8
votes
What the expectation of S^2 is from GBM?
As Sanjay said, you can apply Itô's Lemma to $f(t,x)=x^2$ and obtain
\begin{align*}
\mathrm{d} S^2_t=\left(2\mu S_t^2+\sigma^2S_t^2\right)\mathrm{d}t+\left(2\sigma S_t^2\right)\mathrm{d}W_t.
\end{...
- 14.8k
7
votes
Proof that no trading system always wins
At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with ...
- 1,031
7
votes
Accepted
Understanding the solution of this integral
Let $\tau = T-t$. Then
\begin{align*}
S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z},
\end{align*}
where $Z$ is a standard normal random variable, independent of $\mathcal{F}...
- 20.6k
6
votes
Accepted
Clarify a derivation in Pat Hagan's Convexity Conundrums
Swap
Just to be clear, (3.4c) leads to (3.5a) when we assume lognormal $R(\tau)$. Lognormal $R(\tau)$ means we can write
$$R(\tau) = R_0 e^{-\frac{1}{2}\sigma^2 \tau + \sigma \sqrt{\tau} Z}$$
with $...
- 1,859
6
votes
Book recommendation: math toolkit for quantitative finance and statistics
All the topics you've mentioned are wonderful and shouldn't be eschewed by reading some finance-oriented review book. I recommend these instead.
Linear algebra: Hoffman and Kunze and Halmos
Set ...
- 5,113
6
votes
How can I go about applying machine learning algorithms to stock markets?
Sorry, but despite being used as a popular example in machine learning, no one has ever achieved a stock market prediction.
It does not work for several reasons (check random walk by Fama and quite a ...
- 61
6
votes
Accepted
Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 4
Under the risk-neutral measure the discounted (under some numéraire) price process is a martingale. If we have a bank account with dynamics $dB_t = r B_t dt$ then the discounted asset $X_t = \frac{S_t}...
- 1,057
5
votes
Is a linear combination of GARCH processes also a GARCH process?
No, a sum of two GARCH processes is generally not a GARCH process.
(I am not even sure whether there exists a nontrivial special case where the opposite holds.)
By GARCH I mean the classic ...
- 1,955
5
votes
Accepted
derivation of heston pde in gatheral
1) Gatheral expresses everything in forward terms: forward value of the spot and of the call.
Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want ...
- 3,876
5
votes
Accepted
Put-Call Parity Application
First, we have $P(t)+S(t)=C(t)+B(t,T)\cdot K$,
Then, $\frac{\partial P(t)}{\partial S(t)} + \frac{\partial S(t)}{\partial S(t)} = \Delta^{\text{put}}_{t}+1$ and $\frac{\partial C(t)}{\partial S(t)} +...
- 106
5
votes
Accepted
Preparation for interview: influx of power of the moon
Here's how i'd have at it;
* I happen to know these are okay guesses.
** Let's assume it's just the potential energy, and that ...
- 2,446
5
votes
Accepted
Formula for conditional expectation. Related to the Fundamental Theorems of Asset Pricing
Let define $\mathbb{Q}$ and $\mathbb{P}$ two equivalent probabilities on a filtered space $(\Omega,(\mathcal{F}_t)_{t\geq 0})$
Let define $Z_T=\frac{d\mathbb{Q}}{d\mathbb{P}}$ restricted to $\mathcal{...
- 2,382
5
votes
Accepted
Do we have a Brownian motion
Aside from the independence requirement for the increments, that is, the independence of $X_{s+t}-X_s$ and $\mathcal{F}_s$, you can check whether the increment $X_{s+t}-X_s$ has the distribution of $N(...
- 20.6k
5
votes
Accepted
Swap contract comparative advantage
It is actually rather simple.
Lets start with the fixed rate market. A can borrow at 5% while B can borrow at 7%. Simply said, A has a comparative advantage of 2% in the fixed rate market.
In the ...
- 173
5
votes
Accepted
Finding the process of $X/Y$
You are right about the dropped $\sim$, it's probably just a typo. Furthermore, remember that in stochastic calculus, you have to take into account second order derivatives, i.e.
$$d\left(\frac{1}{...
- 1,497
5
votes
Do quants need to know bloomberg terminal and VBA?
(these are just my random opinions. Sorry, I expect many people to disagree with some of them.)
Bloomberg makes no effort to make its terminals available to students - quite the opposite. Hence lots ...
- 10.4k
5
votes
Accepted
How can I express this sum in a easier way?
As you mentioned, we have
$$\sum_{l=0}^{k}{p}=\frac{k(k+1)}{2}$$
You want to know
$$\sum_{k=0}^{n}{\sum_{l=0}^{k}{p}}=\sum_{k=0}^{n}{\frac{k(k+1)}{2}}=\frac{1}{2}\sum_{k=0}^{n}{k^2}+\frac{1}{2}\...
- 743
5
votes
Abstract algebra in economics and finance
Yes, I've seen some interesting papers that improve one's insight into how things work, even if it is not clearly applicable to practice.
Belal Ehsan Baaquie published several books on applications ...
- 10.4k
5
votes
Accepted
How do you derive this Carr-Madan-like equation?
Equation (11) in Kammeyer and Kienitz' paper is a very well-known and popular option pricing formula. It goes back to the work from Lewis (2001), see Theorem 3.2 in Lewis' paper.
Original Formula ...
- 14.8k
5
votes
Reflection principle of the Brownian motion
I'll only show it for $M_T = \max_{u\leq T} B_u$ and $(x,h)$-domain
$$ \{ h> 0, h > x \}. $$
By the reflection principle we have:
$$ P\left( B_T < x, M_T > h \right) = P\left( 2h - B_T &...
- 4,908
4
votes
Book recommendation: math toolkit for quantitative finance and statistics
I would recommend the books from Steven Shreve.
Here is a link to some one of his older online pdf's (1997 but nevertheless true) so you can check if that fits the bill.
http://citeseerx.ist.psu.edu/...
- 226
4
votes
Risk-Neutral Probabilities, Trinomial Model
Trinomial trees give incomplete markets so there is a range of possible risk neutral prices. So you have to find the possible probabilities that make the tree risk-neutral and see what prices you get. ...
- 6,813
4
votes
Perpetual American options
Use Dynkin's formula to write the expectation: $\mathbb{E}[e^{-r\tau} \phi(S_\tau)]= g(S_0)+\mathbb{E}[\int_ 0 ^ \tau (A g -rg) dt]$ where $\phi$ is the payoff.
Use the infinitismal generator $A$ to ...
- 1,359
4
votes
Understanding the solution of this integral
Another take on the question which uses stochastic calculus
[Digression]
Assume deterministic and constant rates without loss of generality. Also assume the absence of arbitrage opportunities and ...
- 14.2k
4
votes
Accepted
taylor expansion of PnL
$\require{cancel}$
$$\text{PnL} = -[P(t+\delta t,S+\delta S)-P(t,S)] + rP(t,S)\delta t + \Delta(\delta S - rS \delta t + q S\delta t)$$
Assuming a pure diffusion, at the order 1 as $\delta t \to 0$
$$...
- 14.2k
4
votes
Accepted
4
votes
Accepted
Intuition behind log return of portfolio = weighted sum of log returns
The above relation really only approximately. If you consider arithmetic retunrs then it is exact.
For the approximation you just need to look at the Taylor series of the exponential:
$$
e^x = 1 + x +...
- 13.3k
4
votes
Accepted
Mark Joshi, Quant Interview Question problem 2.34; replicating a digital option on a 4-step symmetric binomial tree
The answer above is only confusing because it is missing the the bet amounts.
You have a series of events, you are only allowed to bet on single events.
You want to construct something such that the ...
- 2,446
4
votes
Accepted
Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance
It is a complete solution.
Bearing in mind the SDE verified by $(X_t)_{t \geq 0}$, applying Itô's lemma to compute the (stochastic) differential of $f(X_t)$ yields
\begin{align}
df(X_t) &= \...
- 14.2k
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