Questions tagged [martingale]
The martingale tag has no usage guidance.
177
questions
3
votes
0answers
50 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
86 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
44 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
152 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
83 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 ...
4
votes
2answers
277 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
42 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
81 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,
$$E[exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)] = 1$$
$$E[exp(-\frac{1}{2}\sigma^2 \tau_m)] = e^{-\sigma m}$...
1
vote
1answer
59 views
If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?
I am trying to understand the connection between market completeness and risk neutral measures.
A market is complete if and only if the equivalent martingale measure is unique.
But if I change to the $...
2
votes
1answer
82 views
Martingale pricing with time-dependent risk-free rate
I want to find the price of a European call-option under the assumption that the risk-free rate $r$ is time-dependent, i.e.
$$
d\beta = r(t)\beta dt \leftrightarrow \beta(T) = e^{\int_0^T r(u)du}
$$
I ...
2
votes
1answer
129 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
99 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
74 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
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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
219 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
153 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
149 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
92 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
69 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
votes
0answers
65 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
219 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
111 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:
$$
\tag{1}\label{1}
L(t, T, T + \...
0
votes
0answers
87 views
Martingale optimal transport with two different nature of assets
In most of the litterature , for solving the optimal transport problem $sup_{Q\in \mathcal{M}}E^{Q}[c(S_{1},S_{2})]$ where $\mathcal{M}$ is the set of probability coupling such that the marginals of Q ...
0
votes
0answers
35 views
Equivalent martingale measure for Levy processes
Hey what is the easiest way to find equivalent martingale measure for Levy processes (in Merton model and Kou model for example)? I would like to write the dynamics of the stock price process under ...
2
votes
1answer
153 views
Clarification of Ito's lemma
I was looking at the various examples provided in the discussion Worked examples of applying Ito's lemma
One such example is 9.1 (c). This states that -
if $S_t =\! S_0 + \int\limits_{0}^{t} \mu_u ...
1
vote
1answer
312 views
Power Options & Forwards on Stock Squared
Short story: the process for Stock price squared is not a martingale when discounted by the money-market numeraire under the risk-neutral measure. How can we then compute derivative prices on $S_t^2$ ...
1
vote
1answer
114 views
Are there stocks dynamic that cannot be represented by Generalized Black Scholes model?
The generalized Black Scholes Model refers to a stock dynamic that satisfy
$$
dS(t)=S(t)(\mu_t dt+ \sigma_t dW(t))
$$
By martingale representation theorem, it seems that if there is a risk neutral ...
5
votes
1answer
262 views
Fama: Efficient Capital Markets: A Review of Theory and Empirical Work - are martingales incorrect?
In his paper, Eugene Fama gives the definition of a "fair game" as given below. I disagree. AFAIK, a martingale has the following property: $E[X_{t+\tau} | X_t] = X_t$. What am I missing?
...
13
votes
3answers
4k views
What is the Risk Neutral Measure?
What is the Risk Neutral Measure?
I don't believe this has been answered on the internet well and with all the parts connecting.
So:
What is the risk neutral measure/pricing?
Why do we need it?
How ...
1
vote
1answer
73 views
Covariation of Ito semimartingales
If we have two Ito semimartingales over $[0,T]$:
$$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
116 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
85 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
515 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
52 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
63 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
1answer
182 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
430 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 ...
1
vote
0answers
76 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
80 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
666 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
223 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
627 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
112 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
164 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 ...
