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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, \...
LocalVolatility's user avatar
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). ...
Dom's user avatar
  • 2,167
8 votes
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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 ...
AKdemy's user avatar
  • 9,014
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 ...
LocalVolatility's user avatar
6 votes
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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}\...
spaceisdarkgreen's user avatar
6 votes
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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 $$F=\int_0^t g(s) dW_s.$$ ...
Gordon's user avatar
  • 21.2k
6 votes
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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 ...
byouness's user avatar
  • 2,220
6 votes
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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. (...
Kevin's user avatar
  • 16k
6 votes
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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 ...
Kevin's user avatar
  • 16k
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 $p_t$, ...
RRL's user avatar
  • 3,700
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 ...
Mark Joshi's user avatar
  • 6,983
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 ...
Jan Stuller's user avatar
  • 6,178
5 votes
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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(...
Magic is in the chain's user avatar
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 $e^{-\mu t}$ ...
Gordon's user avatar
  • 21.2k
4 votes
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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= ...
Gordon's user avatar
  • 21.2k
4 votes
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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 ...
Daneel Olivaw's user avatar
4 votes
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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 ...
Kevin's user avatar
  • 16k
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 ...
Daneel Olivaw's user avatar
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)....
Gordon's user avatar
  • 21.2k
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. ...
Canardini's user avatar
  • 553
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 ...
ir7's user avatar
  • 5,043
4 votes
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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 $$ X_t = F^{-1}_t X_0 + rF^{-1}_t \int_0^t F_u du $$
ir7's user avatar
  • 5,043
4 votes
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Pricing PDE of Asian option by Shreve

Intuitively, once a stock hits value zero the underlying company is bankrupt and the value remains zero (this might not be true in the real world but it's a common assumption). So, once $S_t$ is zero ...
Bob Jansen's user avatar
  • 8,562
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 \...
rvignolo's user avatar
  • 741
3 votes
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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*...
Gordon's user avatar
  • 21.2k
3 votes
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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, ...
ZRH's user avatar
  • 1,671
3 votes
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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 ...
Quantuple's user avatar
  • 14.7k
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 ...
Danny Pak-Keung Chan's user avatar
3 votes
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Expectation in a stochastic differential equation

Assuming you are talking about unconditional expectation, in general you have $$ \mathbb{E}[X_t] = \mathbb{E}[e^{W_t}] = e^{\mathbb{E}[W_t] + \frac{1}{2}\text{Var}(W_t) } $$ which yields $$ \mathbb{...
rafaelc's user avatar
  • 146

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