All Questions
Tagged with finance stochastic-analysis
52
questions
3
votes
1
answer
147
views
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
0
votes
0
answers
111
views
Is this the correct proof of Proposition 10.23 in Björk's ''Arbitrage theory in continuous time''?
I want to prove Proposition 10.23 from Tomas Björk's ''Arbitrage theory in continuous time'' in the snippet below.
My attempt: For simplicity, assume everything is one-dimensional, with one risky ...
- 135
3
votes
1
answer
107
views
Functional Analytical definition of no arbitrage
Let $ {(S_t)}_{t\in[0,+\infty[} $ be a semimartingale and ${(x_t)}_{t \in[0,+\infty[}$ an admissible strategy. We denote by $(x.S)_{+\infty}=\lim \int_{0}^{t} x_u dS_u$ if such limit exists, and by $...
- 3,961
1
vote
1
answer
41
views
Returns of an asset in risk-neutral measure and its PDE
I am a bit confused regarding how an asset returns in a risk-neutral measure (say $\mathbb{Q}$), and subsequently its Black-Scholes-esque PDE.
In class, we learned about the approach to take when ...
1
vote
1
answer
126
views
Is Heston model an affine jump-diffusion?
In Duffie, Pan and Singleton's paper "Transform Analysis and Asset Pricing for Affine Jump-diffusions" (2000) they define affine jump-diffusion (AJD) a process of the following form:
$$dX_t=\...
- 392
0
votes
1
answer
61
views
Application of Ito's lemma to consol price process
I have a question related to this paper https://www.jstor.org/stable/2245302.
Given a process for the short rate $r$, the authors consider the price process $Y$ for a consol bond that satisfies
\begin{...
- 1
3
votes
1
answer
122
views
On the plot of Black-Scholes-Merton formula
The price $C(t,S_t)$ of a European call option is given by the famous Black-Scholes formula
\begin{equation}
C(t,S_t)=S_{{t}}{\mathrm{N}}(d_{{1}})-Xe^{{-r(T-t)}}\mathrm{N}(d_{{2}})\tag{1}
\end{...
- 141
2
votes
1
answer
155
views
Prove that $V(t)=e^{-r(T-t)} \mathbb E\left[S_{t}\right]$ satisfies the Black–Scholes PDE
Let us consider the geometric Brownian motion:
$$
d S_{t}=\mu S_{t} d t +\sigma S_{t} d B_{t}
$$
where $\mu$ is the drift, $\sigma \in \mathbb{R}^{+}$ is the volatility and $B_{t}$ is the Wiener ...
- 7,880
0
votes
0
answers
33
views
Stochastic differential notation in Standard Brownian Market Models
I am trying to become familiar with stochastic integration and stochastic differential notation. I tried to do the following little exercise. In my lecture notes the risky Asset is defined in the ...
- 63
1
vote
1
answer
128
views
Solving stochastic control problems using Hitsuda representation
I would like to solve the following problem.
Consider a financial market with quadratic transaction costs, one risky asset with price dynamics:
$S_t = s_0 + \mu t + \sigma W_t$, for $t \geq 0 , \...
- 19
2
votes
1
answer
211
views
Black Scholes PDE in forward log space
In BS world, we have the stock process in log space $dS_t=(r-\frac{1}{2}\sigma^2)dt+\sigma dW$. Let's say we want to price $f(t,x)=\mathbb{E}_{t,x}[h(S(T)]$. Using Feynman-Kac, we get
\begin{equation}
...
- 43
3
votes
0
answers
217
views
Apply the Girsanov theorem, determine the stochastic dynamics of $S^{(1)}$ and determine the risk-neutral price of $X$
I'm not sure if I'm applying Change of Numeraire and Girsanov correctly in part c) and d). Also with the information I got, I don't know how to get a result for e).
Consider a financial market with 2 ...
- 147
6
votes
1
answer
1k
views
Why is the drift of the stock price not important for options pricing?
This question is motivated by MSE question 4199364: Bachelier model option pricing.
There, one considers the price of a stock depending on time $t$, given by the family of random variables $(S_t)_{t\...
- 13.8k
4
votes
1
answer
186
views
Will there be arbitrage if we don't have the $dB_t$ term?
Assume you have a filtered probability space $(\Omega,\mathcal{F},P,\mathcal{F}_t)$. Lets say you have the financial market with the bank process:
$$dR_0(t,\omega)=\rho(\omega,t)R_0(t,\omega)dt,$$
...
- 1,262
0
votes
1
answer
55
views
Øksendal: Expression for the bank process.
In Øksendals Stochastic differential equations on page 269 he defines the bank proess as a process satisfying:
$$dX_0(t)=\rho(t,\omega)X_0(t)dt; X_0(0)=1.$$
He later says that the bank process ...
- 1,262