Questions tagged [equivalent-measure]
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15
questions
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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 ...
6
votes
0answers
158 views
Let $\mathbb{P} \sim \mathbb{Q} \sim \mathbb{R}$ be equivalent probability measures on some measurable space
Let $\mathbb{P} \sim \mathbb{Q} \sim \mathbb{R}$ be equivalent probability measures on some measurable space $(\Omega, \mathcal{F})$, and let $\mathcal{G} \subset \mathcal{F}$ be a sub- $\sigma$-...
2
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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 ...
3
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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, ...
5
votes
1answer
267 views
Is first order stochastic dominance conserved under change of measure?
As the title states, my question is whether first order stochastic dominance is conserved under change of measure, for instance from the $\mathbb{P}$ measure to $\mathbb{Q}$ measure and change of ...
3
votes
2answers
172 views
Change of measure for a stochastic process to be a martingale
$\text { Give a measure change so that } X_{t}=e^{B_{t}}\left(B_{t}-t / 2\right) \text { is a martingale, } 0 \leq t \leq T$
My attempt
Using Ito's lemma on $X_{t}$ we get:
$-\frac{e^{B t}}{2} d t+\...
3
votes
1answer
94 views
EMM for Bachelier model
The stock price is assumed to evolve as $S_{t}=S_{0}+\mu t+\sigma B_{t}$, where $S_{0}>0, \mu>0$ and the process $B_{t}$ is Brownian motion.
The saving account is assumed to be $\beta_{t}=e^{r t}...
1
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1answer
66 views
How to prove that the following is still a Brownian motion [closed]
Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
3
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0answers
64 views
Is $E_t^{Q}(g(Y))=E_t^{Q^Z}(g(Y))$?
Consider $$Z(t)=\left(\frac{S(t)}{H}\right)^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$
and a simple claim with a pay-off ...
1
vote
1answer
123 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....
6
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67 views
Can the risk-neutral measure depend on the option type?
In an ideal Black-Scholes setting, the Risk-Neutral measure $Q$ is unique and so, obviously, does not depend on what derivative instrument we want to price.
Assume some deviation from perfect markets (...
1
vote
1answer
68 views
We have a two LIBOR contracts, how to compare their values by change of change of numeraire
We have two LIBOR contracts:
contract 1 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{1}$
contract 2 pays $L\left(T_{1},\:T_{2}\right)-K$ at time $T_{2}$.
Now, $F_{1}$ is the par strike such that ...
1
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0answers
51 views
Replicating portfolio of an option and to find inital price
I am very new to financial math so I am not sure how to do with this question. A friend sent me this question to practice but I am unsure how to begin. I read about call option . Can that be used for ...
2
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0answers
105 views
Last step step in Girsanov's theorem proof
I consider the version of Girsanov's theorem presented in this question.
Let us take the particular case that $\mathbb{F}$ is the filtration generated by standard Brownian motion $(W)_{t\in[0;T]}$ ...
2
votes
1answer
106 views
Market Price of Risk for Consumption Asset - Hull's Example 28.1
In Hull's Options, Futures, and Other Derivatives, he gives an example 28.1 as below.
Consider a derivative whose price is positively related to the price
of oil and depends on no other ...
