.Deshpande (2015), On the role of F\"ollmer-Schweizer minimal martingale measure in Risk Sensitive control Asset Management,Vol. 52, No. 3, Journal of Applied Probabilit

Sufficient stochastic maximum principle for the optimal control of semi-Markov modulated jump-diffusion with application to Financial optimization

The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem and to a quadratic loss minimization problem.Comment: Forthcoming in Stochastic Analysis and Application

Game-theoretic approach to risk-sensitive benchmarked asset management

In this article we consider a game theoretic approach to the Risk-Sensitive Benchmarked Asset Management problem (RSBAM) of Davis and Lleo \cite{DL}. In particular, we consider a stochastic differential game between two players, namely, the investor who has a power utility while the second player represents the market which tries to minimize the expected payoff of the investor. The market does this by modulating a stochastic benchmark that the investor needs to outperform. We obtain an explicit expression for the optimal pair of strategies as for both the players.Comment: Forthcoming in Risk and Decision Analysis. arXiv admin note: text overlap with arXiv:0905.4740 by other author

Validation of a questionnaire to measure success in financial computing literacy

An embedded model of teaching financial computing within a course on numerical analysis in finance has been proposed recently in (Deshpande, 2017). It consists of only 10 steps that are aimed at programming beginners. These steps expect students only to be self-motivated to learn.  Hence other attributes like pre-knowledge of programming and cleverness aren’t expected to influence the learning outcome. Through qualitative assessment via laboratory observation this was indeed found to hold true. In order to understand the outcome of these 10 steps on a much finer scale, we develop here a questionnaire that measures success in financial computing literacy (SFCL) via quantitative assessment.  Four scales were developed: self-efficacy or computing confidence, active learning strategy/pro-activeness, learning environment stimulation and an achievement goal in terms of student satisfaction. Findings of this pilot study confirm construct validity of the questionnaire. Importantly we conclude that self-motivation is not enough and that tenacity is a vital component to keep motivation going. Tenacity can be induced via providing credit for attempting steps

Financial computing literacy: 10 steps.

It is often the case that in a financial engineering/mathematics master's curriculum, computer programming if taught, is before the start of an important course typically titled Numerical analysis of financial derivatives. Typically in this computer programming course, C++ is taught, and the material spans from basic constructs to an introduction to advanced programming such as design patterns. A major reason for its early introduction is that students would subsequently be able to use the skills to computationally solve problems occurring in numerics. We believe this curriculum strategy to be an element of computer literacy which has been criticized in our context as one that discourages students with lower programming abilities who otherwise may have predisposed mathematical abilities. We aim to make fundamental financial computing inclusive via a series of 10 steps thereby leading to what we refer to as financial computing literacy

Topics in risk-sensitive stochastic control

This thesis consists of three topics whose over-arching theme is based on risk sensitive stochastic control. In the �first topic (chapter 2), we study a problem on benchmark out-performance. We model this as a zero-sum risk-sensitive stochastic game between an investor who as a player wants to maximize the risk-sensitive criterion while the other player ( a stochastic benchmark) tries to minimize this maximum risk-sensitive criterion. We obtain an explicit expression for the strategies for both these two players. In the second topic (chapter 3), we consider a finite horizon risk-sensitive asset management problem. We study it in the context of a zero-sum stochastic game between an investor and the second player called the "market world" which provides a probability measure. Via this game, we connect two (somewhat) disparate areas in stochastics; namely, stochastic stability and risk-sensitive stochastic control in mathematical finance. The connection is through the Follmer-Schweizer minimal martingale measure. We discuss the impact of this measure on the investor's optimal strategy. In the third topic (chapter 4), we study the sufficient stochastic maximum principle of semi-Markov modulated jump diffusion. We study its application in the context of a quadratic loss minimization problem. We also study the finite-horizon risk-sensitive optimization in relation to the underlying sufficient stochastic maximum principle of a semi-markov modulated diffusion

Comparing the value at risk performance of the CreditRisk+ and its enhancement : a large deviations approach

The standard CreditRisk⁺ (CR⁺) is a well-known default-mode credit risk model. An extension to the CR⁺ that introduces correlation through a two-stage hierarchy of randomness has been discussed by Deshpande and Iyer (Central Eur J Oper Res 17(2):219-228, 2009) and more recently by Sowers (2010). It is termed the 2-stage CreditRisk⁺ (2-CR⁺) in the former. Unlike the standard CR⁺, the 2-CR⁺ model is formulated to allow correlation between sectoral default rates through dependence on a common set of macroeconomic variables. Furthermore the default rates for a 2-CR⁺ are distributed according to a general univariate distribution which is in stark contrast to the uniformly Gamma distributed sectoral default rates in the CR⁺. We would then like to understand the behaviour of these two models with regards to their computed Value at Risk (VaR) as the number of sectors and macroeconomic variables approaches infinity. In particular we would like to ask whether the 2-CR⁺ produces higher VaR than the CR+ and if so then for which type of credit portfolio. Utilizing the theory of Large deviations, we provide a methodology for comparing the Value at risk performance of these two competing models by computing certain associated rare event probabilities. In particular we show that the 2-Stage CR⁺ definitely produces higher VaR than the CR⁺ for a particular class of a credit portfolio which we term as a "balanced" credit portfolio. We support this statistical risk analysis through numerical examples.15 page(s

Asymptotic Stability of Semi-Markov Modulated Jump Diffusions

We consider the class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator. We find conditions under which the solutions of this class of switching jump-diffusion processes are almost surely exponentially stable and moment exponentially stable. We also provide conditions that imply almost sure convergence of the trivial solution when the moment exponential stability of the trivial solution is guaranteed. We further investigate and determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equations /=() is almost surely exponentially stable. It is observed that for a one-dimensional state space, a linear unstable system of differential equations when stabilized just by the addition of the jump part of an sMMJD process does not get destabilized by any addition of a Brownian motion. However, in a state space of dimension at least two, we show that a corresponding nonlinear system of differential equations stabilized by jumps gets destabilized by addition of Brownian motion

Applications of asymptotic methods in quantitative finance and insurance

Thesis by publication.Includes bibliographical references.1. Introduction -- 2. Value at risk performance comparison of the CreditRisk+ and the 2-stage CreditRisk+ : a large deviations approach -- 3. On the existence of an asymptotic options price in a Markov modulated economy -- 4. Asymptotic stability of semi-Markov modulated jump diffusions.This thesis deals with three essays related to studying asymptotic behavior of a portfolio tail loss, asymptotic behavior of options price and asymptotic stability of a class of jump diffusion process.In the first article that constitutes our first essay, we study an enhancement to the CreditRisk+ model termed as the 2-stage CreditRisk+. We determine under what conditions on the portfolio does the 2-stage CreditRisk+ credit risk model gives higher Value at Risk than the CreditRisk+. This entails studying rare event probability of large portfolio loss event. For the same we use technique from the theory of large deviations.In the second article, we consider an asymptotic options pricing problem in a Markov modulated regime switching market. In such market, the key model parameters are modulated by a continuous-time, finite-state, Markov chain. Such a market is incomplete and hence there exist a range of options price. For an asymptotic analysis, we consider two variations of the chain, namely, a slow chain and a fast chain. It has been observed that there exists an asymptotic option price for the slow chain case while it is been argued that such price may not exist for the fast chain case. In this article, we attempt to show why this is so by determining the range of options price for the slow chain and the fast chain.In the third and the last article, we consider a jump-diffusion process whose drift, diffusion and the jump kernel is modulated by a semi-Markov process. The semi-Markov process is a generalization over the Markov chain case since its sojourn time need not be exponentially distributed. We study the issue of asymptotic stability of this process with regards to almost sure and moment exponential sense. We study this issue here because of its motivational connection to the ruin theory in insurance. We also study the issue of stabilization and de-stabilization of a non-linear system of differential equation perturbed by a semi-Markov modulated jump diffusion process. We thereby comment on the interesting behaviour that we observe with regards to (de)-stabilization of the system of differential equation in one and in higher dimension.Mode of access: World wide web1 online resource (vii, 87 pages