Quantitative Finance

Questions tagged [martingale]

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3
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1answer
105 views

I am trying to solve this question about optimal stopping theory. I don't know how to get started. Any hints would be very helpful

Let $Z = (Zn)_{n=0,1,...,N}$ be the Snell envelope of $X = (Xn)_{n=0,1,...,N}$ and $τ ∈ T_{0,N}$. Let $Z_n = M_n − A_n$ be the Doob decomposition of Z, then $Z_n^τ = M_n^τ − A_n^τ$ is the Doob ...
2
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1answer
53 views

Is the Non-discounted Bachelier call option price a Martingale? [duplicate]

My math finance professor once said someting that I can't make sense of. Hope you can answer: For a foward process the non-discounted price for a European call option under Bachelier is $$C_t = \...
1
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2answers
56 views

Intuitive view of conditional expectation

I would like someone to give me an intuitive view of conditional expectation. I mean: I have always understood it through formulas but I don't "see" what it is yet. Thank you
1
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1answer
81 views

Proof standard Brownian Motion under change of measure

Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
3
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1answer
112 views

How to prove martingality of forward rate under T-forward measure

Let $P(t,T)=\mathbb{E}_{Q_{R}}[e^{\int^{T}_{t}r(u)du}|\mathcal{F}_{t}]$ be the price of a 1-euro zero-coupon bond with maturity $T$ and $r(u)$ the interest rate process. Consider the the forward rate $...
1
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0answers
48 views

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 ...
2
votes
1answer
105 views

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 ...
1
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0answers
47 views

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 ...
2
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1answer
151 views

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 ...
2
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1answer
68 views

Finding the limit $\lim_{n \to \infty} P_0^n$ for a European Cash-or-Nothing put option with $P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$

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 $...
6
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0answers
113 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,...
3
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0answers
47 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 ...
4
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1answer
92 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)}...
0
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1answer
67 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 ...
3
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0answers
100 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 ...
3
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1answer
84 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 ...
3
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1answer
234 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^...
2
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1answer
125 views

Showing that a market model has arbitrage and describing martingales

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 $\...
4
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1answer
306 views

Change of numeraire between T-forward and Bank Account

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 ...
3
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1answer
193 views

Libor Market Model (LMM) under risk neutral measure

I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get : Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus: $$ ...
3
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0answers
100 views

Equivalent martingale measure in time changed Levy models

I am investigating time changed Levy models. As far as I have seen, these models are usually directly described under the risk neutral measure $\mathbb{Q}$. However, I'm interested in first modelling ...
3
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0answers
33 views

Utility Maximization on a finite Probability Space. Possible mistakes in a paper?

I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the ...
9
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3answers
773 views

Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to ...
5
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0answers
46 views

Martingale property of inhomogenous poisson process

I have found this martingale property for an inhomogenous poisson process with intensity $\lambda(s)$ which I don't know how to prove. The text itself advises: "proceed using Monotone class theorem". ...
2
votes
1answer
346 views

Calculating the stochastic integral of $\exp(-rt)S_t$

I am currently reading lecture notes which aim to show that if $$ S_t = S_0 \exp (\mu t + \sigma W_t) $$ then, under the probability measure $\tilde{\mathbb{P}}$ with density $$ \gamma_T = \exp (c W_T ...
2
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1answer
92 views

Changing to martingale probability world

This question is really getting me annoyed and I'm struggling to do the final proof, I have no problem obtaining the adjustment to the drift rate necessary to collapse the drift term to make it a ...
1
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0answers
124 views

Martingale approach - Option pricing with Radom-Nikodym

I would like to get the price of an option which pays at time T the minimum between the logarithm of (S(1,T) / S(1,0)) and the logarithm of (S(2,T) / S(2,0), with the following processes: (The two ...
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0answers
104 views

Hull Martingales and measures problem 27.16 7e?

Here's a question from Hull's Options Futures and Other derivatives which I'd appreciate if someone helped me to clarify. The question is from the chapter "Martingales and Measures" Suppose that the ...
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0answers
69 views

Does pricing contingent claims under the EMM require us to define the distribution?

I am familiar with martingale pricing as primarily a notational abstraction which allows us to price contingent claims on $X_\tau$ by its conditional expectation. Usually, we interpret this to mean ...
0
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1answer
202 views

Equivalent Martingale Measure result Hull?

I've been reading Hull's chapter about Martingales and measures where he states that if you have the dynamics of two securities as follows: \begin{align} \frac{df}{f} = (r + \lambda \sigma_f) dt + \...
11
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2answers
2k views

Intuitive Explanation for Shannon's Demon?

I am reading Fortune's Formula by William Poundstone, and I am puzzled by a phenomenon called "Shannon's Demon", which Claude Shannon allegedly proposed in a series of lectures, and preserved only by ...
2
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0answers
148 views

Is the VIX a Martingale?

Say the S&P500 follows a Gaussian diffusion process, so that: $$ VIX^2_{T,t}=\frac{1}{T}\mathrm{E}_t^\mathbb{Q}\left[\int_t^{t+T}\sigma_s^2ds\right] $$ where $T$ is the tenor (assume fixed), $t$ ...
2
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1answer
271 views

Test if a process (with no drift) is a martingale

Consider the process $$Z(t)=\int_{0}^{t} \frac{u^a}{t^a}dW_u$$ for some real constant $a$ and $W_t$ is a wiener process. I want to check whether this process is a $F_t^W$-martingale. I noticed Lemma 4....
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0answers
129 views

Constant volatility and risk-free rate assumptions of Black Scholes

I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems ...
2
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2answers
123 views

The two fundamental theorems of Finance, as they relate to the martingale measure

I RECENTLY read this in an article by Battig and Jarrow, "the first fundamental theorem relates the notion of no arbitrage to the existence of an equivalent martingale measure, while the second ...
3
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0answers
55 views

Martingales with power-law tails and CLT

I'm writing a course paper on stable distributions and I couldn't find any source discussing limits of Martingales with power-law tails. Suppose we have a Martingale that produces IID observations at ...
3
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1answer
202 views

Equality under T-forward measure for convexity adjustment

I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,...
5
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1answer
344 views

Martingale measure result application for interest rates under T-forward measure?

I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality: \begin{align*} f_o = g_0 E^{g}\big(\frac{f_T}{...
1
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1answer
551 views

Change of measure between T-forward and T*-forward contract?

I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation: \begin{align*} P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{...
1
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1answer
313 views

Write expectation of brownian motion conditional on filtration as an integral?

Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is $f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So $$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz =\...
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0answers
588 views

Proof that integral of Brownian motion wrt time is not a martingale

Let $X_t=\int_0^t W_s ds$ where $W_s$ is Brownian motion, so $E[W_s]=0$. Then $E[X_t]=\int_0^t E[W_s] ds=\int_0^t 0 ds=0$. So $E[X_t|{\cal F}_s]=0\neq X_s$, almost everywhere. So by previous ...
4
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1answer
167 views

Ito representation unique up to indistinguishability? Proof?

Given an Ito-process $X(t)$, $t\in[0,T]$ $$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$ with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$. It is now often claimed that this ...
2
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0answers
35 views

Infinite Horizon Barrier Option Paradoxe [duplicate]

I've came across this question which is puzzling me. Imagine that interest rates are zero and that you observe a stock $S_t$ whose value today $S_0$ is equal to 1\$. We consider the derivative that ...
0
votes
1answer
118 views

Why calibration in $Q$ against option prices without showing that $Q$ is equivalent to $P$?

In practice, I have seen articles and financial textbooks on calibration of processes directly under the risk neutral world without showing that the measure is equivalent to a physical measure $P$. ...
2
votes
2answers
274 views

Pricing Equation for Best of Options

I am trying to derive a martingale pricing equation (closed form solution) for a best-of option. But I am getting stuck at a point. There are 2 stocks $U(t)$ and $V(t)$ they both follow GBM with a ...
2
votes
2answers
158 views

Calculate $E^{\mathbb{Q}}\left[e^{-\int_{0}^{T_2}r_t\,dt} \frac{S\left(T_2\right)}{S\left(T_1\right)}\right]$

Let $S\left(t\right)$ be a tradable financial security that doesn't generate cash flow (eg no dividend). $S\left(t\right)$ follows an unknown stochastic process. We now have a financial derivative ...
3
votes
2answers
246 views

Equivalent martingale measure price dynamics

Assume $S_0(t)=\exp(\int_0^t r(s) ds)$. Then $\mathbb{Q}\sim \mathbb P$ is a martingale measure $\iff$ every asset price process $S_i$ has price dynamics under $\mathbb Q$ of the form $dS_i(t)...
7
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3answers
3k views

Why discounted derivative price is a martingale?

Usually after showing that discounted stock price process is martingale under the risk-neutral measure, most authors say that this implies that the discounted derivative price process is a martingale ...
2
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0answers
76 views

An arbitrage strategy involving forward contracts to show that LIBOR rates are martingales

I note $L_{t}^{[T_s, T_e]}$ the forward rate at time $t$ for the period $[T_s, T_e]$. Recall it is the strike making equal to $0$ the value at time $t$ of a forward contract for the period $[T_s, T_e]$...
3
votes
1answer
1k views

Girsanov Theorem application to Geometric Brownian Motion

I recently read this from a book on mathematical finance The important example for finance the (unique) EMM for the geometric Brownian. Let $S_{t}$ be the price of an asset, $${{d{S_t}} \over {{...

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