All Questions
Tagged with finance martingales
62
questions
0
votes
1
answer
29
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Analyzing Expected Profit in a Symmetric Random Walk with Trading Actions
Problem Formalization:
I am examining a problem where a stock price $X_t$ follows a symmetric random walk starting at 10, and increments or decrements by 1 unit at each step with equal likelihood. The ...
4
votes
0
answers
43
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Question regarding a value in a one-period model
There is a script at my university (can't post it for copyright reasons) for a course on discrete time financial mathematics. I decided to give it a try and found this problem:
Consider a financial ...
- 51
4
votes
1
answer
109
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Showing a basic market admits no arbitrage
Setting
We work in $\left(\Omega, \mathcal{F},\left(\mathcal{F}_t\right)_{t=0}^1, \mathbb{P}\right)$. Let $d=1, T=1$ and assume the discounted price equals the non-discounted price.
Take $S_0^1 \in \...
- 131
1
vote
1
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99
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Seeking clarity on the concept of equivalent martingale measures
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
$$S_T =
\begin{cases}
S_0u,& ...
- 333
3
votes
1
answer
147
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Why are there different definitions of admissibility in the literature, and why do we need admissibility?
Wikipedia essentially defines an admissible trading strategy as a stochastic process $H = (H_t)_{t\geq 0}$ such that the associated value process $\int H(u) d S(u)$ is lower bounded. As I understand ...
- 135
1
vote
1
answer
146
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Arbitrage-free market
Consider a single period market consisted of the zero-risk asset $S^1$ and two risky assets $S^2,S^3$ with states
$$S^1_0 = S^2_0=S^3_0=1, P(S^1_1 = 1 + r, S^2_1 = S^3_1 = u) = \frac{1}{3},
P(S^1_1 = ...
- 317
1
vote
1
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402
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Showing that a process is a supermartingale using Ito's formula
Consider a stock with price dynamics $$dS_t=S_t\sigma_tdW_t$$ where $(W_t)_{t\geq0}$ is a Brownian motion and $(\sigma_t)_{t\geq0}$ a bounded and continuous process adapted to the filtration $(\...
- 1,023
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0
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125
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A discrete local martingale which is integrable for some random $T>0$ is a true martingale up to time $T$.
Let $X$ be a discrete local martingale such that $X_T$ is integrable for some non-random
time $T > 0$. I am tasked with showing that $(X_t)_{0≤t≤T}$ is a true martingale.
The hint is to show that $...
- 1,023
2
votes
1
answer
206
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The payoff in binomial model is a martingale.
Let $V_N$ the payoff of a security at time $N$, recurssvely define
\begin{equation}
V_n=\frac{1}{r+1}(\tilde{p}V_{n+1}(H)+\tilde{q}V_{n+1}(T))
\end{equation}
where $\tilde{q},\tilde{p}$ are the risk ...
- 366
1
vote
0
answers
353
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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 ...
- 409
5
votes
1
answer
570
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General version of the fundamental theorem of asset pricing in continuous time
I know the fundamental theorem of asset pricing in discrete time which says that there are no arbitrage opportunities if and only if there exists a equivalent martingale measure. As far as I ...
- 409
3
votes
1
answer
282
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Quadratic variation of a stochastic integral w.r.t. a local martingale
I am trying to prove the (seemingly simple) property: for a continuous local martingale $M$ and an $M$-integrable process $H$, the quadratic variation $\langle\int H\,dM\rangle$ of $\int H\,dM$ is ...
- 249
6
votes
3
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622
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Bachelier model option pricing [closed]
Consider a Brownian motion $W_t$ and Bachelier model $S_t = 1 + \mu t + \sigma W_t$ for the stock price $S_t$. Find the value of an option that pays $1(S_1 > 1)$.
Attempt: As I understand it, the ...
- 61
2
votes
1
answer
124
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Derive Delta (Greeks) using change of probability measure (Mathematical Finance)
This question is not for homework, it is just for personal curiosity.
I am aware that we can calculate the greeks using basic calculus, and simplification. I am also aware that there is a great video ...
- 123
1
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1
answer
65
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How to price a contract that is denominated in another currency using the Martingale Approach to the Black and Scholes theory?
I am taking a course in asset pricing and I have the following problem at hand:
Suppose that the level of the UK FTSE100 index (in British pounds) evolves according to
$$\frac{\mathrm{d}S_t}{S_t}=\...
