References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt155",{id:"formSmash:upper:j_idt155",widgetVar:"widget_formSmash_upper_j_idt155",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt156_j_idt159",{id:"formSmash:upper:j_idt156:j_idt159",widgetVar:"widget_formSmash_upper_j_idt156_j_idt159",target:"formSmash:upper:j_idt156:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

Asymptotic Methods for Pricing European Option in a Market Model With Two Stochastic VolatilitiesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Mälardalen University, Västerås, Sweden , 2016.
##### Series

Mälardalen University Press Dissertations, ISSN 1651-4238 ; 219
##### Keyword [en]

Asymptotic Expansion, European Options, Stochastic Volatilities
##### National Category

Mathematics
##### Research subject

Mathematics/Applied Mathematics
##### Identifiers

URN: urn:nbn:se:mdh:diva-33475ISBN: 978-91-7485-300-1 (print)OAI: oai:DiVA.org:mdh-33475DiVA: diva2:1040251
##### Public defence

2016-12-07, Kappa, Mälardalens högskola, Västerås, 13:15 (English)
##### Opponent

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt399",{id:"formSmash:j_idt399",widgetVar:"widget_formSmash_j_idt399",multiple:true});
##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt406",{id:"formSmash:j_idt406",widgetVar:"widget_formSmash_j_idt406",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt412",{id:"formSmash:j_idt412",widgetVar:"widget_formSmash_j_idt412",multiple:true});
Available from: 2016-10-28 Created: 2016-10-26 Last updated: 2016-11-10Bibliographically approved
##### List of papers

Modern financial engineering is a part of applied mathematics that studies market models. Each model is characterized by several parameters. Some of them are familiar to a wide audience, for example, the price of a risky security, or the risk free interest rate. Other parameters are less known, for example, the volatility of the security. This parameter determines the rate of change of security prices and is determined by several factors. For example, during the periods of stable economic growth the prices are changing slowly, and the volatility is small. During the crisis periods, the volatility significantly increases. Classical market models, in particular, the celebrated Nobel Prize awarded Black–Scholes–Merton model (1973), suppose that the volatility remains constant during the lifetime of a financial instrument. Nowadays, in most cases, this assumption cannot adequately describe reality. We consider a model where both the security price and the volatility are described by random functions of time, or stochastic processes. Moreover, the volatility process is modelled as a sum of two independent stochastic processes. Both of them are mean reverting in the sense that they randomly oscillate around their average values and never escape neither to very small nor to very big values. One is changing slowly and describes low frequency, for example, seasonal effects, another is changing fast and describes various high frequency effects. We formulate the model in the form of a system of a special kind of equations called stochastic differential equations. Our system includes three stochastic processes, four independent factors, and depends on two small parameters. We calculate the price of a particular financial instrument called European call option. This financial contract gives its holder the right (but not the obligation) to buy a predefined number of units of the risky security on a predefined date and pay a predefined price. To solve this problem, we use the classical result of Feynman (1948) and Kac (1949). The price of the instrument is the solution to another kind of problem called boundary value problem for a partial differential equation. The resulting equation cannot be solved analytically. Instead we represent the solution in the form of an expansion in the integer and half-integer powers of the two small parameters mentioned above. We calculate the coefficients of the expansion up to the second order, find their financial sense, perform numerical studies, and validate our results by comparing them to known verified models from the literature. The results of our investigation can be used by both financial institutions and individual investors for optimization of their incomes.

1. Pricing European Options Under Stochastic Volatilities Models$(function(){PrimeFaces.cw("OverlayPanel","overlay1034032",{id:"formSmash:j_idt452:0:j_idt456",widgetVar:"overlay1034032",target:"formSmash:j_idt452:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Perturbation Methods for Pricing European Options in a Model with Two Stochastic Volatilities$(function(){PrimeFaces.cw("OverlayPanel","overlay1040244",{id:"formSmash:j_idt452:1:j_idt456",widgetVar:"overlay1040244",target:"formSmash:j_idt452:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Numerical Studies on Asymptotics of European Option under Multiscale Stochastic Volatility$(function(){PrimeFaces.cw("OverlayPanel","overlay882730",{id:"formSmash:j_idt452:2:j_idt456",widgetVar:"overlay882730",target:"formSmash:j_idt452:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Second Order Asymptotic Expansion for Pricing European Options in a Model with Two Stochastic Volatilities$(function(){PrimeFaces.cw("OverlayPanel","overlay1040242",{id:"formSmash:j_idt452:3:j_idt456",widgetVar:"overlay1040242",target:"formSmash:j_idt452:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Numerical Methods on European Options Second Order Asymptotic Expansions for Multiscale Stochastic Volatility$(function(){PrimeFaces.cw("OverlayPanel","overlay1040245",{id:"formSmash:j_idt452:4:j_idt456",widgetVar:"overlay1040245",target:"formSmash:j_idt452:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Approximation Methods of European Option Pricing in Multiscale Stochastic Volatility Model$(function(){PrimeFaces.cw("OverlayPanel","overlay1040247",{id:"formSmash:j_idt452:5:j_idt456",widgetVar:"overlay1040247",target:"formSmash:j_idt452:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1138",{id:"formSmash:lower:j_idt1138",widgetVar:"widget_formSmash_lower_j_idt1138",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1139_j_idt1141",{id:"formSmash:lower:j_idt1139:j_idt1141",widgetVar:"widget_formSmash_lower_j_idt1139_j_idt1141",target:"formSmash:lower:j_idt1139:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});