Questions tagged [martingale]
The martingale tag has no usage guidance.
193
questions
0
votes
0answers
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{...
1
vote
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 ...
1
vote
0answers
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} < ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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, ...
0
votes
0answers
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 ...
3
votes
0answers
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 (...
2
votes
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 ...
10
votes
2answers
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}...
2
votes
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 ...
3
votes
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 ...
1
vote
0answers
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,\...
2
votes
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 ...
2
votes
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 ...
0
votes
0answers
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 ...
3
votes
2answers
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, ...
1
vote
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:
$$dS_{t} = \mu S_{t} + \...
3
votes
1answer
126 views
replicating self-financing portfolio for risk neutral measure
Let the price process $S_{t}, 0 \leq t \leq T$, be a diffusion, and savings account be $\beta_{t}$ such that the Equivalent Martingale Measure $Q$ exists. Let $C_{T}=g\left(X_{T}\right)$ be the claim ...
3
votes
0answers
63 views
Derivation of option pricing PIDE: Why does the drift need to be zero?
I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero.
...
1
vote
1answer
99 views
Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
1
vote
1answer
80 views
Martingale problem on biased random walk
I am struggling to understand the martingale property of exponential of a biased random walk. For example, in the following problem how do I verify whether the following is a martingale, submartingale ...
2
votes
2answers
262 views
Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
4
votes
0answers
86 views
If arbitrage can happen exactly at one moment, is it really arbitrage?
There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk:
If $S=\left(S_{t}\right)_{t=0}^{T}$ is a ...
7
votes
2answers
586 views
Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
3
votes
0answers
45 views
Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale
I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that:
The existence of an equivalent martingale measure $Q$ on the discounted price ...
1
vote
1answer
96 views
How to take the expectation of an exponential martingale? And an exponential with a random value?
I am reading Shreve's Stochastic Calculus for Finance II. He states on pages 110 and 111 that,
$$E[exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)] = 1$$
$$E[exp(-\frac{1}{2}\sigma^2 \tau_m)] = e^{-\sigma m}$...
1
vote
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 $...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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....
0
votes
0answers
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 &...
0
votes
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 $...
5
votes
2answers
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:
$$...
1
vote
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 ...
6
votes
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 ...
1
vote
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 + \...
0
votes
0answers
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 ...
2
votes
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 ...
1
vote
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 $S_t^2$ ...
1
vote
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 ...
5
votes
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?
...
18
votes
3answers
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 ...
