# Questions tagged [martingale]

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### What is the Risk Neutral Measure?

What is the Risk Neutral Measure? I don't believe this has been answered on the internet well and with all the parts connecting. So: What is the risk neutral measure/pricing? Why do we need it? How ...
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### Stock Price as Numeraire, Two Stocks & One Money Market Account

We have two uncorrelated Stock price processes and the classical Money-Market (MM) account. Under the MM Numeraire, both stocks are Martingales when discounted by the MM, as usual. Question: I would ...
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### Numeraire correlated to the traded asset

Edit: is there any work published on Numeraire being correlated to one of the traded assets? I haven't found a single paper online on this topic. If anyone has links to any resources on this topic, ...
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### How to understand “the world defined by specific numeraire”?

I am reading John Hull's book. When talking about "equivalent martingale measure result", the author introduces the concept of "the world defined by specific numeraire g" to describe the world in ...
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### Martingale optimal transport

I'm a student and currently studying martingale optimal transport for deriving upper and lower contract bounds but i happen struggling on the fact that in most papers , interest rates are not taken in ...
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### Prove that it is possible to make a self-financing portfolio

I would like to prove that if there exists $(X_1,\ldots,X_n)$ satisfying $\mathbb E[\int |X(s)|^2 d[Y]_s]<\infty$ (for a standard filtered probability space $(\Omega, F,( F_t)_{t\ge 0},\mathbb P)$ ...
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### How to expand lognormal approximation of Brownian motion

How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion. If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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### Reflection principle on continuous process and reflection property for Gaussian matrix

In my note it said that the reflection principle holds for continuous process but the reflection property only holds for a Gaussian martingale. I am wondering why is that? Is it somehow related to the ...
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### Simple application of the fundamental theorem of asset pricing

From what I understood the fundamental theorem of asset pricing (FTAP) details that discounted asset prices are martingales under the risk neutral mesure. As an example: We consider an ATM call ...
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### Black-Scholes Model with Cost of Carry

Good day, I was going over some exercises for my course and found this rather long one but I dont know how to start. Consider a commodity whose unit price at time t is St. Ownership of a unit of ...
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### Show that a zero-coupon bond discounted by a bond with mautrity $T$ is a martingale under the $T$-Forward measure

Here's the exact question: Show that for any $s>0$, $\frac{P(t,s)}{P(t,T)}$ is a $Q^T$-martingale. Here's my attempt: Let $t^\prime < t$. First consider the case $s>T$. \begin{aligned} \...
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### Three proofs regarding brownian motions and martingales

1. Let $(B_t)_{t \geq 0}$ and $(W_t)_{t \geq 0}$ be two standard Brownian motions and let $X_t := B_t W_t$. Is $(X_t)_{t \geq 0}$ a martingale? The easiest way to proceed seems to be to apply Ito's ...
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### Pricing European call with Feynman-Kac

I am trying to calculate the solution to the Black-Scholes (BS) equation using the Feynman-Kac (FK) formula for a simple European call. According to FK, the solution to BS is the discounted average of ...
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### Unconditional Expectation vs. Conditional Expectation at time $0$

In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
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### Fair price of a coupon paying bond

Consider a coupon paying bond with a maturity of $3$ years, that pays coupon annually. Let $c$ be the coupon rate (percentage) and let $F$ be the face value. This means that the holder of the bond ...
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### What is the true “value” process of American derivatives?

Consider a continuous-time market where LOOP (law of one price) holds. The first fundamental theorem of asset pricing states explicitly that in the absence of arbitrage, the risk-neutral measure ...
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### Risk neutral modelling of a stock

Suppose a stock $S$ follows $$dS(t) = \alpha(t)S(t)dt + \sigma(t)S(t)dW(t),$$ where $W(t)$ is a Brownian motion under $P$. Also suppose there is a short rate process $r(t)$. My question would be is ...
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Exercise : Let $K>0$. A European Cash-or-Nothing put option $P$ has the following pay-out profile : $$P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$$ Let $P_0^n$ be the no-arbitrage value at time $... 0answers 135 views ### Random variable minus Integral of Ito Generator is a Martingale under what conditions? I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,... 0answers 48 views ### Martingale positive price process I hope you can help me with this problem. In my lecture notes, my professor stated that for a state price deflator \phi\in L_{n+1}^2(P, F) (F being a filtration) and a strictly positive price ... 1answer 180 views ### Discounted asset price is martingale in BS model I want to verify that the discounted stock price process \mathrm{e}^{-r(T-t)}V(S_t,t) is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that$$ \mathrm{d}\mathrm{e}^{-r(T-t)}... 1answer 82 views ### Risk neutrality coherence with risk aversion I haven't been able to find an understandable explanation why the risk neutrality is coherent with the risk aversion implication of the expected utility hypothesis. I can see that when using the risk ... 0answers 111 views ### How to justify the martingale condition By Radon-Nikodym theorem, the conditional expectation of$X$with respect to a$\sigma$-algebra$\mathscr F$is a nonnegative random variable denoted by$\def\E{\mathbf E}\E(X\mid \mathscr F)$, such ... 1answer 124 views ### Hedging Value-Financial Mathematics EXERCISE We consider a free from arbitrage financial market$(Ω,F,P,S_0,S_1)$with$α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$ Let C be a ... 1answer 358 views ### Equivalent martingale measure exists if and only if$a < S_0^1(1+r)< b$Exercise : We consider a market of one period$(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space$\Omega$has a finite number of elements and the$\sigma-$algebra$\mathcal{F} = 2^\...
This is an exercise which I came upon while studying an introduction to financial mathematics. Exercise : Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let \$\...
I follow a course, and get to the point that one bond price discounted by another one is a martingale: $$\frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale }$$ I can not ...