Questions tagged [martingale]

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27 views

$aS^0 + bS^1$ is a $Q$-martingale does not imply discounted market is arbitrage-free

In the following framework : let $(S_0^{'} , S_1^{'} )$ be an undiscounted financial market in discrete time on $(\Omega, F, \mathbb{F}, P)$ with a finite time horizon $T \in \mathbb{N}$ and $\mathbb{...
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1answer
35 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 ...
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32 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} < ...
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36 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 ...
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30 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 ...
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29 views

Martingale-equivalent compound Poisson process

My question is related to the paper "a Martingale approach to premium calculation principle in an arbitrage-free market" by Delbaen and HAEZENDONCK (1989). In short, they characterized all ...
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40 views

Probablity distributions of zero crossings in 1D random-walk

Consider a simple 1D random walk that starts at position zero, and each second changes position by either +1 or -1 with 50-50 probabalities. I know it is proven to cross zero infinitely many times, ...
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92 views

Can grid strategy make profit on a random walk?

I've seen this thread, but it's a little too advanced for me. I haven't studied finance, just recently had some experience with grid strategy bots on cryptocurrency exchanges (in future markets), and ...
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89 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 (...
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1answer
195 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 ...
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796 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}...
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1answer
71 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 ...
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2answers
192 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 ...
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58 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,\...
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1answer
125 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 ...
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1answer
92 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 ...
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23 views

Q determined by the market in Binomial Model

I read in a book about change of measure, so that the discounted stock price in a binomial model is equal to the current price. Namely: $$E_{Q}[S_{1}/ \beta |S_{0}]= S_{0} $$ It then says: " Q is ...
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118 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, ...
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2answers
157 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} + \...
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1answer
126 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 ...
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63 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. ...
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1answer
99 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 ...
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1answer
80 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 ...
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2answers
262 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 ...
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86 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 ...
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586 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}, ...
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45 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 ...
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1answer
96 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}$...
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1answer
84 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 $...
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1answer
175 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 ...
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1answer
183 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 ...
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1answer
126 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 ...
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1answer
126 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....
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55 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 &...
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1answer
62 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 $...
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257 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: $$...
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1answer
189 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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1answer
186 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 ...
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1answer
114 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 ...
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1answer
86 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 ...
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109 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 ...
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79 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 ...
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1answer
332 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 ...
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1answer
151 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 + \...
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92 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 ...
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1answer
203 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 ...
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1answer
521 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 $S_t^2$ ...
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1answer
126 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 ...
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1answer
274 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? ...
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6k 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 ...