Questions tagged [martingale]
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184
questions
2
votes
1answer
59 views
Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup
Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent ...
3
votes
2answers
163 views
Ito's lemma $f(t,W_t^2)$
Let $f$ be a function of $t$ and $W_t^2$.
a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration.
b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
1
vote
0answers
58 views
Finding the PDE and replicating strategy of a european contigent claim [duplicate]
Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying:
$dB(t)=r(t)B(t)dt$ and
$dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$
where $r,b,\...
2
votes
1answer
104 views
Solving an SDE using Ito's Lemma
Suppose that
$Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$
with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process
There is also ...
2
votes
1answer
87 views
Martingale proof: Call-prices must be increasing in maturity
I have observed that IV is increasing with time to maturity by using market prices and plotting IV (from Black-Scholes) against log-moneyness, $\log(S_t/K)$. $S_t$ being the price of the stock at time ...
0
votes
0answers
23 views
Q determined by the market in Binomial Model
I read in a book about change of measure, so that the discounted stock price in a binomial model is equal to the current price. Namely:
$$E_{Q}[S_{1}/ \beta |S_{0}]= S_{0} $$
It then says: " Q is ...
3
votes
2answers
102 views
Poisson process under equivalent martingale measure
I have a stochastic process $N(t)$ which is equal to $n$ with probability
$P\{N(t) = n\}=\frac{\left(\lambda t \right)^{n}}{n!}e^{-\lambda t }$
where $t$ represents the time period. In other words, ...
1
vote
2answers
82 views
Drift Term in Black-Scholes Model Martingale
How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof).
For instance if Im looking at:
$$dS_{t} = \mu S_{t} + \...
3
votes
1answer
101 views
replicating self-financing portfolio for risk neutral measure
Let the price process $S_{t}, 0 \leq t \leq T$, be a diffusion, and savings account be $\beta_{t}$ such that the Equivalent Martingale Measure $Q$ exists. Let $C_{T}=g\left(X_{T}\right)$ be the claim ...
3
votes
0answers
60 views
Derivation of option pricing PIDE: Why does the drift need to be zero?
I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero.
...
1
vote
1answer
96 views
Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
1
vote
1answer
51 views
Martingale problem on biased random walk
I am struggling to understand the martingale property of exponential of a biased random walk. For example, in the following problem how do I verify whether the following is a martingale, submartingale ...
2
votes
2answers
210 views
Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
4
votes
0answers
84 views
If arbitrage can happen exactly at one moment, is it really arbitrage?
There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk:
If $S=\left(S_{t}\right)_{t=0}^{T}$ is a ...
6
votes
2answers
404 views
Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
3
votes
0answers
43 views
Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale
I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that:
The existence of an equivalent martingale measure $Q$ on the discounted price ...
1
vote
1answer
87 views
How to take the expectation of an exponential martingale? And an exponential with a random value?
I am reading Shreve's Stochastic Calculus for Finance II. He states on pages 110 and 111 that,
$$E[exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)] = 1$$
$$E[exp(-\frac{1}{2}\sigma^2 \tau_m)] = e^{-\sigma m}$...
1
vote
1answer
71 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
103 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 ...
2
votes
1answer
147 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
106 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
95 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
53 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 &...
0
votes
0answers
71 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
238 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
173 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
171 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 ...
2
votes
1answer
103 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
78 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
102 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
votes
0answers
72 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
248 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
122 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
88 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
37 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
168 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
386 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
116 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
266 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?
...
14
votes
3answers
5k 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
76 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
$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
3
votes
1answer
118 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
0answers
87 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
543 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
64 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
33 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
1answer
198 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
473 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
53 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 ...
