Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
563
questions
1
vote
1answer
63 views
Transition density of geometric Brownian motion with time-dependent drift and volatility
Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE
...
2
votes
0answers
41 views
Correct application of Feynman Kac formula
I have a question on Feynman-Kac formula but can I ask the community if I have done it correctly? If no, may you point out to where I went wrong? Thanks!
The original FK formula states: Assume $f(t,x)$...
3
votes
1answer
249 views
Clarification on Deriving Ito's Lemma
The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the ...
1
vote
0answers
54 views
How to solve this particular PDE using Feynman-Kac formula?
I have to solve the PDE
$$
\begin{align}
\frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
1
vote
1answer
132 views
Black Scholes to Heat Equation
Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1).
Can someone please help me understand all the steps for how we get from the heat equation under time ...
0
votes
1answer
73 views
Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
1
vote
0answers
71 views
Option that never expires
I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
2
votes
4answers
392 views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
4
votes
0answers
68 views
Why is the Schöbel-Zhu model affine?
In the Schƶbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$.
The characteristic function of the stock process can be found by arguing that the model is ...
0
votes
0answers
58 views
Arbitrage free pricing of option to trade stocks
Consider Black-Scholes model with constant interest rate r and stocks with prices $S_t^A$ and $S_t^B$ that satisfy the SDE's $dS_t^A = S_t^A(\mu^A dt + \sigma^A dB_t)$ and $dS_t^B = S_t^B(\mu^B dt + \...
1
vote
0answers
39 views
Solution to Stock Price SDE with mean reversion [duplicate]
Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part):
$$ S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h) $$.
I actually don't know how ...
3
votes
2answers
544 views
Intuition for Martingale Representation Theorem
Can you please explain Martingale Representation Theorem in a non-technical way that what is it and why is it required?
Most of the stuffs I studied so far are ...
6
votes
2answers
96 views
Solution for a SDE for a Bond found in Bugard & Kjaer
I'm going over the paper -Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs- from Burgard and Kjaer. There the following SDE is given for ...
4
votes
0answers
60 views
Confused about discretization
I am reading a paper here: https://pdfs.semanticscholar.org/5f91/2d46b02b03230a4ffaaa42d655b2b6147d56.pdf
The following is my confusion.
The paper has the following continuous time model for the price ...
2
votes
0answers
23 views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
1
vote
1answer
50 views
Covariation of Ito semimartingales
If we have two Ito semimartingales over $[0,T]$:
$$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$
What is the relationship between
$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
2
votes
2answers
122 views
Itos Lemma Derivation notation
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
1
vote
1answer
66 views
Taking Expectation of Stopping Time and Integral Manipulation
Consider a stopping time $\tau$ that represents the point in time when the first credit event (e.g. default) occurs on a compact interval $[0,T]$.
Consider the expectation of the indicator function, $\...
4
votes
0answers
70 views
mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
3
votes
1answer
174 views
Application of Ito's Lemma in expected utility theory
An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
2
votes
1answer
98 views
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 ...
0
votes
0answers
23 views
Ito's formula with a random jump measure
Suppose all processes and functions defined are nice enough such that all the following definitions make sense.
On a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $\...
3
votes
0answers
114 views
Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyƶngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
1
vote
1answer
61 views
Serial correlation, quadratic variation and variance of returns
On p. 3 of Lorenzo Bergomi's book on Stochastic Volatility Modeling, there is the following assertion:
Indeed, to a good approximation, the variance of returns scales linearly with their time scale, ...
3
votes
1answer
103 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
0answers
19 views
Statistical test for comparing two different speed of mean reversion parameters for CIR model
I am trying to compare two different values of speed of mean reversion parameter for CIR model.
I would like to know if there exists a statistical test for comparing these two parameters.
the estimate ...
1
vote
0answers
113 views
Dynamic programming and Bellman equation to obtain the maximum
This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM"
Suppose an endowment economy where the representative ...
0
votes
1answer
107 views
How to replicate the future instantaneous short rate?
Suppose we have an interest rate model $R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$, where the brownian motion is under the risk neutral measure. Suppose $S(t)$ is the price at time $t$ for a contract ...
2
votes
0answers
27 views
Differentiation of value function in perpetual american option
I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.
$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $
where $...
3
votes
0answers
93 views
Boundary condition in perpetual american option problem
I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.
$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $
where $...
0
votes
0answers
36 views
Stochastic differential equation of Avellaneda model
I was reading this paper and page 14 the model is given.
I'm trying to find the steps to get to the SDE given.
OI is the open interest, E is the elasticity of demand.
$$
\frac{\Delta S}{S} ā¼ EĀ·Q^p \...
3
votes
0answers
226 views
Black and Scholes equation for portfolio **with** arbitrage
I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
1
vote
1answer
63 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
1
vote
1answer
65 views
Properties of integrated GBM
(I asked this question on MSE but I think it might have more success here)
Good day,
I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
0
votes
1answer
80 views
Stochastic Interest Rates in Option pricing
My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
2
votes
1answer
77 views
Brownian motion and Stochastic Integration
I have two questions relating stochastic integration which perhaps could be answered together.
First question:
First of all, I don't really understand why we can't use Riemann-Stieltjes integration ...
0
votes
1answer
85 views
Normal vs. Lognormal Greeks for Negative Rates Options
My understanding is that for some of the G10 currencies with negative rates (CHF, EUR), Swaption and Cap / Floor prices are quoted in terms of BOTH, normal and log-normal Vols. That in itself is not ...
2
votes
0answers
71 views
Option on $ \left( \int_0^T dW_t \right)^2$
Silly question, but how would you actually price
$$
E_0 \left( \left( \int_0^T dW_t \right)^2 - K \right)_+
$$
where $dW_t$ are standard Brownian motions. Is there a closed form analytical solution?...
0
votes
0answers
43 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
1
vote
1answer
38 views
Differential bond price stochastic rates
Suppose that the short rate follows the process
$$dr(t) = a(t, r(t))dt + \sigma(t, r(t))dW(t)$$
If $B(t) = exp(-\int_0^t r(u) d u)$, can one still write the differential $dB(t)$ a-la-Ito? Thanks.
1
vote
2answers
74 views
Proof that $f$ is continuous if and only if it has 0 quadratic variation?
I understand that $f$ continuous $\Rightarrow Q(f) = 0$ where this is defined over a bounded interval [0,T] as then we may use uniform continuity and the mean value theorem. But I am not sure how the ...
0
votes
0answers
55 views
Hedging a short position in the Lookback Option
SOLUTION
I got the correct answer using this formula
$X_2(HH)=(1+r)*[X_1(H)-\Delta_1(H)*S_1(H)]+\Delta_1(H)*S_2(HH)$
$(1+0.25)[2.24-(.06667*8)]+0.06667*16=3.20$
6
votes
2answers
139 views
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: $$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(...
2
votes
1answer
76 views
Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this ...
2
votes
1answer
59 views
Exact solution stock price with Vasicek interest rate model
Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(...
1
vote
0answers
41 views
Simulate correlated Brownian motions conditioned on future state(s)
Consider a model defined by 2 geometric Brownian motions
$$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$
$$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$
with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
2
votes
1answer
56 views
Generalization of Ito's Lemma to composite function
Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$
Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a ...
4
votes
1answer
189 views
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be ...
1
vote
1answer
64 views
Derivation of stock price formula John C. Hull 9th Ed p309
It says assuming a no-uncertainty Weiner process that models stock price:
$$
\Delta S = \mu S\Delta t
$$
Can be rearranged to (after taking the limit of $\Delta t \to 0$...
$$
\frac{dS}{S}=\mu dt
$$
...
1
vote
1answer
112 views
Integration of a deterministic function w.r.t. a Brownian motion
Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose
$X_t = \int_{0}^{t}\delta _{s}dW_{s}$
where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
