15
votes
Accepted
Solution of Merton's Jump-Diffusion SDE
Let
$$ dS_t = \mu S_t dt + \sigma S_t dW_t + S_{t^-} dJ_t $$
where
$$ J_t = \sum_{j=1}^{N_t} (V_j - 1) $$
is a compound Poisson process, with $V_j$ i.i.d. jump sizes (positive random variables) whose ...
- 14.5k
13
votes
Accepted
Clarification on Deriving Ito's Lemma
Just a few notes
How to make sense of $\text dW_t$ is the entire point of stochastic calculus. It's far beyond the scope of any answer here. You should read some introductory lecture notes/books on ...
- 15.2k
10
votes
Accepted
List: Behavioural characteristics of key Ito processes used in finance
I will provide some references such that you can see where the different processes are used. These papers typically motivate their models and show which effect the single paramaters have and what ...
10
votes
Accepted
Deriving the solution for European call option in the Heston Model
Itô's Lemma
The standard version of Itô's Lemma applies to a single Itô process $\text{d}X_t=\mu(t,X_t)\mathrm{d}t+\sigma(t,X_t)\mathrm dW_t$. Then,
$$\mathrm{d}f(t,X_t) = \left(f_t+\mu(t,X_t)f_x + \...
- 15.2k
9
votes
More questions about integral of Brownian Motion w.r.t time
It is indeed Riemann integrable, so you don't need stochastic integration. For a given path, you can interpret the integral in the Riemann sense. For a given t, the paths are random, so it is a random ...
- 6,204
9
votes
Accepted
More questions about integral of Brownian Motion w.r.t time
As usual with those kind of integrals, another way to reach the result is to:
Express $W_s$ in integral form as $\int_0^s dW_u$
Use Fubini theorem to change the integration bounds of the resulting ...
- 14.5k
9
votes
Accepted
Gamma PnL from Itô's Lemma derivation
$$ \frac{1}{2} \frac{\partial^2 f}{\partial S^2} dS^2 \approx \frac{1}{2} \sigma^2 S^2\frac{\partial^2 f}{\partial S^2} dt$$
(for small $dt$, ignoring $(dt)^2$ terms )
$\sigma$ is embedded in $dS = \...
- 5,008
8
votes
Accepted
2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?
$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$.
Instead one should turn to Itô's lemma, ...
- 14.5k
8
votes
How can I learn stochastic process & stochastic calculus in two weeks?
This is impossible unless you are very intelligent with good memory-retention skills and already mathemathically proficient in the field of analysis and statistics (and no, a single course in basic ...
- 101
7
votes
Accepted
Baxter & Rennie HJM: differentiating Ito integral
Let
$$Z_t = \exp(-X_t)$$
with
$$X_t = \sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)du ds $$
and $W_t$ a standard Brownian motion, along with the usual assumptions.
...
- 14.5k
7
votes
Accepted
Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?
We don't model the prices, we model the returns.
The stock prices aren't explicitly modelled as log-normal, but rather this is a consequence of the actual model used to describe the returns. The core ...
- 1,389
7
votes
Can I write Ito's Lemma as a taylor expension?
Ito Lemma (as 'Taylor expansion'): For $X$ an Ito process and $f =
f(t, x) ∈ C^{1,2}(\mathbb{R}^2)$ a deterministic function, the stochastic process
$$Y_t = f(t,X_t)$$
is an Ito process and we have
$$...
- 5,008
7
votes
Accepted
conditional expectation of stochastic integral
What a great question! I've had a go at it below, I'd say I'm about 75% sure of the result I've got to but I'd love feedback from others.
I'm going to use the definition of the Ito integral,
\begin{...
- 2,996
7
votes
Ito calculus is Gaussian (using method of characteristic function)
Hints:
First show (using Ito Lemma) that
$$ \exp(iuX_t) = 1 + iu \int_0^t\exp(iuX_s) h(s) d W_s -2^{-1}u^2 \int_0^t\exp(iuX_s) h(s)^2ds$$
Then show (by taking expectations):
$$ E[\exp(iuX_t)] = 1 - 2^...
- 5,008
7
votes
Accepted
Ito multiplication
If $M$ and $N$ are independent (your references appear to make this assumption), then $M+N$ is also a Poisson process. So, using the polarization identity:
$$ dMdN = 2^{-1}\left[(d(M+N))^2 - (dM)^2 - (...
- 5,008
6
votes
How is the Wiener integral $\int{WdW}$ calculated?
You are "deriving" with respect to $t$ (the time index in your stochastic process).
$f(t,x) = x^2$ so $f(t,W_t) = W_t^2$. And Ito's lemma tells you
$W_b^2 - W_a^2 = \int_{t=a}^b d(W_t^2) = \int_{t=...
- 3,936
6
votes
Accepted
Stochastic differential equation of a Brownian Motion
In stochastic calculus, only stochastic integrals are defined. The differential form is just a notation. That is, $$dF=g(t)dW_t$$ is just another expression for the integral $$F=\int_0^t g(s) dW_s.$$ ...
- 21k
6
votes
Accepted
Quantile normal and lognormal
Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance).
Put ...
- 14.5k
6
votes
Accepted
exercise on multivariate Ito's lemma + jumps (Poisson)
Answer
Assuming the Poisson process $N_t$ is independent from the Brownian motions $(W_{1,t},W_{2,t})$, you'll have
\begin{align}
df(X_{1,t},X_{2,t}) &= \frac{\partial f}{\partial X_{1,t}} dX_{1,...
- 14.5k
6
votes
Basic question on Ito integrals
To verify @AntoineConze's suggestion, the variance should be:
$$\int_0^4 (2_{[0,1]}(t)+3_{(1,3]}(t)-5_{(3,4]}(t))^2\,dt.$$
Since the supporting domains are disjoint, the product of any two of the ...
6
votes
Accepted
Application of Ito's lemma
Consider OP's general formula $f(g(t),X_t)$. In case of ambiguity, let us claim that
$f=f(t,x)$ is defined with variables $t$ and $x$,
$g=g(s)$ is defined with the variable $s$, and
$h=h(u,x)=f(g(u),...
- 396
6
votes
Accepted
Ito`s Lemma problem
write down Ito's lemma for the function X:
$$dX=\frac{\partial X}{\partial Y}dY+\frac{1}{2}\frac{\partial^2 X}{\partial Y^2}(dY)^2+\frac{\partial X}{\partial c}dc+\frac{1}{2}\frac{\partial^2 X}{\...
- 1,651
6
votes
Variance of a time integral with respect to a Brownian Motion function
Using Fubini's argument, assuming that $f$ is deterministic
$$E(I_t^2) = E\left(\int_0^t f(s) W_s ds\int_0^t f(u) W_u du\right)=\int_0^t\int_0^t{f(s)f(u)min(s,u)duds}$$
If $f$ is continuous(even ...
- 753
6
votes
Variance of a time integral with respect to a Brownian Motion function
As @Canardini pointed out,
\begin{align*}
E\big(I_t^2\big) &= E\left(\int_0^t f(s) W_s ds\int_0^t f(u) W_u du\right)\\
&= \int_0^t\!\int_0^t f(s)f(u)\min(s,u)dsdu\\
&= \int_0^t\left(\int_0^...
- 21k
6
votes
Accepted
Pricing call option using risk-neutral martingale approach with squared stock price boundary?
You do not really need the dynamics of $S_t^2$. You can simply apply your standard technique from risk-neutral pricing. The time zero price of a European-style contract with payoff $X$ is given by $$...
- 15.2k
6
votes
Application of Ito's Lemma in expected utility theory
The risky and riskless assets follow processes,
$$\frac{dS_t}{S_t}= \mu \, dt + \sigma \, dB_t, \,\,\, \frac{dM_t}{M_t}= r \, dt$$
If the proportion invested in the risky asset at time $t$ is $p_t$, ...
- 3,595
6
votes
Itos Lemma Derivation notation
The theory behind the actual reasoning is a bit complicated than the coverage in Hull's, but staying within the simple reasoning, the difference comes down to the following:
The Brownian increments ...
- 6,204
6
votes
Derivative of Stochastic Integral
No. Itō’s formula helps you derive the dynamics of $f (S_\cdot )$ given the SDE followed by $S$. Here this is not the case. You simply have:
$$
\mathrm{d} \left[\int{g(S_t)\mathrm{d}S_t}\right] = g(...
- 2,570
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