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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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votes
1answer
148 views

Problem with derivating integral

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
2
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1answer
251 views

HJM framework problem - showing that HJM drift condition implies that $b(z)=b+βz$ and $(ρ)^2=α$

Hi I am looking for some general clarification to Heath–Jarrow–Morton framework. I am analyzing a problem where the forward rate is modeled as $$ f(t,T)=e^{\beta(T-t)} Z_t+h(T-t) \tag{1}$$ for some ...
2
votes
2answers
414 views

Question about the martingale property of stochastic integral

Let $W_{t}$ be a Wiener process, and let $$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$ Is $X_{t}$ a martingale? We can rewrite in differential form as $$dX_{t} = W_{t}dt$$ ,which means $X_{t}$ is a diffusion ...
2
votes
2answers
295 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
2
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2answers
214 views

How can I make this portfolio self-financing?

$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$ $b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate So the value of the portfolio: $$V_t = ...
2
votes
1answer
90 views

forward option, stochastic calculus

I encounter a problem to understand this: The price of a forward option is : $C(K,t,T)=\mathbb{E}[((S_{T}/S_{t})-K)+]$ OK The option should only depend on $T-t$ because the yield randomness (for a ...
2
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2answers
2k views

How to express the Black Derman & Toy Model in a $dr=A\,dt+B\, dW$ form?

The Black Derman & Toy (BDT) model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t))}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, ...
2
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2answers
55 views

Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price

(My Question) Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M. $$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
2
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1answer
54 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 ...
2
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1answer
74 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 $...
2
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1answer
62 views

Characteristic function and distribution of a random variable

This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time. $$ X_t = \int^t_0 \sigma(s)dW_s $$ $\sigma$ is a deterministic function and $W_t$ is brownian motion. I am asked to find the ...
2
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1answer
93 views

Distribution in Heston

$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$ $W_t$ is wiener process and the rest is just some parameters. For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
2
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1answer
69 views

Stochastic Calculus: How to test for dependency of random variables

If I let $g(x)$ be a deterministic function of a real variable $x$ and define $X(t)$ as: $$X_T=\int_{0}^{T}f(u)dW_u$$ with $W_t$ being a wiener process. For $s<t$, Will $X_s$ and $X_s-X_t$ then be ...
2
votes
1answer
281 views

Test if a process (with no drift) is a martingale

Consider the process $$Z(t)=\int_{0}^{t} \frac{u^a}{t^a}dW_u$$ for some real constant $a$ and $W_t$ is a wiener process. I want to check whether this process is a $F_t^W$-martingale. I noticed Lemma 4....
2
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1answer
83 views

Spot Interest Rate at time $t$

I know that the general model for the dynamics of the spot interest rate is $$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$ My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?
2
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1answer
230 views

Closed- solution for Convertible bond price two factor model

I am trying to find the closed- solution of convertible bond $V(s,r,t)$ under Vasicek model of two factor model of PDE shown in below link Ito lemma of Convertible Bond under Two-factor Model ...
2
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1answer
191 views

Analytic ZCB call option under Vasicek

The call option with strike $X$ and maturity $T$ on a ZCB maturing at time $S$, where $T\le S$, is $$ZBO(t,T,S,X)=E_t[e^{-\int_t^Tr_sds}(P(T,S)-X)^+]$$ The ZCB price is denoted by $$P(t,T)=E_t[e^{-\...
2
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1answer
138 views

stochastic interest rate $r_t=x_t+y_t$

Let $$dr_t=(\alpha(t)-\beta r_t)dt+\sigma dW_t$$ where $\alpha$ is non stochastic process and $\beta$ and $\sigma$ are constant. Can we write process $r_t$ in the form $$r_t=x_t+y_t$$ where the ...
2
votes
1answer
295 views

“Expectation” of a FX Forward

I have an FX process $X_t = X_0 \exp((r_d-r_f)t+ \sigma W_t)$. Now clearly $E[X_t] = F_{0,t}^X$. i.e. a forward contract of the process $X$ starting at time 0 and maturing at time $t$. What if I ...
2
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2answers
536 views

Asymptotic behavior property of geometric Brownian Motion proof

Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as: If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \...
2
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1answer
139 views

Second Moment of Stock Process

I have a stock process which I have decided to model as $$S_T=S_t\exp((r-q-\frac{1}{2}\sigma^2)(T-t)+\sigma(W_T-Wt))-D_T$$ where $D_T$ is a cash dividend at time $T$. This dividend is known. I then ...
2
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1answer
210 views

Brownian motion. Solve stoc. integral by using Ito's lemma

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^...
2
votes
1answer
256 views

Difference between stochastic calculus and newton calculus

As I am not a student of hard core mathematics,I just want to know how stochastic calculus is different from newton calculus. What make stochastic calculus different from simple newton calculus ?
2
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1answer
224 views

Probability of Stock breaching barrier

If a stock has a process: $dS(t) = sigma*dB(t)$, where $B(t)$ is a standard Brownian motion, and current stock price is $S(0)$. There is a barrier $H>S(0)$. What is the probability that the stock ...
2
votes
1answer
60 views

equality in distribution

I encounter the following problem : I have the equality in distribution: for all $\lambda >0, ((1/\lambda)*\int_{0}^{\lambda t}\sigma_{u}^{2}du,t\geq0)=(\int_{0}^{t}\sigma_{u}^{2}du,t\geq0)$ ...
2
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1answer
117 views

What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S: $dS=\mu S dt + \sigma S dX$ Clearly $S$ ...
2
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1answer
67 views

The Riccatti equation for The Cox-Ingerson-Ross Model

(My Question) I went through the calculations halfway, but I cannot find out how to calculate the following Riccatti equation. Please tell me how to calculate this The Riccatti equation with its ...
2
votes
1answer
112 views

The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
2
votes
1answer
77 views

How to deduce the formula of the wealth process of a stochastic volatility model?

I am reading the paper Solution of the HJB Equations Involved in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu. The authors consider the utility function $U: \mathbb{R} \to \...
2
votes
1answer
211 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
votes
1answer
313 views

Show that the Ito integral is Gaussian

Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
2
votes
1answer
180 views

How to find correct change of measure

I'm trying to figure out how to find the correct equivalent martingale measure to change into. First of, since I am on mobile and find it hard to write LaTeX here, I will refer to Wikipedia's version ...
2
votes
1answer
169 views

How to solve one-touch American call

I want to solve the one-touch American call at $t = 0$ with level $B,$ maturity $T$ under the following assumption: $$d S= rSd t + \sigma SdW,\quad S_0<B.$$ We have following formula: $$V(S_0,0) = \...
2
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1answer
82 views

Utility-optimal leverage with costs

Say I have a portfolio, $X_t$, using a leverage of $f$, such that the dynamics are given by \begin{equation} dX_t = \mu f X_t dt + \sigma f X_t dW_t \end{equation} I want to optimize the expected ...
2
votes
1answer
662 views

Ito's Lemma, differentiating an integral with Brownian motion

In How were these SDE derived? I don't understand one part of Gordon's answer, specifically: $$\ln S_t=\ln F_{0,t}-\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda t})+\sigma e^{-\lambda t}\int_0^t e^{\...
2
votes
1answer
190 views

Computing Correlation between Forward Rates

I have the feeling this question has an extremely simple answer but I'll put it out to the group anyway. Imagine I have data for 3M and 6M forward rates following a lognormal process, and that I ...
2
votes
1answer
166 views

How PCA is performed in the paper “Markov Models…”

can anyone explain in a bit detail on how PCA is performed in the paper "Markov Models for Commodity Futures: Theory and Practice" by Leif B. G. Andersen. I'm not clear on how the high dimension ...
2
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1answer
88 views

investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...
2
votes
1answer
165 views

Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale

We are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb ...
2
votes
2answers
737 views

Discounted Stock Price

I have the following Question : Prove that under the risk-neutral probability p the stock and the banjaccount have the same average rate of growth. In other words, if $ S_0 , S_N $ are the initial ...
2
votes
2answers
208 views

Transformation into Martingale

If $f$ is some function of BV on $\mathbb{R}$ and $dZ_t = f(W_t)dW_t + \mu_t dt$ ($W_t$ is a $1$-dimensional standard Brownian Motion), then what choice of real valued function $F$ makes: \begin{...
2
votes
2answers
446 views

Bachelier model: number of stocks in replicating strategy

Given: Consider a two-asset, continuous time model (B,S) where \begin{equation} dB_t = B_t r dt, \quad dS_t = \mu dt + \sigma dW_t. \end{equation} The question is: Show that there exists a trading ...
2
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0answers
46 views

Interchange Expectation and Supremum in Snell Envelope/American Options

I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
2
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0answers
37 views

Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$

Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as $$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\...
2
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0answers
55 views

The Ho-Lee Model (1986)'s Bond Call Option Pricing [closed]

(My Question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (the details in this ...
2
votes
0answers
39 views

How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?

The integral is shown below: And how to use python to calculate pi (better if we don't need to code for each pi)?
2
votes
0answers
50 views

The Ho-Lee Model (1986)

(My question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (Thank you for your ...
2
votes
0answers
100 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 ...
2
votes
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 ...
2
votes
0answers
56 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)}. \...