Questions tagged [martingale]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
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
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
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 ...