Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
746
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How to hedge a dual digital option
Let us assume we have two FX rates: $ 1 EUR = S_t^{(1)} USD$ and $ 1 GBP=S_t^{(2)} USD $. Let $K_1>0, K_2>0$ be strictly positive values and a payoff at some time $ T>0 $ (called maturity) ...
- 576
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1
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221
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Question on Merton's self financing derivation
I'm reading Merton's Optimum Consumption and Portfolio Rules in a Continuous-time Model, and don't understand the step where he goes from discrete to continuous time. Specifically, my confusion is ...
- 165
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106
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Stochastic integral involving Poisson Process
Consider an (inhomogeneous) Poisson process $N_t$ with intensity $\lambda_t$. Then I want to compute the following integral
$\mathbb{E} \left(\int f(t,N_{t-}) d\tilde{N}_t\right)^2$
for some smooth ...
- 143
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68
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Volatility of the product of two correlated asset following a log normal distribution [duplicate]
I am trying to solve the problem: Given two assets X and Y that follow a log normal distribution with volatility $\sigma_1$ and $\sigma_2$ respectively and with correlation $\rho$, what is the ...
- 11
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Affine Jump Diffusion
I'm currently looking into affine jump-diffusions. I would like to get to know the literature better and I know the paper by Duffie, Pan, and Singleton (2000) is a very celebrated paper. Although I ...
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240
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On first and last zeros before t in a Brownian Motion
Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$:
\begin{align*}
\alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \...
1
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1
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103
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How is variance derived in BS?
The realized variance under classical Black Scholes where the stock price process follows a GBM is given as
$$V_T = \frac1T\int_0^T\sigma_s^2ds\qquad (1)$$
however, the texts I have been reading do ...
- 51
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109
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How calculate expectation and variation of stochastic integral Based on Heston model?
I was calculated Heston volatility model. But I think it is wrong.
$dS_t = \mu dt + \sqrt V_t dW_t^s$
$dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$.
$dW^s_t dW^v_t = \rho dt$
take integral to ...
- 43
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1
answer
128
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What is the PDE for this interest rate derivative?
We have the following model for the short rate $r_t$under $\mathbb{Q}$:
$$dr_t=(2\%-r_t)dt+\sqrt{r_t+\sigma_t}dW^1_t\\d\sigma_t=(5\%-\sigma_t)dt+\sqrt{\sigma_t}dW^2_t$$
What is the PDE of which the ...
- 11
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179
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Volatility swaps hedging
I have heard that traders use a straddle to hedge volatility swaps (in the FX context), although I could not figure out the specifics. Is this type of hedge used in practice? And if yes, how does it ...
- 576
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104
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Smile wings and varswap pricing
Is it true that far wings of the volatility smile have an outsized influence on the price of a variance swap? Is there a mathematical argument demonstrating this idea? What do we generally refer as ...
- 576
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62
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how to calculate pdf and cdf for an Ornstein-Uhlenbeck process
I have the
Task. For Ornstein-Uhlenbeck process generate a path and plot a)
cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile.
My solution.
From the literature we ...
- 219
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Transform non-linear HJB PDE into system of linear ODEs [closed]
I am reading this market making paper, and am trying to understand the transformation presented on page 6. A good resource for background relevant to the transformation is this other market-making ...
- 11
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1
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77
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Is this the correct discretisation of the Hull-White SDE for building a python model?
I've tried to build a basic one-factor Hull-White model using python, which I've done by trying to discretise the characteristic SDE.
According to my notes, the Hull-White SDE is
$$
dr_t = \alpha (\mu(...
- 135
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62
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Mixing formula for SVJ models
I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to ...
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What happens trying to price derivatives starting from a non-geometric brownian motion?
To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question.
The set-...
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146
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Calibrating Hull-White model using historical data
I'm in search of a way to calibrate a very simple Hull-White model with a constant volatility and a constant mean-reversion speed, purely based on historical zero rates.
$$dr(t) = (\theta(t) - \alpha ...
- 101
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2
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444
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Is Local Volatility a function of the Strike or the Underlying price?
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 a function of the strike.
Additionally, I wonder ...
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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 ...
- 576
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58
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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}{...
- 163
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1
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77
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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 $(\...
- 237
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1
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55
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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{...
- 61
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1
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139
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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 ...
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91
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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:
$$ B(t,T) \, = \,\frac{1}{D(t)}. \tilde E~[D(T)\mid F(t)], 0\,\leq \,t\,\leq\,T\,\leq\,\...
- 11
2
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0
answers
58
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Munk (2011) exercise 3.6
I'm trying to solve the exercise in Munk (2011). The exercise reads:
"Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
- 61
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92
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Analytical expression for SDE
I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
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101
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How to derive the HJM drift condition?
I'm trying to derive the Heath Jarrow Morton drift condition (from Björk, page 298) and this equation is the part that I'm not able to derive:
$$ A(t,T) + \frac{1}{2} ||S(t,T)||^2 = \sum_{i=0}^d S_i(t,...
- 61
1
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1
answer
142
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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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3
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1
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144
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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
$$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
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0
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66
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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
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68
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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 ...
- 11
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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 ...
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Locally riskless
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
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140
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optimal stopping time problem
I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
2
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2
answers
177
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Black-Scholes PDE derivation gap
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
- 33
2
votes
1
answer
120
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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}...
- 303
1
vote
1
answer
340
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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 ...
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0
answers
221
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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 ...
- 21
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2
answers
418
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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 ...
- 43
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39
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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 $\sigma^{0}_{i}$...
- 773
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1
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194
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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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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 ...
- 303
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0
answers
98
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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 ...
- 773
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votes
1
answer
131
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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.
- 5
2
votes
0
answers
126
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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 ...
- 773
4
votes
0
answers
135
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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 ...
- 773
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votes
2
answers
227
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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 ...
- 303
0
votes
1
answer
141
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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 ...
- 3
1
vote
1
answer
140
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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 ...
- 11
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vote
0
answers
75
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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 ...
