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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.
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| 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>
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| ==Measure theory representation==
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| 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,
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| <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>
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| where <math>\tilde{\mathbb{P}}</math> is the [[risk-neutral measure]].
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| <math>(r(t))_{t\in [0,T]}</math> is an <math>\mathcal{F}_t</math>-measurable (risk-free, possibly stochastic) interest rate process.
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| 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.
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| 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
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| <math>dX(t)=\Delta(t) \ dS(t) + r(t)(X(t)-\Delta(t)S(t)) \ dt</math>
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| 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.
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| 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.
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| ==See also==
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| *[[Martingale (probability theory)]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Martingale Pricing}}
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| [[Category:Finance theories]]
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| [[Category:Mathematical finance]]
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| [[Category:Pricing]]
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