Questions tagged [martingale]

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Martingale under risk neutral probability

I have a question to prove martingale under risk-neutral measure: Question Consider a discrete time process $S$, which at time $n \in \mathbb{N}$ has value $S_n = S_{0}\prod_{j=1}^{n}Z_j$, $S_0>0$ ...
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multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
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Discounted price of an option

If the discounted price of any asset is a martingale under risk neutral measure, why is $E^Q[e^{-rT} (S_T-K)_+ | F_t]$, not merely $e^{-rt} (S_t-K)_+$? This is something I wanted to clarify, since ...
LAC's user avatar
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Minimal entropy martingale measure and Bayes estimated under Kullback-Laibller divergence loss function

We know that no unique equivalent measure exists in an incomplete market. Therefore, we need to choose a pricing measure equivalent to the physical measure based on a criterion. One typical approach ...
sss's user avatar
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integral of adapted process with respect to semimartingale is a martingale

Fix $T > 0$ a finite time horizon. Let $H$ be an adapted (or progressively measurable, if needed) continuous process and S be a continuous semi martingale, both on $[0,T]$. Under what conditions is ...
yrual's user avatar
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3 votes
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Is this an optimal stopping problem?

I am trying to work out how to approach a machine learning problem of 'learning' an optimal liquidation time/threshold, under some conditions, from historic data. The idea is a trader armed with this ...
Zac's user avatar
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Standard Brownian Motion and Exponential Martingale calculation [closed]

Let $W(t)$ be a standard brownian motion and let $Z(t) = \exp (\lambda W(t) - \frac{1}{2}\lambda^2 t).$ In Xinfeng Zhou's Green Book in the section on Brownian Motion (p.130) he writes as part of the ...
phhhlpfk's user avatar
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Expectation of Bt^4 given BS [closed]

What is the expectation of Bt^4 and Bt^3 given Bs? Given t>s. I understand that the expectation of Bt given Bs is Bs and that the expectation of Bt^2 given Bs is something like Bs - s + t.
Lawrence Chun's user avatar
1 vote
1 answer
236 views

4th Order Brownian Motion Martingale [closed]

I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
Lawrence Chun's user avatar
2 votes
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First known reference using martingale theory to derive BS formula

What is the first known paper which derives the Black-Scholes valuation formula for an option (1973) using martingale machinery - instead of PDEs?
Daneel Olivaw's user avatar
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What happens trying to price derivatives starting from a non-geometric brownian motion?

To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question. The set-...
Emanuele's user avatar
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Understanding the asset pricing theory and numeraire

While reading about asset pricing theory and numeraire, I had faced some confusion. Short summary of asset pricing theory from my book We start our journey with a risky asset $S_t=\mu S_tdt+\sigma ...
emonhossain's user avatar
2 votes
1 answer
432 views

How is an exchange rate process a martingale under any measure?

Suppose a process for a stock price of a US-based company traded in the USA is, under the USD money-market numeraire: $$dS_t=S_tr_{USD}dt+S_t\sigma_SdW_1(t)$$ Using fundamental theorem of asset ...
Conductor's user avatar
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1 answer
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Does every process need to be a martingale under martingale measure?

From the pricing theory, processes need to be martingales when divided by the numeraire asset. A classical example is a stock option: Consider a money market $B$ being the numeraire asset. When we ...
user2743931's user avatar
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1 answer
159 views

Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach

I am looking to compute the following using Ito's formula. $$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$ Knowing the properties of brownian motion, it is rather easy to show that the ...
ilikemath3.14's user avatar
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1 answer
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European option with payoff $X_T^2$ [closed]

I have been ask to price a European option with payoff $H(X_T,T) = X_T^2$ using the equivalent martingale measure (EMM). For this I used the process: \begin{equation} dX_t = r X_t dt + \sigma X_t d\...
Alejandro Andrade's user avatar
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Good performance of naive forecasting in efficient markets

I am doing spot price forecasting for a market, and so far, the naive forecasting model, which forecasts with the last observed prices, is the best forecasting model. I know that it might be because ...
BSel's user avatar
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3 votes
1 answer
165 views

Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...
user9078057's user avatar
1 vote
1 answer
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Floating swap payoff with rate determined on current instead of previous date

I am attempting to determine the payoffs a modified swap, in which the floating payments at a time $T_k$ are made on the current date (i.e. $L(T_k,T_{k+1})\equiv L_{k+1}(T_k)$) rather than at the ...
Ice Tea's user avatar
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How to find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$

How do I find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$, where $P(T_{1})$ and $P(T_{2})$ refer to the $T_{1}$ and $T_{2}$ zero coupon bond with $0 < T_{1} < ...
user9078057's user avatar
1 vote
1 answer
156 views

Why the Esscher transform is the right transform for pricing formula?

A Wiener process has infinitely many states of the world at any time step. Does that not mean that there are infinitely many EMM's for any model that uses the Wiener process? But then if there is only ...
user53249's user avatar
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Why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?

why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire? For example, if I have the price of a forward price $f_t$ and a ...
junyaozheng98's user avatar
3 votes
0 answers
102 views

Is this term structure model valid? (Modeling the Zerobonds directly)

Let us define the dynamics of the discounted Zerobonds as $$ \tilde{P}(t,T) = \int \sigma(t,T) dW_t + \tilde{P}(0,T)$$ Lets assume $\sigma(t,T)$ is s.t. $\tilde{P}(t,T) $ is a martingale and positive (...
algebruh's user avatar
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2 votes
1 answer
331 views

Proving the discounted stock price is martingale

Let $\mathcal{K}_s$ be $$ \mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$ where $\tilde{V}_t(\theta)$ is the discounted value process of the self financing ...
abc's user avatar
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10 votes
2 answers
1k views

Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete. (a) $\beta_{t}=e^{t}, S_{t}...
randorando's user avatar
2 votes
1 answer
189 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 ...
user394334's user avatar
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2 answers
497 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 ...
Mathxx's user avatar
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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,\...
Martin_Gale's user avatar
2 votes
1 answer
232 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 ...
Martin_Gale's user avatar
2 votes
1 answer
195 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 ...
Landscape's user avatar
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4 votes
2 answers
215 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, ...
user9875321__'s user avatar
1 vote
2 answers
708 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} + \...
Sam Loi's user avatar
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3 votes
1 answer
248 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 ...
Stochastichelp's user avatar
3 votes
0 answers
78 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. ...
Leguan3000's user avatar
1 vote
1 answer
111 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 ...
qszbwldxz's user avatar
1 vote
1 answer
217 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 ...
noisyoscillator's user avatar
2 votes
2 answers
489 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 ...
Martin_Gale's user avatar
4 votes
0 answers
95 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 ...
W. Volante's user avatar
10 votes
2 answers
3k 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}, ...
quantmath's user avatar
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3 votes
0 answers
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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 ...
MinaThuma's user avatar
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1 answer
203 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}$...
cona's user avatar
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1 vote
1 answer
171 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 $...
user92234's user avatar
2 votes
1 answer
579 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 ...
Tyler D's user avatar
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2 votes
1 answer
405 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 ...
novo's user avatar
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1 answer
360 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 ...
J Dash's user avatar
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1 vote
1 answer
228 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....
stats19's user avatar
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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 &...
MinaThuma's user avatar
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0 votes
1 answer
72 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 $...
user avatar
5 votes
2 answers
328 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: $$...
Jan Stuller's user avatar
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1 vote
1 answer
242 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 ...
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