Yeliz Yolcu Okur

WHITE NOISE GENERALIZATION OF THE CLARK-OCONE FORMULA UNDER CHANGE OF MEASURE

We prove the white noise generalization of the Clark-Ocone formula under change of measure by using Gaussian white noise analysis and Malliavin calculus. Let W(t) be a Brownian motion on the filtered white noise probability space (Ω, ℬ, {ℱ t }0≤t≤T , P) and let be defined as , where u(t) is an ℱ t -measurable process satisfying certain conditions for all 0 ≤ t ≤ T. Let Q be the probability measure equivalent to P such that is a Brownian motion with respect to Q, in virtue of the Girsanov theorem. In this article, it is shown that for any square integrable ℱ T -measurable random variable, where 𝔼 Q is the expectation under Q and D · F(ω) is the (Hida) Malliavin derivative. The important point in this settlement is F does not have to be in stochastic Sobolev space 𝔻1, 2 ⊂ L 2(P). This makes the formula more useful in applications of finance. As an example, the replicating portfolio for a digital option with the payoff χ[K, ∞) W(T) ∉ 𝔻1, 2 is calculated by using this generalized Clark-Ocone formula under change of measure.

Analysis of volatility feedback and leverage effects on the ISE30 index using high frequency data

In this study, we employ the techniques of Malliavin calculus to analyze the volatility feedback and leverage effects for a better understanding of financial market dynamics. We estimate both effects for a general semimartingale model applying Fourier analysis developed in Malliavin and Mancino (2002) [10]. We further investigate their joint behaviour using 5 min data of the ISE30 index. On the basis of these estimations, we look for the evidence that volatility feedback effect rate can be employed in the stability analysis of financial markets.

A Malliavin calculus approach to general stochastic differential games with partial information

In this article, we consider stochastic differential game where the state process is governed by a controlled Ito–Levy process and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functional are assumed to be random. We use Malliavin calculus to derive a maximum principle for the optimal control of such problem. The results are applied to solve a worst-case scenario portfolio problem in finance.

On the single name CDS price under structural modeling

Regulators, banks and other market participants realized that true assessment of the credit risk is more critical and complex than their ex-ante appraisals after the US Credit Crunch. They have turned their attention to complex credit risk models and credit instruments such as credit derivatives. Credit default swap contracts (CDSs) are the most common credit derivatives used for speculation and hedging purposes in the credit markets. Thus, in this paper we fundamentally study the pricing of a single name CDS via the discounted cash flow method with survival probability functions of two pioneer structural credit risk models, Merton model and Black-Cox model with constant barrier. Hence, this approach is not only a new one, but also provides a practical technique to price CDSs using publicly available data of equity returns.

An extension of the Clark–Ocone formula under benchmark measure for Lévy processes

The classical Clark–Ocone theorem states that any random variable can be represented as where denotes the conditional expectation, is a Brownian motion with canonical filtration and D denotes the Malliavin derivative in the direction of W. Since many applications in financial mathematics require representation of random variables with respect to risk neutral martingale measure, an equivalent martingale measure version of this theorem was stated by Karatzas and Ocone (Stoch. Stoch. Rep. 34 (1991), 187–220). In this paper, we extend these results to be valid for square integrable pure jump Lévy processes with no drift and for square integrable Itô–Lévy processes using Malliavin calculus and white noise analysis. This extension might be useful for some applications in finance. As an application of our result, we calculate explicitly the closest hedge strategy for the digital option whose pay-off, , is square integrable and the stock price is driven by a Lévy process.The classical Clark–Ocone theorem states that any random variable can be represented as where denotes the conditional expectation, is a Brownian motion with canonical filtration and D denotes the Malliavin derivative in the direction of W. Since many applications in financial mathematics require representation of random variables with respect to risk neutral martingale measure, an equivalent martingale measure version of this theorem was stated by Karatzas and Ocone (Stoch. Stoch. Rep. 34 (1991), 187–220). In this paper, we extend these results to be valid for square integrable pure jump Levy processes with no drift and for square integrable Ito–Levy processes using Malliavin calculus and white noise analysis. This extension might be useful for some applications in finance. As an application of our result, we calculate explicitly the closest hedge strategy for the digital option whose pay-off, , is square integrable and the stock price is driven by a Levy process.