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Questions tagged [martingale]

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26
votes
4answers
6k views

Strictly local martingales: what is the intuition behind them?

A process $X_t$ is a local martingale if for each increasing sequence of stopping times $\{\tau_k,k=1,2,...\}$ the stopped process is a martingale. All true martingales are local martingales, but the ...
23
votes
6answers
14k views

What is a martingale?

What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?
23
votes
5answers
11k views

Is the stock price process a martingale or a Markov process?

Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property". Are they unrelated? Is it that the Markov property implies some sort of ...
12
votes
2answers
408 views

Realized variance in SVJJ (Heston with jumps) model

I am working with the stochastic volatility model with jumps in both the price and volatility dynamics, ie. the risk neutral dynamics are of the form: $$\mathrm{d}V_t = \kappa(\theta - V_t)\mathrm{d}t ...
11
votes
2answers
2k views

Intuitive Explanation for Shannon's Demon?

I am reading Fortune's Formula by William Poundstone, and I am puzzled by a phenomenon called "Shannon's Demon", which Claude Shannon allegedly proposed in a series of lectures, and preserved only by ...
10
votes
2answers
7k views

Why is this stochastic integral a martingale?

Suppose that: $W^*_t$ is a Wiener process under probability measure $\mathbb{P}^*$ and; $\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$. In my lecture notes, it says that $\tilde{S}_t$ is a ...
9
votes
3answers
821 views

Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to ...
9
votes
2answers
703 views

Change of measure discrete time

Suppose I have a random walk $X_{n+1} = X_n+A_n$ where $A_n$ is an iid sequence, $\mathsf EA_n = A>0$. How to construct a martingale measure for this case?
8
votes
6answers
779 views

Kurtosis in asset logarithmic returns

Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval $n$ are the sum of smaller logarithmic returns in $1/n$ time ...
8
votes
1answer
198 views

FTAP a-la Harrison, Kreps and Pliska

I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some ...
7
votes
1answer
4k views

Convexity Adjustment for Futures

Let $B_t$ be the cash account numeraire. The future and forward prices at time t are expressed as: $$ Fut = E_t^Q\left[S_T\right],$$ $$ Fwd = \frac{E_t^Q[S_T/B_T]}{E_t^Q[1/B_T]}.$$ Where $$ \frac{...
7
votes
4answers
10k views

How to use Itô's formula to deduce that a stochastic process is a martingale?

I'm working through different books about financial mathematics and solving some problems I get stuck. Suppose you define an arbitrary stochastic process, for example $ X_t := W_t^8-8t $ where $ W_t ...
7
votes
3answers
3k views

Why discounted derivative price is a martingale?

Usually after showing that discounted stock price process is martingale under the risk-neutral measure, most authors say that this implies that the discounted derivative price process is a martingale ...
7
votes
3answers
1k views

Difference betweem martingale property and adapted filteration

What is the difference between a random process that is adapted to a filteration and one that had the martingale property. It seems the two notions are quite similar and would be helpful to construct ...
7
votes
1answer
635 views

How do equivalent martingale measures arise in pricing?

I'm studying for an exam in financial models and came across this question: "An agent with $C^2$ strictly increasing concave utility $U$ has wealth $w_0$ at time 0, and wishes to invest his wealth in ...
6
votes
0answers
115 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,...
5
votes
1answer
446 views

Martingale representation theorem

Let $r_t, \theta_t$ denote some stochastic processes driven by a $N$ dimensional Brownian motion $W_t$ (they are of course assumed adapted to the natural filtration $\mathcal{F}_t$ of that Brownian ...
5
votes
2answers
1k views

Uniqueness of equivalent martingale measure in Black Scholes-Model

Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
5
votes
1answer
350 views

Martingale measure result application for interest rates under T-forward measure?

I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality: \begin{align*} f_o = g_0 E^{g}\big(\frac{f_T}{...
5
votes
1answer
172 views

Ito representation unique up to indistinguishability? Proof?

Given an Ito-process $X(t)$, $t\in[0,T]$ $$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$ with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$. It is now often claimed that this ...
5
votes
2answers
385 views

Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
5
votes
1answer
223 views

What is the stochastic differential of a general semimartingale?

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3: $$H = B + H^c + h(x) \...
5
votes
1answer
221 views

Parameter estimation using martingale measures - include real world data?

Please note: I posted this in nuclearphynance first, but didn't get any replies. For desks which sell exotics it is common practice (as far as I know it) to calibrate the model (Stochastic Volatility,...
5
votes
0answers
46 views

Martingale property of inhomogenous poisson process

I have found this martingale property for an inhomogenous poisson process with intensity $\lambda(s)$ which I don't know how to prove. The text itself advises: "proceed using Monotone class theorem". ...
4
votes
1answer
461 views

unique equivalent martingale measure in incomplete markets

Do you have any idea about how we can prove, and under which conditions, that an equivalent martingale measure (EMM) in an incomplete market is unique? The assumptions we have made are: 1) that the ...
4
votes
1answer
98 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)}...
4
votes
1answer
320 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 ...
4
votes
0answers
264 views

Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
3
votes
3answers
4k views

Understanding the concept of Martingale pricing

I am a bit confused about how to formulate a problem where I have to price an option on a stock. Many papers say that stock prices are best modeled using a geometric Brownian motion (GBM), and I ...
3
votes
2answers
275 views

Is $\frac{P(t,S)}{P(t,T)}$ martingale?

Assume $r_t$ follow the CIR process and $P(t,T)=E[exp(-\int_{t}^{T}r_s ds)|F_t]$.I am going to show $\frac{P(t,S)}{P(t,T)}$ ($S<T$) is an $F_t$-martingale under Forward Measure but So confused! Do ...
3
votes
1answer
107 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 $τ ∈ T_{0,N}$. Let $Z_n = M_n − A_n$ be the Doob decomposition of Z, then $Z_n^τ = M_n^τ − A_n^τ$ is the Doob ...
3
votes
1answer
240 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^...
3
votes
1answer
217 views

On an application of Ito's lemma

Assume that instantaneous returns are generated by the continuous time martingale: $$dp_t = \sigma_t dW_t$$ where $W_t$ denotes a standard Weiner process and One day returns are denoted by $r_{t+1} =...
3
votes
1answer
136 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 $...
3
votes
1answer
84 views

Hedging Value-Financial Mathematics

EXERCISE We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$ Let ...
3
votes
2answers
259 views

Equivalent martingale measure price dynamics

Assume $S_0(t)=\exp(\int_0^t r(s) ds)$. Then $\mathbb{Q}\sim \mathbb P$ is a martingale measure $\iff$ every asset price process $S_i$ has price dynamics under $\mathbb Q$ of the form $dS_i(t)...
3
votes
2answers
98 views

Integration to calculate expected value of swap rate

In Hagan's paper on valuing CMS swaps (Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors), there is: So the swap rate must also be a Martingale, and $$E \big[ R_s(\tau) \big| \mathcal{...
3
votes
1answer
203 views

Libor Market Model (LMM) under risk neutral measure

I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get : Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus: $$ ...
3
votes
1answer
360 views

Calculating the stochastic integral of $\exp(-rt)S_t$

I am currently reading lecture notes which aim to show that if $$ S_t = S_0 \exp (\mu t + \sigma W_t) $$ then, under the probability measure $\tilde{\mathbb{P}}$ with density $$ \gamma_T = \exp (c W_T ...
3
votes
2answers
145 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
3
votes
2answers
545 views

Martingale Stock Prices

In http://www.principlesofforecasting.com/files/pdf/Granger-stockmarket.pdf Granger makes survey of some arguments. In section I there are two hypothesis H01, and H02. H01: Stock prices are a ...
3
votes
2answers
180 views

What impact does arbitrage have on realised volatility estimates?

Doing some research modeling/estimating volatility in the bitcoin market. There is quite a bit of scope for arbitrage within crypto-currency markets. Wonder if this has any impact on my volatility ...
3
votes
1answer
207 views

Equality under T-forward measure for convexity adjustment

I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,...
3
votes
1answer
1k views

Girsanov Theorem application to Geometric Brownian Motion

I recently read this from a book on mathematical finance The important example for finance the (unique) EMM for the geometric Brownian. Let $S_{t}$ be the price of an asset, $${{d{S_t}} \over {{...
3
votes
1answer
66 views

Show that Z(t)/Z(0) is a positive mean-1 martingale

We look at a standard no dividends Black-Scholes model and here we have a process Z, which is defined by: Z(t)=(S(t)/H)^p , where H is a positive constant and p=1-2r/sigma^2 I am now asked to show ...
3
votes
1answer
337 views

Normalized price process $Z(t)=\frac{\Pi(t)}{B(t)}$

If an interest rate model with the following $P$-dynamics for the short rate. $$dr(t)=\mu(t,r(t))dt+\sigma(t,r(t))d\bar{W}(t)$$ Now consider a $T$-claim of the form $\chi = \Phi(r(T))$ with ...
3
votes
1answer
400 views

Which measure to determine Risk?

Say I hold an equity and I want to calculate the Value-at-Risk over some period. Would one calculate the Value-at-Risk of the equity under a risk-neutral (as in martingale) measure or under the ...
3
votes
1answer
165 views

Assumptions based on non-martingale?

Quantitative finance formular are mostly based on martingales, Poisson jump, GBM, CEV, etc.. The logic behind it is that martingale means the future could not be predicted, or, EMH (Efficient-market ...
3
votes
0answers
47 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 ...
3
votes
0answers
102 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 ...