Questions tagged [martingale]
The martingale tag has no usage guidance.
151
questions
6
votes
3answers
418 views
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 ...
1
vote
1answer
43 views
Covariation of Ito semimartingales
If we have two Ito semimartingales over $[0,T]$:
$$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$
What is the relationship between
$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
3
votes
1answer
101 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
0answers
55 views
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 ...
9
votes
1answer
326 views
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, ...
0
votes
0answers
25 views
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 ...
0
votes
0answers
42 views
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 ...
0
votes
0answers
55 views
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 $(\Omega, F,( F_t)_{t\ge 0},\mathbb P)$ ...
1
vote
0answers
28 views
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 $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
0
votes
0answers
12 views
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 ...
0
votes
0answers
47 views
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 ...
0
votes
0answers
140 views
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 ...
0
votes
1answer
60 views
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}
\...
4
votes
1answer
294 views
Pricing call option using risk-neutral martingale approach with squared stock price boundary?
I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0:
$$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
1
vote
0answers
49 views
Martingale Property {Proof [closed]
Can someone assist with this proof? I apologize for such a vague post. I have no idea where to begin. I am in a class a little above my level with this stuff. I have added a picture of the proof he is ...
1
vote
0answers
66 views
Some basics of option pricing
I am a mathematician trying to learn finance on my own. Try to avoid financial lingo in your answer when not necessary.
So I am trying to understand (European) option pricing under the no free lunch ...
-1
votes
1answer
75 views
How to show if this is Martingale or not?
Consider the outcome of a game played by repeatedly tossing a fair coin, where you win a dollar if heads appears and you lose a dollar if tails appear, the outcome is denoted $X_1$, $X_2$, $X_3$,...,$...
3
votes
3answers
163 views
How do we determine the “correct measure”?
Frequently I come across the statement that the "correct measure" for a product is this-or-that measure. For example,
Eurodollar Futures or Stock returns - Risk neutral measure
Libor forward rate - T-...
2
votes
1answer
180 views
Steven Shreve: Stochastic Calculus and Finance
The lecture notes have the following theorem:
Let $\theta\in \mathbb{R}$ be given and $B(t)$ stands for the Brownian motion which is a martingale, then $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
7
votes
3answers
299 views
Intuition for Stock Price Numeraire Drift
I would like to ask whether there is an intuition for the drift of price processes under the Stock numeraire.
I find it intuitive that the martingale measure under the Money Market numeraire induces ...
2
votes
1answer
84 views
Risk Neutral Pricing and the Drift
For risk neutral pricing, why do we want to compute expectation of a martingale? why is this so important? Why do we dislike the drift so much?
Avoid math heavy answers please.
0
votes
0answers
45 views
How to proof the formula to be martingale under ITO process?
How can implies that is a martingale when using the defaultable bond price?
3
votes
1answer
98 views
Forward rates are martingale under the T-forward measure
Forward rates are martingale under the $T$-forward measure but this derivation is suggesting otherwise. Could anyone please point out the mistake ?
Let $dW_Q$ be a Brownian Motion in the risk ...
3
votes
2answers
152 views
Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale
I'm trying to prove that $Z(t)=\exp(aW(t)-0.5a^2t)$ is a martingale where $W(t)$ is a Wiener process and $a$ is a constant. Here is my attempt:
$$E[Z(t+s)] = E\left[\exp\left(aW(t+s)-0.5a^2(t+s)\...
5
votes
1answer
76 views
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 ...
0
votes
1answer
183 views
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 ...
0
votes
2answers
83 views
Forward Rates are martingal under Forwar Measure detailled proof
So I read an other post about this :
How to prove martingality of forward rate under T-forward measure
But I can't see how to get from there to there :
$F \left(t,T_n \right)P \left(t,T_{n+1}\...
2
votes
2answers
192 views
Proving a process is martingale under the Risk Neutral Measure
Show that for any $\lambda \in \Re$, the process $Y_{\lambda,t}$ defined as:
$$Y_{\lambda,t} = (S_t/S_0)^\lambda e^{-(r\lambda-\lambda(1-\lambda)\sigma^2/2)t}$$
is a martingale under the risk ...
0
votes
0answers
83 views
Swap rate in the annuity measure and Martingale Representation Theorem
As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ...
1
vote
0answers
34 views
Hedged portfolio dynamics under T-forward measure
I'm looking to find the hedging PDE for a multi-currency derivative $u(F_d, F_f, X,t, T)$ under the T-forward measure, using the delta-hedging argument (F - forward rate, X - forward FX rate).
...
1
vote
1answer
84 views
Calculating the value of Beta - Martingales
Assume a risk free bond $B_t$and the stock St follow the dynamics of the Black & Scholes model.
(with interest rate r, stock drift $\mu$ and volatility $\sigma$).
Find $\beta$ such that the ...
3
votes
1answer
112 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 $Ļ ā T_{0,N}$. Let $Z_n = M_n ā A_n$ be the Doob decomposition of Z, then $Z_n^Ļ = M_n^Ļ ā A_n^Ļ$ is the Doob ...
2
votes
1answer
96 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
$$C_t = \...
1
vote
2answers
64 views
Intuitive view of conditional expectation
I would like someone to give me an intuitive view of conditional expectation. I mean: I have always understood it through formulas but I don't "see" what it is yet.
Thank you
2
votes
1answer
101 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
3
votes
2answers
544 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
vote
0answers
100 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
124 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
vote
0answers
58 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
votes
1answer
162 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
votes
1answer
84 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 $...
7
votes
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: $$M_s = U(s,...
3
votes
0answers
48 views
Martingale positive price process
I hope you can help me with this problem.
In my lecture notes, my professor stated that for a state price deflator $\phi\in L_{n+1}^2(P, F)$ (F being a filtration) and a strictly positive price ...
4
votes
1answer
180 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
2
votes
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 ...
3
votes
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 ...
3
votes
1answer
124 views
Hedging Value-Financial Mathematics
EXERCISE
We consider a free from arbitrage financial market $(ā¦,F,P,S_0,S_1)$ with $Ī±<S_0^{1}\cdot(1+r)<Ī²$,where $$0<Ī±:=min_{Ļ \in Ī©} S_1^{1}(Ļ), Ī²:=max_{Ļ \in Ī©}S_1^{1}, Ī±<Ī²$$
Let C be a ...
3
votes
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^\...
2
votes
1answer
182 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 $\...
5
votes
1answer
491 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 ...
