Questions tagged [martingale]

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Question about the integrand space of stochastic integral wrt martinagles

I am reading the book "Introduction to Stochastic Integration" by Hui-Hsiung Kuo. In Chapter 5, he introduces the definition of stochastic integral w.r.t martingale: $$I(f) = \int_a^b f(t) ...
Mingzhou Liu's user avatar
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1 answer
51 views

Martingale property of the CEV model

I am a bit confused about the martingale property of the CEV model. Given dS(t)=σS(t)^βdW(t), is S a martingale for values of β<1?
Atone202's user avatar
3 votes
1 answer
156 views

How to prove that a market is incomplete using the concept of EMMs?

Question Consider a one-step trinomial tree, where there are two traded assets, a bond with risk-free rate, $r$, a stock with initial price, $S_0$, and terminal price $$S_T = \begin{cases} S_0u,& ...
Hmmmmm's user avatar
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1 vote
2 answers
139 views

Is this arbitrage? Infinite payoff / infinite loss (energy generation investment problem)

I'm a student using stochastic optimization in energy systems and I have a particular phenomena in an optimization problem that I think must occur in finance aswell, so I have been trying to find ...
waxcomb's user avatar
  • 11
2 votes
1 answer
97 views

Why A Derivative With Intrinsic Arbitrage Cannot Be Valued & Hedged With Assets In Risk Neutral?

I'm attempting to concisely show why a derivative that, by nature, introduces arbitrage cannot be valued using risk neutral pricing tools. Derivative: Buyer is sold a 'call option', with time 0 value ...
TheOneTwoThreeForPumpkin's user avatar
0 votes
0 answers
59 views

Risk-Neutral Non-Linear Option Pricing Black Scholes Model

Looking for some help on this question. Suppose the Black-Scholes framework holds. The payoff function of a T-year European option written on the stock is $(\ln(S^3) - K)^+$ where $K > 0$ is a ...
Kai's user avatar
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61 views

modelling time series using semi-martingale process

During this week lecture my professor said that the semimartingale( brownian motion contamined by noise) is a model in reduced form because we do not specify the dynamic which leads to price ...
XY0's user avatar
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2 votes
2 answers
105 views

Conditional expectation of increments of stochastic process [closed]

I have come across the following result in my book on stochastic finance and I have trouble understanding the proof. On a filtered probability space with filtration $(\mathcal{F}_t)_{t \in \mathbb{R}^+...
Michaël's user avatar
0 votes
1 answer
110 views

Black Scholes/American Put/Martingale Condition

Consider a Black Scholes model with $r \geq 0$. Show that the price of an American Put Option with maturity $T > 0$ is bounded by $\frac{K}{1 + \alpha} {(\frac{\alpha K}{1 + \alpha})}^{\alpha}{S_{0}...
Parinn's user avatar
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0 answers
85 views

When Forward is a martingale under risk-neutral measure?

Why is such a proof for futures not suitable for a forward? For futures we have: $V_{t}$ is self-financing portfolio: $V_t = \frac{V_t}{B_t}B_t$, where $B_t$ is a riskless asset Suppose $H_t$ - number ...
Strike's user avatar
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1 answer
191 views

Geometric Brownian motion and semi-martingality

I am trying to learn by myself stochastic calculus, and I have a question regarding semi-martingale stochastic process and geometric Brownian motion (GBM). We know that a stochastic process $S_t$ is ...
XY0's user avatar
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1 answer
98 views

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 $S_n = S_{0}\prod_{j=1}^{n}Z_j$, $S_0>0$ ...
joshdalton's user avatar
1 vote
0 answers
117 views

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 - ...
yrual's user avatar
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1 answer
218 views

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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1 vote
1 answer
61 views

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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0 answers
29 views

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
  • 141
3 votes
0 answers
202 views

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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1 answer
216 views

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
0 votes
1 answer
247 views

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
378 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
0 answers
116 views

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
1 vote
0 answers
69 views

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
2 votes
0 answers
381 views

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
508 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
1 vote
1 answer
304 views

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
2 votes
1 answer
227 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
0 votes
1 answer
136 views

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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0 answers
87 views

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
171 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
66 views

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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1 vote
0 answers
46 views

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
222 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
  • 419
0 votes
0 answers
148 views

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
104 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
  • 171
2 votes
1 answer
383 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
  • 165
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
307 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
3 votes
2 answers
724 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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1 vote
0 answers
66 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,\...
Martin_Gale's user avatar
2 votes
1 answer
248 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
288 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
  • 548
4 votes
2 answers
252 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
1k 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
  • 21
3 votes
1 answer
290 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
82 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
131 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
249 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
696 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
96 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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