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7
votes
2answers
169 views

how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?

I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure $P$: \begin{equation} ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ...
6
votes
2answers
283 views

Ito, Stochastic Exponential and Girsanov

This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential. Whilst reading notes relating to Girsanov it is stated that the ...
4
votes
1answer
224 views

Closed form solution of PDE of Option Price

Let $V=V(S_t,t)$ be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ...
6
votes
0answers
97 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
4
votes
1answer
89 views

Why $W_{t}^3$ is not a martigale?(by Definition)

If $W_t$ be a wiener process then,how can i show that $W_{t}^{3}$ is not a martingale by definition?
4
votes
1answer
134 views

How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$

How can I calculate? \begin{align} Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right) \end{align} Thank you for your attention.
4
votes
1answer
167 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
3
votes
1answer
128 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
1
vote
1answer
212 views

Girsanov theorem in CMS convexity derivation

I am going through the derivation of CMS convexity from the notes of Lesniewski There is a transformation from $T_p$ forward measure to annuity measure $Q$ as $$ ...
1
vote
1answer
91 views

Stochastic calculus: what am I doing wrong?

it is just the computation of a second moment but however is creating debate !!... Can someone spot the error?
2
votes
2answers
152 views

Is the Brownian motion multiplication rule a definition or is it a theorem?

Is the Brownian motion multiplication rule a definition or is it a theorem? Refer to the highlight part of http://i.stack.imgur.com/doQuT.png where $dw_1(t)dw_1(t)=dt$
2
votes
0answers
110 views

Multivariate Itô's lemma

Hey guys I'm looking for worked examples who show how to apply Itô's lemma in several variables, starting from the very basics. Thank you in advance!
2
votes
0answers
60 views

Variance of a stochastic integral

Ok guys, I'm new to stochastic calculus and I did an exercise that I don't know if it is correct, so I need somebody with more experience to check if it is true. Compute the variance of the R.V. ...
5
votes
1answer
198 views

Multidimensional Ito's Lemma for Vector-Valued functions

Consider the vector of $n$ Ito processes $$ d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t $$ where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in ...
2
votes
2answers
131 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
88 views

Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
1
vote
2answers
106 views

Why does the short rate in the Hull White model follow a normal distribution?

Consider Hull White model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha ...
3
votes
3answers
152 views

For $B_t$ a Brownian motion what is the probability that $B_1>0$ and $B_2<0$?

Let $B_t$ be a Brownian Motion. What's the probability that $B_1>0$ and $B_2<0$?
3
votes
1answer
100 views

stochastic calculus - brownian motion

I don't know how to prove this : let be $X_t = \int_{0}^{t}\sigma_{u}dW_{u}$ where $\sigma_{t}$ is a predictable process. If $|\sigma_{t}| = c$ a.s. how can I prove that $X_{t}=c*\beta_{t}$ ...
1
vote
0answers
66 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + ...
2
votes
1answer
57 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
votes
1answer
55 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 ...
3
votes
2answers
137 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...
2
votes
1answer
66 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
1
vote
0answers
26 views

Stochastic Optimal Control for ratios

Do you know any good papers on methods of Stochastic Optimal Control and Hamilton-Jacobi-Bellman(HJB) for optimization of different ratios(Sharpe, M2, Sortino, Sterling, etc.)? Meaning that using ...
4
votes
6answers
598 views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
1
vote
1answer
27 views

Complete Multiperiod Binomial model

I have the following deifnition of a Complete multiperiod binomial model: A multi period binomial model can be called complete if every derivative security can be replicated by trading in the ...
2
votes
2answers
116 views

Using Black-Scholes to price a geometric average price call

Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway. I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...
2
votes
2answers
144 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: ...
4
votes
1answer
239 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
6
votes
2answers
1k views

Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
5
votes
0answers
83 views

Stochastic control (HJB) for wealth process involving stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ where $\theta_t$ is the worth of holding at time $t$ and $c_t$ is the consumption stream. ...
2
votes
1answer
69 views

What is the filtration described?

What is the filtration $(\mathfrak{F}_t)$ encircled below? Is it $(\mathfrak{F}_t) = (\sigma(W_t)) = (\sigma(\tilde{W_t})), t \in [0,T]$? Or is it $(\mathfrak{F}_t) = (\sigma(\hat{W_t})), t \in ...
5
votes
4answers
678 views

Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
9
votes
2answers
7k views

Worked examples of applying Ito's lemma

In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
3
votes
1answer
156 views

How to get Black Scholes' Geometric Brownian Motion differential form form the closed form?

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that: $$S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu dt + ...
3
votes
2answers
67 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
3
votes
1answer
457 views

Covariance matrix and Cholesky decomposition

I am simulating a spread option with stochastic volatility using Monte Carlo simulation. I have the positive-definite covariance matrix $$ \rho = \left( \begin{array}{cccc} 1 & \rho_{1,2} & ...
11
votes
4answers
853 views

Solving Path Integral Problem in Quantitative Finance using Computer

I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again. We know that for option value calculation, path ...
3
votes
2answers
246 views

Libor Market Model: numeraire change

I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
3
votes
1answer
184 views

Ho and lee derivation for short rates model

A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
5
votes
2answers
506 views

Malliavin Calculus

From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing. ...
2
votes
1answer
76 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 $S$ ...
5
votes
0answers
180 views

Integral-differential equation for forward rates

I am struggling in this question: Let $P(t,T)$ denote the price of a zero-coupon bond (with marturity at time $T$) at time $t \in [0,T]$. As usual, at time $t$ for maturity $T$, the forward rate is ...
7
votes
2answers
332 views

Why is this stochastic integral a martingale?

Suppose that: $W^*_t$ is a Wiener process under probability measure $\mathbb{P}^*$ and; $\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$. In my lecture notes, it says that $\tilde{S}_t$ is a ...
6
votes
1answer
240 views

Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$ ...
2
votes
2answers
206 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 ...
2
votes
1answer
95 views

Black Scholes model: condition of payout function

Given: Consider a two-asset, continuous time model (B,S) where $$dB_t = B_t r dt, \quad dS_t = S_t ( \mu dt + \sigma dW_t)$$ Clearly, the martingale deflator is: $$Y_t = e^{(-r - ...
7
votes
1answer
405 views

Girsanov Theorem and Quadratic Variation

Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation. Below is the ...
1
vote
1answer
171 views

FX Rate dynamics

Let's suppose USD/EUR price in USD follows a GBM with $$ dS_t = rS_tdt + \sigma S_tdW_t $$ What process does EUR/USD follow in EUR?