Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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12
votes
1answer
464 views

Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev \begin{equation} \sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}} \end{...
0
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1answer
43 views

Why is the value of the Brownian motion bounded by the maximum value of this square difference?

This comes from Taleb and Madeka's paper (https://www.academia.edu/39998351/All_Roads_Lead_to_Quantitative_Finance_Response_to_Clayton_?auto=download) regarding arbitrage restrictions on binary ...
2
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0answers
36 views

Discretisation of OU (mean reverting) process with a jump process

I have a question about how to apply the Euler approximation on OU process with a jump process. The stochastic process $X_t$ has dynamic $$dX_t=\alpha(\beta-X_t)dt+\sigma dW_t+dY_t$$ where $dY_t=...
0
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1answer
60 views

Not clear on an SDE solution example on YouTube [closed]

This video, from about 6 to 12 minutes: https://youtu.be/qdbkvD4N-us I feel like I’m following him ok, but then at the end his f(t,B(t)) has become an f(t,x) and there is no B(t) in his result, so it ...
3
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1answer
157 views

Bayes Theorem with change of measure

Tomas bjork- arbitrage theory in continuous time. Appendix B, proposition B41 says: The proof is not clear to me. Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
2
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2answers
61 views

For Ito Integrals with respect to a Brownian motion, why would the amount of stock held be a stochastic process?

Suppose that $B$ is a Wiener process and suppose $H$ is a right-continuous, adapted, and locally bounded process. Suppose $$\int_0^t H dB$$ is the Ito integral of $H$ with respect to the Wiener ...
1
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1answer
44 views

$\beta = 1$: Simulation of SABR and whether a solution is *exact*

Quick question regarding the conditional distributions (SABR is just an example here) Consider $$dS_t = \sigma_tS_tdW_t$$ $$d\sigma_t = \alpha\sigma_tdV $$ $$dW_tdV_t=\rho dt$$ Hence a SABR process ...
0
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2answers
78 views

Advantage of continuous time stochastic calculus over discrete version?

I'm new to the stochastic calculus, and I keep converting the continuous stochastic differential equation to its counterpart in discrete time, such as the autoregressive models. I wonder in practice, ...
2
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2answers
370 views

Black and Normal Model for Caplet using Python

I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ? I am valuing caplet that caps interest rate on ...
1
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0answers
64 views

Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$

https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf In the above document stochastic taylor expansions are nicely explained. Let us now consider a typical SDE model in finance ...
4
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1answer
74 views

Bond-price dynamics in the Vasicek model

Hello I am studying about interest rate modeling There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...
1
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2answers
100 views

Why is Delta Hedging a Hedge Against Short Position? [closed]

Consider the usual one-period binomial model. The delta-hedging formula, following Shreve's convention, is: $$\Delta_0=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)}$$ Shreve states: "The agent has ...
1
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1answer
61 views

Example of complex structured products on FX market?

Lately I have been working a lot with the vol smile and different stochastic volatility models with FX forwards data. Now I want to work with pricing examples through simulations. Can you suggest some ...
3
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1answer
82 views

Problem at deriving Bachelier formula with interest rates

In the Bachelier model, I have difficulties with a certain step. I want to figure out the distribution of $S_T$, which is the price process in the Bachelier model. So far I could state that ($\mathbb{...
0
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1answer
55 views

Stochastic solution (mean, variance) to lognormal drift and normal volatility

I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model I ...
1
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0answers
46 views

Symbol “.” in the derive of Quanto Adjustment

I am reading "Analysis, Geometry and Modeling in Finance". In section 2.10.2 which derives the quanto adjustment, it states that (in page 46) by definition the process $S_t^{d/f}S_t^f$ is the foreign ...
3
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1answer
120 views

What is the easiest way to learn Option pricing with PDE?

I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
1
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1answer
41 views

Why the variance of a process is $\left( \frac{dS_T^2}{dt}\right)^2$?

Consider an Ito process $dS_t = f(t,S_t) dt + g(t,S_t)dW_t $ What is the reason that we can compute the variance as: $\sqrt{VaR(S_t)} = \frac{(dS_t)^2}{dt}$
0
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0answers
79 views

Term structure equation in the Vasicek model

Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with ...
1
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0answers
67 views

Valuation of Callable Bonds

Is there any way to price American Callable Bonds (those which can be called on any date before expiration) other than basic CRR interest rate trees, since they won't be accurate enough to give ...
3
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1answer
155 views

Dependence of implied volatility on spot-vol correlation

I have the following general SV model: $$ dS = \sigma S dW_S $$ $$ d\sigma = a(\sigma,t) dt + b (\sigma, t) dW_\sigma $$ $$ dW_S dW_\sigma = \rho dt $$ where $a , b$ are deterministic functions of $\...
1
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0answers
28 views

Stochastic process with determinstic frequency of regime changes

Suppose that I have an OU process. For instance, assume that I want to model the interest rates. Suppose that regime change is known ex ante, and is deterministic in terms of frequency (For instance, ...
3
votes
1answer
436 views

Ito`s Lemma problem

Can someone help me with calculus for this problem. I have these 3 equations and with Ito`s Lemma I have to find $dXt$. \begin{cases} dY= μYdt+σYdB \\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
10
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3answers
6k views

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 ...
0
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0answers
26 views

Filtrations and the different “kinds” of pre-knowledge

I am searching for a reference I think I saw in a book by either Shreve or Oskendahl. I am struggling with a theoretical question. As I recall how it was posed, the idea of no prior information (or ...
5
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1answer
111 views

Expected value of exponential of hitting time of GBM

We have a stopping time $$ \tau=\inf\{t\geq 0: S_0e^{\sigma B_t+(r-\sigma^2/2)t}=S^* \} $$ where $S_0,\sigma,r,S^*$ are constants and $S^*<S_0$, and $B_t$ is a brownian motion. I wish to compute ...
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0answers
44 views

Log Contract payoff function

I can’t get where Dr. Rouah gets payoff function of log contract. Could you please take a look at that? https://frouah.com/finance%20notes/Variance%20Swap.pdf It’s on page 2, section 3. I couldn’t ...
3
votes
2answers
688 views

Variance of Multi-Dimensional OU process

I'm trying to implement this model shown here: http://www.sciencedirect.com/science/article/pii/S0304407611000388 As part of the modelling process I have to calculate the unconditional variance of X ...
1
vote
1answer
61 views

Problem finding correct SDE for Stochastic Process

I am really struggling to come up with the correct SDE for the stochastic process: $Y(t) = a[Z(t)]^2$ where $Z(t)$ is a Brownian Motion. According to my Prof, the SDE is: $dY(t) = adt + 2aZ(t)dZt $...
2
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0answers
37 views

How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?

I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ...
1
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0answers
45 views

Unconditional Expectation vs. Conditional Expectation at time $0$

In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
1
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0answers
78 views

The conditional expectation of a geometric brownian motion

In this question it states that $$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$ and I assume that $0 \leq s \leq t$. The accepted answer states that this step is ...
2
votes
1answer
134 views

Solution to a Geometric Ornstein Uhlenbeck Process $dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t$

I've been searching for the solution to the modified Ornstein-Uhlenbeck process \begin{equation*} dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t \end{equation*} but it surprisingly hard to find. The ...
3
votes
2answers
125 views

Find the brownian motion associated to a linear combination of dependant brownian motions

I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
3
votes
4answers
595 views

Log of square of Geometric Brownian Motion

Which of the two calculations below, is wrong? Why? $dF = \sigma F dW$ First: $dF^2 = (F^2)' dF + \frac{1}{2}(F^2)''dF.dF$ $dF^2 = 2F dF + dF.dF$ $dF^2 = 2 \sigma F^2 dW + \sigma^2 F^2 dt$ $\...
1
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0answers
49 views

Self financing strategy and repo rate

I was wondering how to adjust the self financing condition when cash borrowing cas be secured by the stock. Suppose the risk-free money account is $B_t$ and there is a risky asset $S_t$. One have ...
2
votes
0answers
54 views

Novikov condition for Vasicek process

Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \...
3
votes
1answer
186 views

Question on Gÿongy' lemma proof

I have some questions regarding a proof of Gÿongy's lemma given in 1 I would like to understand the following passage: $$ \int_{s=t_0}^{s=t}\mathbb{E}\left[\delta(X_s-K)\langle dX_s\rangle^2 \right]= ...
0
votes
1answer
138 views

When $E[f(\alpha,X)] = f(\alpha, E[X])$

When $E[f(\alpha,X)] = f(\alpha,E[X])$, where $f$ is some convex function of the first and second variables, except when the first variable takes the value $\alpha$ in which case the equality holds, ...
3
votes
1answer
75 views

Derivation and expectation interchange

I would like to know when it is allowed to interchange derivation and expectation. Suppose $X$ is some r.v whose dynamic is controlled by some parameter $\sigma$ and suppose $h$ is some smooth ...
14
votes
3answers
1k views

Deterministic interpretation of stochastic differential equation

In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$...
1
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0answers
43 views

Milstein discretization of the CIR process

Given the CIR process $\ dX_t = (a − bX_t ) dt + \sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
2
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0answers
46 views

Ito Diffusion with Change of Measure

Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
4
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0answers
76 views

Feynman-Kac to derive stochastic representation

$u_t + \frac{1}{2}\sigma^2x^2u_{xx} - \alpha + \lambda((K_d - x)^+ - u) = 0$ with terminal condition $u(T, X) = (K_m - X(T))^+$ $dX = \sigma X(t)dW_t$ $\alpha$ and $\lambda$ are constants Ok so ...
1
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0answers
87 views

Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem

Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but ...
3
votes
0answers
70 views

Stochastic Differential equation: CAPM

Let $R=(R_1, \dots ,R_M)$′ denote a vector of excess returns of M assets observed at $n$ time points, $0<t_1<t_2< \cdots <t_n<T$, within a time span $T>0$. We wish to explain the ...
1
vote
0answers
60 views

Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
2
votes
0answers
78 views

For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$

In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
1
vote
2answers
106 views

How to numerically simulate exponential stochastic integral

For example given an integral $$ \int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+ $$ where $W(t')$ is a standard Wiener process. I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
2
votes
1answer
68 views

If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?

Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say $$ \begin{equation} dS_t = \mu S_t dt + \sigma S_tdW_t \end{equation} $$ My question is what is the distribution of $...