2
votes
1answer
22 views

the relationship between fractional difference and ACF of a time sequence

When reading the GARCH modeling part of book Analysis of Financial Time Series, I read the following statement. In specific, I do not understand how does the author ...
0
votes
1answer
50 views

Derive the Black– Scholes formula for the European call option.

Consider the standard Black–Scholes model. Derive the Black– Scholes formula for the European call option. thanks for help.
0
votes
1answer
28 views

Calculate expectation under risk neutral measure: $\mathbb{E_Q}(\max(S-1,0))$

I am busy with a numerical simulation and I want the calculate the following expectation under the risk neutral measure: $\mathbb{E_Q}(\max(S-1,0))$. $S$ is some variable that I calculated using ...
0
votes
0answers
11 views

Computing the probability that a stock process is more valuable than the bond process

I am currently revising for my exam and I cannot really deal with the following problem (I am a beginner in terms of stochastic processes): $W_t$ is the standard Brownian motion. Consider a stock ...
1
vote
0answers
37 views

Construct an arbitrage opportunity in a multi-period model

I am currently revising for my exam in Financial Mathematics, and I could not solve this question: For $T > 1$, consider a $T$-period model with a single risky asset and a bank account which pays ...
0
votes
0answers
37 views

Simple Stochastic Control Problem

Consider $dX_t = \pi_t X_t dt + \pi_t X_t dW_t, X_0 = x$, where $W_t$ is a standard brownian motion, and $\pi$ is some real valued process. Let T>0. How can we calculate $P[X_T\geq 2x]$, where ...
0
votes
0answers
22 views

Parameter estimation using characteristic function

Is it possible to do parameter fitting using log-returns data & the characteristic function(CF) in Matlab? I have been trying it on the Variance Gamma Scaled Self-Decompasable (VGSSD) model CF for ...
2
votes
2answers
85 views

Understanding basic stochastic differential equations

This is from a physics course in economics, the literature provides a bare minimum of mathematical explanations. I am trying to understand how to work with stochastic differential equations given in ...
1
vote
1answer
38 views

Writing $A(t)=1+S_1S_2^{-1}$ as an Ito diffusion process.

Let $W$ be a Wiener process/Brownian motian and let $$ \begin{align} \mathrm{d}S_1 &= 2S_1(t)dt +3S_1(t) dW\\ \mathrm{d}S_2 &= 4S_2(t)dt +5S_2(t) dW \end{align} $$ Now I'd like to write ...
3
votes
1answer
100 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
0
votes
0answers
82 views

Jump diffusion process with sum of Poisson processes a martingale?

Hi Mathematics community, assume you have dynamics of a jump diffusion process consisting of a Brownian motion and a sum of compensated (not necessarily independent) Poisson processes, i.e. ...
0
votes
0answers
24 views

Weak convergence of discretization scheme with correction

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
2
votes
1answer
58 views

Optimal Investment Strategy

I am not sure to solve the following investment problem: I have an investor which receives an income $I_n\ge 0$ at the start of year $n$. The investor chooses a proportion $p_n\in[0,1]$ of this in ...
1
vote
1answer
82 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
0
votes
1answer
48 views

Compute $d(\log(S_t))$ using Ito's Formula

We are given the following: d$S_t$ = $\sin(S_t)t^2dt + e^{\sqrt{S_t}-t}dB_t$ And are asked to compute several different things, one of which is $d \log(S_t).$ If I'm understanding Ito's formula ...
1
vote
1answer
33 views

How does perfect correlation between Brownian motions imply equivalence?

Assume that both $ B_t $ and $ Z_t $ are standard Brownian motions and that $ Corr(B,Z)=1 $. How do those properties imply that $ B_t = Z_t $ a.s. ? $Cov(B,Z) = t $ is as far as I can get, and I ...
1
vote
0answers
23 views

The impact of jump on the returns of portfolio and asset pricing

There exsits jumps in financial market. What will be the impact of jump on the returns of portfolio and asset pricing? Please explain it both academically and plainly. If you can give some excellent ...
2
votes
0answers
37 views

Stochastic control, numerical, need expectations given coupled SDEs

I'm looking at a trio of processes which arises in a stochastic control situation. I have a process $(V_t)$ which I may control, and $(V_t)$ influences a diffusive stock price process $(S_t)$. The ...
0
votes
0answers
38 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
2
votes
1answer
144 views

Is the following a martingale?

Let $X_{n}$ be a martingale with respect to a filtration $\mathbb{P}_{n}$. Define: $Y_{n}$ := $X_{n}^{3}$ Is $Y_{n}$ a martingale? Supermartingale?
2
votes
1answer
130 views

Black scholes model type

I want to study the following market: $$S_1(t)=S_1(t)(\mu_1dt + \sigma_1dW_1(t))$$ $$S_2(t)=S_2(t)(\mu_2dt+\sigma_2dW_2(t))$$ for $t\in [0,T]$, constants $\mu_i,\sigma_i$, initial values ...
1
vote
0answers
93 views

Compactness of the set of densities of equivalent martingale measures

Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal ...
3
votes
1answer
235 views

Futures pricing and futures price process under the real world measure

This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. ...
4
votes
1answer
383 views

Applications of Compound Poisson Processes

I'm reading the book Non-Life Insurance Mathematics, an introduction with Stochastic Processes by Thomas Mikosch and I'm interested in applications of the Cramer-Lundberg Process to concrete examples ...
3
votes
1answer
617 views

What is the definition of a “predictable process”?

I am reading a book on financial mathematics, and frequently encounter the phrase "predictable process", which I haven't seen definition of, and cannot find the definition online. At first I thought ...
2
votes
2answers
457 views

Using Black-Scholes Equation to “buy” stocks

From what I understand, Black-Scholes equation in finance is used to price options which are a contract between a potential buyer and a seller. Can I use this mathematical framework to "buy" a stock? ...
5
votes
5answers
3k views

Understanding Black-Scholes

Assume I have only basic math knowledge, what specific areas of math would I need to learn in order to understand the following webpage: Black-Scholes Many thanks.