Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
102
questions
57
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5
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Integral of Brownian motion w.r.t. time
Let
$$X_t = \int_0^t W_s \,\mathrm d s$$
where $W_s$ is our usual Brownian motion. My questions are the following:
Expectation?
Variance?
Is it a martingale?
Is it an Ito process or a Riemann ...
6
votes
1
answer
5k
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Extended Hull White Interest Rate Model for Zero Coupon Bond
Let's take the following three SDEs:
$$dr=u(r,t)dt + w(r,t)dX$$
$$u(r,t)=a(t)-br$$
$$w(r,t)=c$$
where $b$ and $c$ are constants and $a(t)$ an arbitrary function of time $t$.
If Zero Coupon Bond $Z(...
12
votes
2
answers
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How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?
I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs
$$(S_T f(S_T))^+$$
How do I express the stock dynamics using the ...
9
votes
2
answers
2k
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Ho and lee derivation for short rates model
A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
3
votes
1
answer
417
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The most general conditions under which Ito lemma holds
Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $f(x)=(x-K)^{+}$, I would be interested in knowing what are the least ...
30
votes
4
answers
10k
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What is a stationary process?
How do you explain what a stationary process is? In the first place, what is meant by process, and then what does the process have to be like so it can be called stationary?
15
votes
6
answers
8k
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Self-financing and Black-Scholes-Merton formula
Self-financing is an important concept in financial product replicating, normally used in pricing.
I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches ...
8
votes
3
answers
9k
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Variance of time integral of squared Brownian motion
I want to calculate the variance of
$$I = \int_0^t W_s^2 ds$$
I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get
$$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
1
vote
1
answer
906
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Illustrating the change of measure in Black-Scholes-Merton
Say that we have the following environment:
\begin{align}
dS_t &= \mu S_t dt + \sigma S_t dZ_t \\
dB_t &= r B_t dt
\end{align}
where $S_t$ is the price of a stock, $B_t$ is the price of ...
13
votes
2
answers
7k
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Distribution of stochastic integral
Suppose that $f(t)$ is a deterministic square integrable function.
I want to show $$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$.
I want to know if the following approach is ...
8
votes
2
answers
986
views
close form for stochastic integral
I am new to stochastic calculus. Can I know how to compute the close-form solution for
$$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$
and
$$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$
I encounter ...
7
votes
1
answer
7k
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Girsanov Theorem for Quanto/Compo adjustment
Assume that I have a foreign asset
$$Y_t = Y_0 \exp \left((r_f-\frac{1}{2}\sigma^2_Y)t+\sigma_Y W_t^1\right)$$ and an exchange rate
$$X_t = X_0 \exp\left((r_d-r_f-\frac{1}{2}\sigma^2_X)t+\sigma_X ...
17
votes
9
answers
10k
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Why the expected return rate of a stock has nothing to do with its option price?
OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
12
votes
3
answers
8k
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What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
11
votes
2
answers
756
views
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
11
votes
2
answers
4k
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FX forward with stochastic interest rates pricing
I would like to extend the following question about FX Forward rates in stochastic interest rate setup: "Expectation" of a FX Forward
We consider a FX process $X_t = X_0 \exp( \int_0^t(r^...
9
votes
1
answer
582
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Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Edit years later: No idea why I'm upvoted. I actually am not sure how I'm correct. But maybe I haven't forgotten conditional expectation as much as I thought I have.
We are given a filtered ...
8
votes
2
answers
5k
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Derivation of Ito's Lemma
My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
8
votes
1
answer
2k
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Multi Fractals Models
From a quant point of view, how would you explain Multi Fractals Models in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing....
7
votes
1
answer
6k
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Can I always use quadratic variation to calculate variance?
Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as
$X_t=X_0 + \mu ds + \sigma dW_t$
The quadratic variation of $X_t$ can be calculated as
$dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$
...
6
votes
1
answer
581
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Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$
I want to solve the following SDE:
$$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$
Integrating, I get:
$$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$
$$ ...
5
votes
2
answers
2k
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Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
4
votes
2
answers
675
views
Ito lemma of Convertible Bond under Two-factor Model Interest Rate
@Behrouz Maleki has provided the PDE of two factor model in other post so
could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ...
3
votes
2
answers
893
views
Dumb question: is risk-neutral pricing taking conditional expectation?
Dumb question: is risk-neutral pricing taking conditional expectation? $\tag{1}$
In trying to recall intuition for risk-neutral pricing, I think I read that we should price derivatives risk-neutrally ...
2
votes
3
answers
2k
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Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
1
vote
1
answer
929
views
FX Rate dynamics
Let's suppose USD/EUR price in USD follows a GBM with
$$ dS_t = rS_tdt + \sigma S_tdW_t $$
What process does EUR/USD follow in EUR?
1
vote
1
answer
957
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What is the link between the SDF in the Black-Scholes-Merton model and the exponential process in Girsanov's theorem?
Question
I have been toying around to get some understanding of what the stochastic discount factor look likes in Black-Scholes-Merton and how it relates to the exponential process in Girsanov's ...
28
votes
2
answers
48k
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Worked examples of applying Ito's lemma
In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
22
votes
1
answer
2k
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What is the trickiest thing to get right in Rates Quant recently (2019)?
What are the biggest challenges for Rates Quants in 2019? Most quants have been through a lot over the past years-shifting their SABR models in JPY swaptions, fixing the FVA models for negative rates, ...
18
votes
8
answers
9k
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Geometric Brownian motion - Volatility Interpretation (in the drift term)
A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution
$$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$
Recently ...
15
votes
1
answer
3k
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Girsanov Theorem and Quadratic Variation
Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation.
Below is the ...
15
votes
2
answers
19k
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Why is this stochastic integral a martingale?
Suppose that:
$W^*_t$ is a Wiener process under probability measure
$\mathbb{P}^*$ and;
$\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$.
In my lecture notes, it says that $\tilde{S}_t$ is a ...
14
votes
3
answers
8k
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How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?
I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
11
votes
5
answers
4k
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Free or Relatively Less Pricey Quant Finance courses online
I am trying to figure out what all online Quant Finance courses are out there which are free or relatively less pricey?
CQF is not less pricey
Financial Engineering course on Coursera - Not so great ...
10
votes
1
answer
1k
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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 ...
10
votes
2
answers
3k
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Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
10
votes
5
answers
1k
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How to compute $E[W(T)\exp(W(T)]$
I have got this interview question twice. Does anyone know from which interview question book or another source this question comes from? It may be some well known source as two different interviewers ...
9
votes
2
answers
1k
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How we can derive the PIDE of double exponential jump-diffusion model (Kou model)?
I'm working in double exponential jump-diffusion model known as the Kou model with following form, under the physical probability measure $P$.
$$ \frac{dS(t)}{S(t-)}=\mu dt+\sigma dW(t)+d(\sum_{...
8
votes
1
answer
1k
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Girsanov's Theorem - Change of Measure
I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by:
$$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$
...
8
votes
1
answer
688
views
How were these SDE derived?
Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)?
\begin{align*}
\frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\
\ln(F_{...
8
votes
2
answers
880
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Anticipating stochastic integral $\int_0^T W_T dW_t$
Using basic techniques from Malliavin calculus it can be shown that
$$
\int_0^T W_T dW_t = W_T^2 - T
$$
As can be seen the above integral is a non-adapted stochastic integral.
We also know using Ito ...
8
votes
2
answers
2k
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Malliavin Calculus
From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing.
...
7
votes
1
answer
1k
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generalized black scholes
I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
7
votes
2
answers
1k
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Radon Nikodym derivative when changing numeraires
I note from Wikipedia that if $Q$ and $Q^N$ are two measures corresponding to numeraires $M$ and $N$, then the Radon Nikodym derivative is given by: $$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(...
7
votes
1
answer
729
views
Baxter & Rennie HJM: differentiating Ito integral
From Baxter and Rennie, page 138:
$$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$
$$Z_t=\exp-\bigg(\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)ds\bigg)$$
$$dZ_t=...
7
votes
1
answer
10k
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Correlation coeffitiont between two stochastic processes
I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$.
I think that these are uncorrelated. But Why?
So thanks
6
votes
1
answer
975
views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
6
votes
1
answer
313
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Application of Vibrato Montecarlo methods
Ciao,
I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain.
In short the main idea of the method is the ...
6
votes
1
answer
313
views
Derivation of the Stochastic Vol PDE
I'm trying to follow the derivation of the stochastic vol pde for an option price - as given in Gatheral (The vol surface), Wilmott on Quant Finance and many other places. As usual one starts off with ...
6
votes
2
answers
750
views
Using Black-Scholes to price a geometric average price call
Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway.
I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...