Value at risk and self-similarity

The concept of Value at Risk measures the "risk" of a portfolio and is a statement of the following form: With probability q the potential loss will not exceed the Value at Risk figure. It is in widespread use within the banking industry. It is common to derive the Value at Risk figure of d days from the one of one–day by multiplying with √d. Obviously, this formula is right, if the changes in the value of the portfolio are normally distributed with stationary and independent increments. However, this formula is no longer valid, if arbitrary distributions are assumed. For example, if the distributions of the changes in the value of the portfolio are self–similar with Hurst coefficient H, the Value at Risk figure of one–day has to be multiplied by dH in order to get the Value at Risk figure for d days. This paper investigates to which extent this formula (of multiplying by √d) can be applied for all financial time series. Moreover, it will be studied how much the risk can be over– or underestimated, if the above formula is used. The scaling law coefficient and the Hurst exponent are calculated for various financial time series for several quantiles

Crash hedging strategies and worst–case scenario portfolio optimization

Crash hedging strategies are derived as solutions of non–linear differential equations which itself are consequences of an equilibrium strategy which make the investor indifferent to uncertain (down) jumps. This is done in the situation where the investor has a logarithmic utility and where the market coefficients after a possible crash may change. It is scrutinized when and in which sense the crash hedging strategy is optimal. The situation of an investor with incomplete information is considered as well. Finally, introducing the crash horizon, an implied volatility is derived

Worst-case scenario portfolio optimization: a new stochastic control approach

We consider the determination of portfolio processes yielding the highest worst-case bound for the expected utility from final wealth if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst-case bounds. A particular application of our setting is to model crash scenarios where both the number and the height of the crash are uncertain but bounded. Also the situation of changing market coefficients after a possible crash is analyzed

On the uniqueness of unbounded viscosity solutions arising in an optimal terminal wealth problem with transaction costs

We study the uniqueness of viscosity solutions of a Hamilton-Jacobi-Bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs. Our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility

Worst-case optimal investment with a random number of crashes

We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. We then verify that this strategy outperforms every other trading strategy using a direct comparison approach. We conclude with numerical examples and calculating the costs of hedging against crashes

"And in another make me understood": Reading George Herbert in the Light of His Contemporaries

This dissertation examines the ways critics have coupled George Herbert with different authors and thinkers of his era and analyzes the effects of these pairings on what Herbert has meant to readers. The specific fellow writers considered are Richard Hooker/John Calvin (in whose company Herbert looks like a religious partisan); Francis Bacon (as modern thinker, examining the physical world separated from a religious interpretation); and John Donne (as artist, creating dramatic speakers in conversation with God). To a great extent, critics have used such couplings to convey the values they wish to impart to readers and build the literary canon thereby. Herbert is a special case because of the sheer variety of appropriations made of his work since its first publication and the often contentious nature of these appropriations. Moreover, Herbert seems aware of his own work's flexibility and describes the uses of this quality in social discourse. The review of the literature traces not only the roller coaster ride that has been Herbert's critical reception but also the dozens of introductions to Herbert's works. Then, for each case, we look first at the shifting scholarship surrounding the fellow authors and then examine how Herbert engages with the questions raised by the other authors' works. Herbert on the Religious Spectrum examines the contentious area of study that places his work on the religious spectrum between presbyterianism and what would later be called Anglo-Catholicism. Herbert and Bacon starts with Herbert and Bacon's mutual admiration and examines how both men's works use nature to talk about God in complicated ways. And Herbert and Donne examines the assumptions and biases surrounding Herbert's coupling with John Donne, the fellow poet-priest and founder of the metaphysical school of which Herbert was considered a member by many critics from the late-nineteenth to the mid-twentieth century. In every case, we parse where critics' concerns start and Herbert's end, concluding, ultimately, that Herbert's intention in using argument, philosophy, and artistry, and in examining his own role in the discourse of religion, science, and art, is to fulfill a pastoral mission

Pricing and hedging of Asian options: Quasi-explicit solutions via Malliavin calculus

We use Malliavin calculus and the Clark-Ocone formula to derive the hedging strategy of an arithmetic Asian Call option in general terms. Furthermore we derive an expression for the density of the integral over time of a geometric Brownian motion, which allows us to express hedging strategy and price of the Asian option as an analytic expression. Numerical computations which are based on this expression are provided

Hosszú távú intravitrealis ranibizumabkezelés exsudativ időskori maculadegenerációban

Bevezetés: Az exsudativ időskori maculadegeneráció kezelésére alkalmazott intravitrealis ranibizumabinjekcióval szerzett eddigi tapasztalatok kedvezőek. Célkitűzés: A szerzők célul tűzték ki az intravitrealis ranibizumabterápia kétéves/hosszú távú funkcionális és anatómiai eredményeinek felmérését. Módszer: Egycentrumú prospektív vizsgálatban 46 beteg (átlagéletkor: 75±9,1 év) adatait dolgozták fel. A kezelési protokoll szerint a betegek az első három hónapban havonta kaptak 0,5 mg ranibizumabinjekciót, majd szükség szerint (pro re nata). A legjobb korrigált visus, illetve a centrális retinavastagság változását követték. Eredmények: A legjobb korrigált látóélesség változása a követési idő végére sem a kezelés előtti értékhez képest (p = 0,760), sem az első év végi eredményhez képest nem volt statisztikailag szignifikáns (p = 0,154). A centrális retinavastagság a kiindulási értékhez képest szignifikánsan csökkent (p = 0,000001), az első év végi értékhez képest nem volt szignifikáns a változás (p = 0,875). Következtetések: Nedves típusú maculadegenerációban szenvedő betegeknél pro re nata protokoll szerinti intravitrealis ranibizumabkezeléssel hosszú távon a látóélesség stabil marad, az exsudatio hatékonyan csökkenthető Orv. Hetil., 2013, 154 (45), 1790–1797