18
votes
Accepted
Processes used in quant finance
Here is a short list (to be edited and improved - community wiki) :
Standard brownian motion (also called Wiener process) for which:
$d\, W_t \sim \mathcal N(0, \sqrt{d t})$
Geometric brownian ...
11
votes
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed
\begin{equation}
\int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
- 5,959
9
votes
What is the purpose of short rate models?
Short rate models were first used in the 1970s and 1980s to try to fit and explain the term structure of interest rates - they went beyond simple parametric shapes (polynomials and exponential forms). ...
- 2,137
8
votes
Accepted
Dynamics of FX rate
I am answering now instead of commenting. The rate of change in FX is naturally forward looking in this case.
What you confuse is what happened to Spot due to changes in interest rate environments ...
- 8,193
7
votes
How do you find variance of a sde?
Here are two approaches that you could take to compute the variance of $X_t$. I am not making the conditioning explicit as it just complicates the notation but doesn't really add any additional ...
- 5,959
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
Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire
Under the stock numeraire measure, $\frac{B_t}{S_t}$ is a Martingale. We can compute $$d\frac{B_t}{S_t}= \frac{1}{S_t}dB_t -\frac{1}{S_t^2}B_tdS_t+\frac{1}{S_t^3}B_t\sigma^2S_t^2dt\\=\frac{B_t}{S_t}\...
- 605
6
votes
Accepted
Expectation of Stochastic Differential
In stochastic calculus, expressions of the type:
$$
dX_t = a(t, X_t)dt + b(t, X_t) dW_t
$$
are called stochastic differential equations.
What the one above means for example is that $X_t$ has the ...
- 2,200
6
votes
Accepted
Simulation of Geometric Brownian Motion in R
The issue is that you do not plot one sample path but for each time point $t$, you simply plot one possible realisation of the random variable $S_t(\omega)$. Thus, you don't get a connected path.
(...
- 15.4k
6
votes
Accepted
Idea of using logarithm for solving SDE in Black-Scholes model
Black and Scholes (1973) were not the first ones to use the geometric Brownian motion as a model for stock prices. For example, Samuelson did it before them.
It all started with a Brownian motion as ...
- 15.4k
6
votes
Application of Ito's Lemma in expected utility theory
The risky and riskless assets follow processes,
$$\frac{dS_t}{S_t}= \mu \, dt + \sigma \, dB_t, \,\,\, \frac{dM_t}{M_t}= r \, dt$$
If the proportion invested in the risky asset at time $t$ is $p_t$, ...
- 3,595
5
votes
What is the purpose of short rate models?
I might get down-voted for this, but in my opinion, short-rate models are not very useful for any practical pricing problems in today's finance. Even for simple vanilla rate derivatives (i.e. Caplet ...
- 5,998
5
votes
Exploding Libor Rates in Libor Market Model
this is a well-known problem. One solution is to make volatility zero when rates exceed a certain high level.
It's less problematic than it looks because any cash-flows generated will be divided by ...
- 6,873
5
votes
Accepted
Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model
Apply the Ito product rule, noting the cov of a deterministic and stochastic term is zero:
$$\begin{align}
d\left(e^{at}r_t\right)&=e^{at} dr_t+r_t de^{at}
\\[6pt]
&=e^{at} dr_t+r_t e^{at} d(...
- 6,204
5
votes
Accepted
Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$
It appears that you need to read some books such as Stochastic Differential Equations. For such type of equations, you need to use something called integrating factor such as the function $e^{-\mu t}$ ...
- 21k
4
votes
Accepted
Methods of SDE Calibration
There is somewhere summary of methods that can be used to estimate parameters of SDE?
If you'd like a brief survey, consider the following packages as well as the accompanying papers (note: you may ...
- 321
4
votes
Accepted
SDE for option value
This has been indirectly discussed in this question. We assume that $\{\mathcal{F}_t, \, t\ge 0\}$ is the natural filtration generated by the Brownian motion $\{W_t,\, t \ge 0\}$. Moreover, let
$B_t= ...
- 21k
4
votes
Accepted
Hawkes process intensity solution
Let us define the auxiliary process $\Lambda_t=e^{\kappa t}\lambda_t$. Note that:
$$ \Lambda_t = \kappa e^{\kappa t} \int_0^t(\rho_s-\lambda_s)ds+\delta e^{\kappa t}\int_0^tdN_t$$
Hence after a jump ...
- 7,999
4
votes
Accepted
Valuation of Cash-Or-Nothing option
You can use such an approximation but there are known analytical prices. You have a special case in which the stock price is normally distributed. See Bachelier Model.
Set $\mu=r-q$ (if you have ...
- 15.4k
4
votes
Evaluating the SDE $dX_t = t\,dS_t$
Using Itô's Lemma, notice that:
$$d(tS_t)=tdS_t+S_tdt=dX_t+S_tdt$$
Hence:
$$X_t=tS_t-\int S_udu$$
Using independence of Brownian increments, $E(S_udW_u)=E(S_u)E(dW_u)=0$, and the chain rule for the ...
- 7,999
4
votes
Problem of stochastic differential equation (SDE)
We assume that the price at time $t$ of a zero-coupon bond, with maturity $u$ and unit face value, is of the form
\begin{align*}
f(u-t, r_t, x_t) = E\left(e^{-\int_t^u r_s ds}\mid \mathcal{F}_t\right)....
- 21k
4
votes
Accepted
Why is the numeraire in the LGM model tradeable?
The confusion is that you think that we define the numeraire as this exponential function... It is not the case. We give the numeraire properties to $N$, then we model it. Similar to any other model.
...
- 753
4
votes
SDE Jump-Diffusion
Let
$$ J_t = \sum_{i=1}^{N_t} Z_i$$ be a compound Poisson process, with $(T_n)_{n\geq 1}$ being the jump times for Poisson process $(N_t)_{t\geq 0}$ and $(Z_i)_{i\geq 1}$ sequence of i.i.d. variables ...
- 5,008
4
votes
Accepted
Help on solving a stochastic differential equation
With your SDE for $F$, I get:
$$ dXdF = -a^2XFdt $$
$$FdX = rFdt + aXFdW $$
$$XdF = a^2XF dt -aXF dW$$
So, adding up:
$$ d(XF) = rF dt, $$
giving
$$ X_t = F^{-1}_t X_0 + rF^{-1}_t \int_0^t F_u du $$
- 5,008
3
votes
What is the purpose of short rate models?
Long story short, the main reason of a short rate model is to provide an analytical solution for the zero coupon bond $P(t, T)$, given by the following expectation:
$$
P(t, T) = E_t^Q \left[ \exp \...
- 741
3
votes
Accepted
Dynamics of LIBOR foward rate under T-forward measure
We assume that, under the risk-neutral measure $Q$,
\begin{align*}
dP(t, T) = P(t, T)(r_t + \sigma(t, T)dW_t),
\end{align*}
where $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. Then
\begin{align*...
- 21k
3
votes
Accepted
How is this SDE interpreted?
$\require{cancel}$
Consider the following SDE
$$ \frac{dF(t,T)}{F(t,T)} = \sigma(t,T) dW_t + (\exp(e^{-a(T-t)}dN_t)-1) + \mu_J(t,T)dt $$
where $N_t$ figures a standard Poisson process, supposedly ...
- 14.5k
3
votes
Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$
This is a corollary of Feynman-Kac theorem. For self-containedness,
I re-produce the proof as follows.
Assume that there exists a $C^{1,2}$-function $F=F(t,x)$ defined
on $[0,T]\times\mathbb{R}$ that ...
3
votes
Accepted
How to calculate mean and volatility parameters for Geometric Brownian motion?
If you want to rely on historical values at all (as opposed to a forward curve and implied volatilities), then $\mu$ would be the annualized exponential growth rate measured over a period T, ...
- 1,651
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