Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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155 views

Hagan's explanation of the Local Volatility model

Long story cut short: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be the function of the strike. Long story: The well-...
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32 views

Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model

I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
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2 votes
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44 views

Feymann Kac pde with correlated process

I have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
  • 153
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1 answer
49 views

Ito's lemma in stochastic volatility models [closed]

I couldn't help but notice that in all stochastic volatility models articles I consulted, whenever Ito lema is applied with a process of the sort $$\frac{d S_t}{S_t} = \sigma_t d W_t $$ With $(\...
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1 vote
1 answer
51 views

Dynamics of discounted prices (multi-dimensional)

My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
1 vote
1 answer
115 views

Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach

I am looking to compute the following using Ito's formula. $$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$ Knowing the properties of brownian motion, it is rather easy to show that the ...
0 votes
0 answers
54 views

Pricing of Zero Coupon bond under Risk-neutral pricing measure

Pg 242 Topic 5.6.2: Futures contract Risk-neutral pricing of a zero-coupon bond is given by the below formulae: $$ B(t,T) \, = \,\frac{1}{D(t)}. \tilde E~[D(T)\mid F(t)], 0\,\leq \,t\,\leq\,T\,\leq\,\...
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2 votes
0 answers
55 views

Munk (2011) exercise 3.6

I'm trying to solve the exercise in Munk (2011). The exercise reads: "Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
1 vote
0 answers
86 views

Analytical expression for SDE

I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
0 votes
0 answers
65 views

How to derive the HJM drift condition?

I'm trying to derive the Heath Jarrow Morton drift condition (from Björk, page 298) and this equation is the part that I'm not able to derive: $$ A(t,T) + \frac{1}{2} ||S(t,T)||^2 = \sum_{i=0}^d S_i(t,...
1 vote
1 answer
112 views

Put price characterisation

I am reading Shreve's Stochastic Calculus for Finance II: Continuous-Time Models. I am trying to understand the below two concepts: Topic 8.3.3 Analytical Characterization of the Put price on Page ...
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2 votes
1 answer
100 views

Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)

I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following $$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
0 votes
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60 views

Backset LIBOR contract

Below is an extract from Steven Shreve’s BK 2, Chapt 10: Term Structure models. LINK I am trying to understand Stochastic Calculus from the above book with the help of a Pure Math PhD student. Despite ...
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1 answer
61 views

Construction of Itos integral

I am trying to understand the below: Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss? In this calculation the purchase price is not taken ...
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2 votes
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151 views

If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows

From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
0 votes
0 answers
60 views

Locally riskless

Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio: $$ \int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
4 votes
0 answers
131 views

optimal stopping time problem

I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
2 votes
2 answers
164 views

Black-Scholes PDE derivation gap

Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio: $$ \int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
2 votes
1 answer
100 views

Deriving the variance of G2++ Model

I'm studying G2++ Model in Brigo(2007)'s book. The model constructed as follows, $$ r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\ $$ with the dynamics of $dx(t)$ and $dy(t)$ described by: \begin{align}...
1 vote
1 answer
190 views

Euler Discretization python code

Write the Euler discretization of the 1-dimensional stochastic equation $dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$ For this part I would say all right because it is a purely ...
2 votes
0 answers
139 views

Local volatility implied spot vol correlation

I have a question about local volatility models. In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility ...
4 votes
2 answers
400 views

Transformation of local volatility model

Assume we have an SDE $$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$ where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
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Maximum entropy probability distribution for $S_T$ implied from discrete market quotes

Consider a maturity $T$, for this maturity I have some implied volatility from market denoted $\sigma^{0}_{i}$. I want to interpolate these volatility using Entropy approach, by using $\sigma^{0}_{i}$...
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0 votes
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73 views

How to calculate mean and variance in Vasicek Model

In the Vasicek model, the short rate of interest under the risk-neutral probability measure is given by: where k, θ, σ > 0 and W is a standard Brownian motion. Consider the related process where ...
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3 votes
1 answer
186 views

Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$

Assume $(X_t)_{t\geq 0}$ follows an SDE of the form: $$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$ where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
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1 answer
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I want to know stochastic derivation of zero coupon bond formula

I'm elementary level of stochastic calculus. In the above picture, from equation (11) to (12) I don't know what is the clue of $μ(t)$ is the expectation of $r(t)$ and how from this identity we can get ...
2 votes
0 answers
67 views

Relation between SABR parameters and Taylor expansion parameters

Suppose a SABR model framework (with $\beta=1$) $$dF_t=\sigma_t S_t dW^{S}_{t}$$ $$d\sigma_t=\alpha \sigma_t dW^{\sigma}_{t}$$ $$dW^{S}_{t}dW^{\sigma}_{t}=\rho dt$$ I know that the Implied Volatility ...
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0 votes
1 answer
75 views

Integration of exponential raised with Brownian Motion wrt the Brownian Motion

I have to derive several things for my thesis, however, I have the following expression: $$ \int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t} $$ Does anyone know what the solution for this is? Kind regards.
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3 votes
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93 views

Implied Volatility is the harmonic average of Local Volatility

I am trying to demonstrate the famous result that states that when $T \rightarrow 0$, the Implied Volatility is the harmonic average of Local Volatility. I am st the final stage, and I have the ...
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5 votes
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105 views

Closed formula for computing Implied Volatility from Local Volatility function

The main result of this paper (Asymptotics and Calibration in Local Volatility Models, Berestycki, Busca, and Florent. Quantitative Finance, 2002) is equation (16) on page 63, that states that: In the ...
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0 votes
2 answers
158 views

The meaning of filteration ( coin toss example )

Reference book is 'Steven Shreve: Stochastic Calculus and Finance' What I don't understand is $F_3$ below picture I understand that 'filteration' have accumulative information. So when we tossed the ...
0 votes
1 answer
123 views

Black-Scholes differential equation rewritten [closed]

I have seen that the Black-Scholes equation $$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S}-rV=0$$ can also be written in the ...
1 vote
1 answer
126 views

Calculating Expectation of Stochastic Volatility

I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM STRUCTURE OF OPTION IMPLIED VOLATILITY AND VOLATILITY COMPONENTS by Guo, Han, and Zhao. I don't understand why the above equations ...
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1 vote
0 answers
67 views

Simultaneous Stochastic Differential Equations

I was thinking about cointegrated time series and came up with the following simultaneous equations model: $dY_t = \alpha (Y_t - \gamma X_t)dt + \sigma dB_t$ $dX_t = \beta (Y_t - \delta X_t)dt + \tau ...
1 vote
0 answers
90 views

Differential vs. derivative in the Vasicek model [closed]

Can anyone help me in understanding how we get the line I have marked with a red arrow? I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
3 votes
1 answer
154 views

Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...
4 votes
1 answer
246 views

How am I supposed to understand the following statement on the convexity adjusted rate

Given, a numéraire $(N(t))_{0\leq t \leq T}$ and an index $(X(t))_{0\leq t\leq T}$ that is a $\mathbb Q^{N}$-martingale, we consider the natural payoff $V_{N}(T)$, where it pays $$V_{N}(T):=X(T)N(T) \...
2 votes
0 answers
26 views

Finding the distribution of $I(T_{1},T_{n})$ under an appropriate measure if the forwards are lognormal? [duplicate]

My question follows beneath the "lengthy" setting I describe: Given a tenor discretization $0 = T_{0}< ... < T_{n} =T$, and under the assumption that under $\mathbb P$, for all $i = 1,....
8 votes
3 answers
679 views

Why does the diffusion term remain the same when we change pricing measure?

Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion In plenty of interest rate examples, I have ...
5 votes
1 answer
617 views

Where does 1/2 in Fourier Transform method of pricing options come from?

I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
2 votes
1 answer
191 views

Obtaining the dynamics of the Vasicek model using Itô

Consider the following expression for the short-term interest rate $$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$ which is ...
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0 answers
68 views

SDE of a Geometric Levy process with compound Poisson process

Suppose that a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is given. A geometric Levy process is defined in the form of $S_t=S_0 exp(X_t)$ where $S_0$, let's say, is the initial price and $...
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2 votes
0 answers
101 views

Largest class of real world probability models admitting explicit risk-neutral change of measure

Assume we have two assets, a random asset $A_t$ and deterministic risk-free bond $B_t = e^{rt}$. Let $P$ be a model of the real-world probabilities of $S$ and $Q$ the unique associated risk-neutral ...
4 votes
1 answer
236 views

Pricing a contract

I'm currently trying to price some different kinds of contracts. I'm stuck on this following exercise, which I can't seems to find a good solution for. The following is assumed: We are in a standard ...
2 votes
3 answers
270 views

Integral of brownian increments

I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following $$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$ I've been ...
1 vote
2 answers
408 views

Why would exchange rates follow a geometric brownian motion?

I'm reading Shreve's Stochastic Calculus for Finance. On page 382, he begins talking about exchange rates: Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
1 vote
0 answers
34 views

From the perspective of a company, when is the right time to start paying dividends?

I am trying to understand geometric Brownian motion as it relates to the present discounted value of future dividend payments. I am supposing that a company has a revenue stream $f(t)$. This is just $...
1 vote
1 answer
241 views

Jump Diffusion Process question

I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
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0 votes
0 answers
194 views

what is $\int t dW$ and $\int W dt$? [duplicate]

More explicitly, if $W(t)$ is Brownian motion, what would be $$f(t) := \int_0^t u dW(u)$$ and $$g(t) := \int_0^t W(u) du$$?
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2 votes
0 answers
132 views

Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?

I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...

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