Questions tagged [martingale]
The martingale tag has no usage guidance.
172
questions
1
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1answer
54 views
If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?
I am trying to understand the connection between market completeness and risk neutral measures.
A market is complete if and only if the equivalent martingale measure is unique.
But if I change to the $...
2
votes
1answer
60 views
Martingale pricing with time-dependent risk-free rate
I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e.
$$
d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du}
$$
I ...
1
vote
1answer
68 views
Efficient market hypothesis and martingales
One of the tasks in the book we´re using in introduction to finance is
Stocks are expected to earn (much) more than the risk-free interest rate. This means
that stock prices are expected to increase ...
0
votes
1answer
89 views
Why do stock prices follow a martingale?
I have a quick question: why does the Efficient Market Hypothesis (EMH) assume that stock prices follow a martingale process?
I understand that discounted prices under the risk-neutral probability ...
0
votes
1answer
62 views
What's the price of a lookback call option in the arbitrage-free CRR-model?
If we consider the CRR-model in two periods, i.e. T=2. Let $S^1$ be the risky asset with $S_0^1=100$ and $S^0$ the bond with $S_0^0=1$. Furthermore, we assume the model is arbitrage-free with $y_b=-0....
0
votes
0answers
50 views
Given the density function of $S^{1}$ in one-period model, find the risk-neutral measure
Consider the one period market model $\left(\overline{\pi},\overline{S}\right)$ consisting of a risk-free asset $\left(\pi^{0},S^{0}\right)=(1,1+r)$ and a risky $\left(\pi^{1},S^{1}\right)$
Let $ r &...
1
vote
0answers
69 views
Martingale stochastic processes
Does anyone know how to do this question?
A player whose initial holding is $N$ bets 1 on each game of a series of independent identical parts. He loses his bet whether he loses or he wins but, if he ...
0
votes
1answer
61 views
Maximal increase payoff
I am interested in the following problem.
We have a Multi-Step Binomial Model with discrete time $T=1,\dots,n$.
We also assume that the stock $S_t$ is a martingale and there is a risk-free bond with $...
5
votes
2answers
194 views
Can a Process with a Stochastic Drift be a Martingale?
I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts?
Suppose I have a process with a stochastic drift:
$$...
1
vote
1answer
140 views
Stochastic volatility Levy models
Hey I have some questions about stochastic volatility for Levy processes. If I understand correctly, if we change the time in Levy's process by CIR process, the newly received process is not Levy's ...
2
votes
1answer
125 views
Discounted price process - martingale
I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
3
votes
1answer
85 views
No-arbitrage Pricing
We have a contract whose value is $A(S_t,t) = S_t^3$ at all times, not just at expiration. $S_t$, the underlying stock, follows a Geometric Brownian Motion, $\frac{dS}{S} = \mu dt + \sigma dB$. How ...
1
vote
1answer
64 views
Why does an autocall on a linear payoff have vega?
Consider a (stochastic) linear index, say $I(t)$, in that it grows at the risk free rate (with some volatility of course). There exists a maturity date $T$ on which I receive $I(T)$; however there is ...
2
votes
0answers
100 views
Recognizing a Martingale
Under which conditions is the stochastic process $\{X_t\}_{t=0,1,...,T}$ a martingale? Demonstrate and explain clearly for each case below. If it is not necessarily a martingale, provide a ...
0
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0answers
59 views
Discounted stock price under a NON risk-neutral measure
Under a risk-neutral measure $\mathbb{Q}$, the discounted stock price is a $\mathbb{Q}$-martingale. Does it mean that under the actual probability measure $\mathbb{P}$ the discounted stock price is ...
5
votes
1answer
177 views
Why aren't american put options martingales?
I don't understand what's wrong in the following argument.
Assume that we have a no-arbitrage market where the following products are traded:
a risky asset $S$,
a risk-free bond $B$,
an American put ...
1
vote
1answer
100 views
Libor rate and martingales
We know that the forward Libor rate $L(t, T, T + \tau)$, in the absence of arbitrage, is a martingale under the measure $T + \tau$, i.e. $Q^{T+\tau}$. In this context:
$$
\tag{1}\label{1}
L(t, T, T + \...
0
votes
0answers
87 views
Martingale optimal transport with two different nature of assets
In most of the litterature , for solving the optimal transport problem $sup_{Q\in \mathcal{M}}E^{Q}[c(S_{1},S_{2})]$ where $\mathcal{M}$ is the set of probability coupling such that the marginals of Q ...
0
votes
0answers
33 views
Equivalent martingale measure for Levy processes
Hey what is the easiest way to find equivalent martingale measure for Levy processes (in Merton model and Kou model for example)? I would like to write the dynamics of the stock price process under ...
2
votes
1answer
132 views
Clarification of Ito's lemma
I was looking at the various examples provided in the discussion Worked examples of applying Ito's lemma
One such example is 9.1 (c). This states that -
if $S_t =\! S_0 + \int\limits_{0}^{t} \mu_u ...
1
vote
1answer
259 views
Power Options & Forwards on Stock Squared
Short story: the process for Stock price squared is not a martingale when discounted by the money-market numeraire under the risk-neutral measure. How can we then compute derivative prices on $S_t^2$ ...
1
vote
1answer
109 views
Are there stocks dynamic that cannot be represented by Generalized Black Scholes model?
The generalized Black Scholes Model refers to a stock dynamic that satisfy
$$
dS(t)=S(t)(\mu_t dt+ \sigma_t dW(t))
$$
By martingale representation theorem, it seems that if there is a risk neutral ...
5
votes
1answer
259 views
Fama: Efficient Capital Markets: A Review of Theory and Empirical Work - are martingales incorrect?
In his paper, Eugene Fama gives the definition of a "fair game" as given below. I disagree. AFAIK, a martingale has the following property: $E[X_{t+\tau} | X_t] = X_t$. What am I missing?
...
12
votes
3answers
3k 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
69 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
$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
3
votes
1answer
112 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
0answers
75 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 ...
11
votes
1answer
478 views
Numeraire correlated to the traded asset
The Fundamental Theorem of Asset Pricing states that:
\begin{align*}
\frac{X_0}{N_0} &= \mathbb{E}^N{ \left[ \frac{X(t)}{N(t)}|\mathcal{F}_0 \right] }
\end{align*}
The usual conditions apply (both ...
0
votes
0answers
28 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
49 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
60 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
30 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
15 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
71 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
287 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
147 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}
\...
5
votes
1answer
388 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
52 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
71 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
79 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$,...,$...
6
votes
3answers
535 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
206 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\}$...
9
votes
3answers
538 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
97 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.
3
votes
1answer
146 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
254 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
103 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
351 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
98 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
239 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 ...
