Classical modular curve - Revision history

Salix alba: /* Mappings */fix tags, replaced: _{0 → ₀ (3) using AWB}

2014-02-28T21:58:02Z

Mappings: fix tags, replaced: 0 → 0 (3) using AWB

← Older revision		Revision as of 23:58, 28 February 2014
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	~~'''Martingale pricing''' is a pricing approach based on~~ the ~~notions of [[Martingale (probability theory)\|martingale]] and [[risk-neutral measure\|risk neutrality]]. The martingale pricing approach~~ is ~~a cornerstone of modern quantitative finance and can be applied to a variety of [[derivative (finance)\|derivatives]] contracts, e~~.~~g. [[option (finance)\|options]], [[Futures contract\|futures]], [[interest rate derivative]]~~s~~, [[credit derivatives]], etc.~~		The title of the writer is Nestor. For years she's been living in Kansas. Camping is some thing that I've done for years. I am a production and distribution officer.<br><br>My webpage; extended car warranty ([http://makemoneyonline.net.in/blogs/post/9758 browse around this web-site])

	~~In contrast to the [[Partial differential equation\|PDE]] approach to pricing, martingale pricing formulae are~~ in ~~the form of expectations which can be efficiently solved numerically using a [[Monte Carlo method\|Monte Carlo]] approach~~. ~~As such, Martingale pricing~~ is ~~preferred when valuing highly dimensional contracts such as a basket of options~~. ~~On the other hand, valuing [[American option\|American-style contracts]] is troublesome and requires discretizing the problem (making it like~~ a ~~[[Bermudan option]])~~ and ~~only in 2001 [[Francis Longstaff\|F. A. Longstaff]] and [[Eduardo Schwartz\|E. S. Schwartz]] developed a practical Monte Carlo method for pricing American options~~.<~~ref~~>{{cite journal\|last1=Longstaff\|first1=F.A.\|first2=E.S.\|last2=Schwartz\|url=http://repositories.cdlib.org/anderson/fin/1-01/\|accessdate=October 8, 2011\|title=Valuing American options by simulation: a simple least squares approach\|journal=Review of Financial Studies\|volume=14\|year=2001\|pages=113–148}}<~~/ref~~>

	~~==Measure theory representation==~~

	~~Suppose the state of the market can be represented by the [[Filtration_~~(~~mathematics)#Measure_theory\|filtered]]~~ [~~[probability space]],<math>(\Omega,(\mathcal{F}_{t})_{t\in[0,T]},\tilde{\mathbb{P}})<~~/~~math>. Let <math>\{S(t)\}_{t\in[0,T]} <~~/~~math> be a stochastic price process on this space~~. ~~One may price a derivative security, <math>V(t,S(t))</math> under the philosophy of no arbitrage as,~~

	~~<center><math>D(t)V(t,S(t))=\tilde{\mathbb{E}}[D(T)V(T,S(T))\|\mathcal{F}_t], \qquad dD(t)=-r(t)D(t) \ dt</math></center>~~

	~~where <math>\tilde{\mathbb{P}}</math> is the [[risk-neutral measure]].~~

	~~<math>(r(t))_{t\in [0,T]}</math> is an <math>\mathcal{F}_t</math>-measurable (risk-free, possibly stochastic) interest rate process.~~

	This is accomplished through [[almost surely\|almost sure]] replication of the derivative's time <math>T</math> payoff using only underlying securities, and the risk-free money market (MMA). These underlyings have prices that are observable and known.
	~~Specifically, one constructs a portfolio process <math>\{X(t)\}_{t\in[0,T]}</math>~~ in continuous time, where he holds <math>\Delta(t)</math> shares of the underlying stock at each time <math>t</math>, and <math>X(t)-\Delta(t)S(t)</math> cash earning the risk-free rate <math>r(t)</math>. The portfolio obeys the stochastic differential equation

	~~<math>dX(t)=\Delta(t) \ dS(t) + r(t)(X(t)-\Delta(t)S(t)) \ dt<~~/~~math>~~

	One will then attempt to apply [[Girsanov theorem]] by first computing <math>\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}</math>; that is, the [[Radon–Nikodym derivative]] with respect to the observed market probability distribution. This ensures that the discounted replicating portfolio process is a Martingale under risk neutral conditions.

	~~If such a process <math>\Delta(t)</math> can be well-defined and constructed, then choosing <math>V(0,S(0))=X(0)<~~/~~math> will result in <math>\tilde{\mathbb{P}}[X(T)=V(T)] = 1<~~/~~math>, which immediately implies that~~ this ~~happens <math>\mathbb{P}</math>~~-~~[[almost surely]~~] ~~as well, since the two measures are equivalent.~~

	~~==See also==~~
	*[[Martingale (probability theory)]]

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Martingale Pricing}}~~
	~~[[Category:Finance theories]]~~
	~~[[Category:Mathematical finance]]~~
	~~[[Category:Pricing]]~~

en>ChrisGualtieri: General Fixes using AWB

2013-10-23T03:29:43Z

General Fixes using AWB

← Older revision		Revision as of 05:29, 23 October 2013
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	~~Friends contact him Royal Cummins~~. ~~Interviewing~~ is ~~what I do~~ in ~~my working day job~~. ~~Years~~ in the ~~past we moved to Arizona but my spouse wants us to move~~. One of ~~my preferred hobbies~~ is ~~camping~~ and ~~now I'm attempting to earn~~ money ~~with it~~.<br><br>~~my webpage: [http:~~//~~Llollz~~.~~com~~/~~blogs~~/~~post~~/~~5238 http:~~//~~Llollz~~.~~com~~]		'''Martingale pricing''' is a pricing approach based on the notions of [[Martingale (probability theory)\|martingale]] and [[risk-neutral measure\|risk neutrality]]. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of [[derivative (finance)\|derivatives]] contracts, e.g. [[option (finance)\|options]], [[Futures contract\|futures]], [[interest rate derivative]]s, [[credit derivatives]], etc.

			In contrast to the [[Partial differential equation\|PDE]] approach to pricing, martingale pricing formulae are in the form of expectations which can be efficiently solved numerically using a [[Monte Carlo method\|Monte Carlo]] approach. As such, Martingale pricing is preferred when valuing highly dimensional contracts such as a basket of options. On the other hand, valuing [[American option\|American-style contracts]] is troublesome and requires discretizing the problem (making it like a [[Bermudan option]]) and only in 2001 [[Francis Longstaff\|F. A. Longstaff]] and [[Eduardo Schwartz\|E. S. Schwartz]] developed a practical Monte Carlo method for pricing American options.<ref>{{cite journal\|last1=Longstaff\|first1=F.A.\|first2=E.S.\|last2=Schwartz\|url=http://repositories.cdlib.org/anderson/fin/1-01/\|accessdate=October 8, 2011\|title=Valuing American options by simulation: a simple least squares approach\|journal=Review of Financial Studies\|volume=14\|year=2001\|pages=113–148}}</ref>

			==Measure theory representation==

			Suppose the state of the market can be represented by the [[Filtration_(mathematics)#Measure_theory\|filtered]] [[probability space]],<math>(\Omega,(\mathcal{F}_{t})_{t\in[0,T]},\tilde{\mathbb{P}})</math>. Let <math>\{S(t)\}_{t\in[0,T]} </math> be a stochastic price process on this space. One may price a derivative security, <math>V(t,S(t))</math> under the philosophy of no arbitrage as,

			<center><math>D(t)V(t,S(t))=\tilde{\mathbb{E}}[D(T)V(T,S(T))\|\mathcal{F}_t], \qquad dD(t)=-r(t)D(t) \ dt</math></center>

			where <math>\tilde{\mathbb{P}}</math> is the [[risk-neutral measure]].

			<math>(r(t))_{t\in [0,T]}</math> is an <math>\mathcal{F}_t</math>-measurable (risk-free, possibly stochastic) interest rate process.

			This is accomplished through [[almost surely\|almost sure]] replication of the derivative's time <math>T</math> payoff using only underlying securities, and the risk-free money market (MMA). These underlyings have prices that are observable and known.
			Specifically, one constructs a portfolio process <math>\{X(t)\}_{t\in[0,T]}</math> in continuous time, where he holds <math>\Delta(t)</math> shares of the underlying stock at each time <math>t</math>, and <math>X(t)-\Delta(t)S(t)</math> cash earning the risk-free rate <math>r(t)</math>. The portfolio obeys the stochastic differential equation

			<math>dX(t)=\Delta(t) \ dS(t) + r(t)(X(t)-\Delta(t)S(t)) \ dt</math>

			One will then attempt to apply [[Girsanov theorem]] by first computing <math>\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}</math>; that is, the [[Radon–Nikodym derivative]] with respect to the observed market probability distribution. This ensures that the discounted replicating portfolio process is a Martingale under risk neutral conditions.

			If such a process <math>\Delta(t)</math> can be well-defined and constructed, then choosing <math>V(0,S(0))=X(0)</math> will result in <math>\tilde{\mathbb{P}}[X(T)=V(T)] = 1</math>, which immediately implies that this happens <math>\mathbb{P}</math>-[[almost surely]] as well, since the two measures are equivalent.

			==See also==
			*[[Martingale (probability theory)]]

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Martingale Pricing}}
			[[Category:Finance theories]]
			[[Category:Mathematical finance]]
			[[Category:Pricing]]

71.229.31.123: /* Parametrization of the modular curve */

2012-04-19T20:49:31Z

Parametrization of the modular curve

New page

Friends contact him Royal Cummins. Interviewing is what I do in my working day job. Years in the past we moved to Arizona but my spouse wants us to move. One of my preferred hobbies is camping and now I'm attempting to earn money with it. my webpage: [http://Llollz.com/blogs/post/5238 http://Llollz.com]

Classical modular curve - Revision history

Salix alba: /* Mappings */fix tags, replaced: 0 → 0 (3) using AWB

en>ChrisGualtieri: General Fixes using AWB

71.229.31.123: /* Parametrization of the modular curve */

Salix alba: /* Mappings */fix tags, replaced: _{0 → ₀ (3) using AWB}