Questions tagged [stochastic-calculus]
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12
questions
16
votes
0answers
499 views
How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?
How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem?
In Duffie's book "Dynamic Asset Pricing," he outlines the "Martingale method" of solving ...
5
votes
2answers
119 views
Find probability that payoff function is in $[10,20]$
In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by:
$$f=(\max_{t\in[0,T]} S_t -110)^{+}$$
where $S_t$ satisfies
$$dS_t=15dW_t$$
$$S_0=95$$
...
5
votes
1answer
1k views
Apply Ito's Lemma to exponential martingale
$\newcommand{\dd}{\, \mathrm{d}}$
Consider the exponential martingale,
$$
\xi_t^\lambda = \exp \left\{ - \int_0^t \lambda_s \dd z_s - \frac 12 \int_0^T \lambda_s^2 \dd s \right\},
$$
that is used in ...
5
votes
2answers
757 views
Intuition of the Kolmogorov Equations
So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008:
"The backward equation involves time $t$ and ...
4
votes
1answer
105 views
One-step Binomial model's Radon-Nikodym derivative
In the one-step binomial model...
Question 1: What exactly is a '$\frac{d \mathbb Q}{d \mathbb P}$'?
I think it's $\frac{d \mathbb Q}{d \mathbb P} = \frac{q_u}{p_u}1_u + \frac{q_d}{p_d}1_d$, so it's ...
3
votes
2answers
774 views
Ito's Lemma derivation
I'm getting into asset pricing and was looking at Ito's Lemma, but cannot understand a few steps that are given.
Ito's Lemma states that given
$$dx_t = \mu dt + \sigma dz_t \\
y_t = f(t, x_t)$$
...
2
votes
1answer
68 views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
1
vote
2answers
88 views
Do real life economists and financial analysts actually use calculus in their jobs?
From what I understand a lot of calculus is used at university level when studying economics and some finance courses (correct me if I'm wrong) but I was just wondering if economists and financial ...
1
vote
3answers
205 views
Urn balls and probabilities
Think of the following balls as individuals of populations.
Say I have $U$ urns, and some balls. Both numbers are really large. So large, that authors like Blanchard and Diamond have approximated ...
1
vote
0answers
19 views
Which Is The Better Way To Calculate Marginal Revenue? Using Derivative Or R(Q)-R(Q-1)?
I Still A Bit Confused About Marginal Revenue.
Since Marginal Revenue Is The Additional Revenue We Get From Producing 1 Additional Good.
That Means Marginal Revenue Is R(Q) - R(Q-1). My Teacher Says ...
1
vote
0answers
139 views
How to solve a variation of Merton's optimal portfolio problem?
Does anyone know how to solve the following problem?
I have tried to solve this but I'm lost since I have never dealt with a Stochastic Dynamic Programming problem with many variables.
$max_{c_{t},\...
1
vote
0answers
87 views
How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?
Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
