0
votes
1answer
63 views

Derivation of Black-Scholes equation by riskless portfolio

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
5
votes
1answer
76 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
0
votes
0answers
9 views

Time homogeneous asset dynamics model

I'm studying asset process. As i know, Black scholes model and CEV model is time homogeneous diffusion model. Are there time homogeneous model ???
0
votes
1answer
30 views

Dynamics of short rate in HJM

According to a simplified HJM framework, we have: Forward Rate: $f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds$, where $W_t$ is brownian motion. Dynamics of forward rate: ...
0
votes
2answers
26 views

Deriving Geometric Brownian Motion's solution?

The Black Scholes model assumes the following underlying dynamics, known as Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: ...
0
votes
0answers
55 views

Second Fundamental Theorem of Asset Pricing

It seems that there is a step missing in the proof of the second Fundamental Theorem of Asset Pricing in Shreve's Stochastic Calculus for Finance II: Does anyone know how to show the following: If ...
0
votes
0answers
25 views

Proposition from Oksendal Stochastic Calculus

I am reading Oksendal's Malliavin Calculus with applications to Finance and there is a part that I do not understand. First we have a proposition which is fine: If $\zeta_1$,$\zeta_2$,... are ...
0
votes
0answers
44 views

Application of Ito's formula

I recently learned about Ito's formula and integral and now i have to do the following exercise, but I actually don't really know, how to start: Apply Ito's formula to prove that $$Z_t=exp(\sigma ...
0
votes
0answers
23 views

Parameter estimation using characteristic function

Is it possible to do parameter fitting using log-returns data & the characteristic function(CF) in Matlab? I have been trying it on the Variance Gamma Scaled Self-Decompasable (VGSSD) model CF for ...
0
votes
0answers
45 views

Girsanov's theorem and simulation of bond prices

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma ...
1
vote
0answers
35 views

Why does the price term in Vega disappear for a European call option?

In my course, I have been asked to prove a number of statements about "the Greeks" from the Black-Scholes model for pricing a European call option with no dividends and a strike price of $K$. One of ...
1
vote
1answer
38 views

Writing $A(t)=1+S_1S_2^{-1}$ as an Ito diffusion process.

Let $W$ be a Wiener process/Brownian motian and let $$ \begin{align} \mathrm{d}S_1 &= 2S_1(t)dt +3S_1(t) dW\\ \mathrm{d}S_2 &= 4S_2(t)dt +5S_2(t) dW \end{align} $$ Now I'd like to write ...
0
votes
0answers
82 views

Jump diffusion process with sum of Poisson processes a martingale?

Hi Mathematics community, assume you have dynamics of a jump diffusion process consisting of a Brownian motion and a sum of compensated (not necessarily independent) Poisson processes, i.e. ...
0
votes
0answers
25 views

Weak convergence of discretization scheme with correction

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
1
vote
1answer
84 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
1
vote
0answers
24 views

How to find sigma-algebra over omega3 generated by the log-return ln(S2/S1) and ln(S3/S2)?

I calculated {S2/S1} = {u, u*u/d, d, d*d/u}, and then get ln(S2/S1)= {ln(u),ln(u*u/d), ln(d),ln( d*d/u)}. I am not sure my way of doing this question is right, because i m confuse about how to get ...
1
vote
1answer
738 views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option princing in Black-Scholes model in pages 93-99, The poof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
1
vote
0answers
23 views

The impact of jump on the returns of portfolio and asset pricing

There exsits jumps in financial market. What will be the impact of jump on the returns of portfolio and asset pricing? Please explain it both academically and plainly. If you can give some excellent ...
1
vote
1answer
128 views

american put option

For a perpetual american put option $v(s)$, satisfies the following problem: $$\frac12\sigma^2S^2\frac{\mathrm d^2V}{\mathrm dS^2}+(r-D)S\frac{\mathrm dV}{\mathrm dS} - rV = 0\quad\text{for ...
2
votes
1answer
131 views

Black scholes model type

I want to study the following market: $$S_1(t)=S_1(t)(\mu_1dt + \sigma_1dW_1(t))$$ $$S_2(t)=S_2(t)(\mu_2dt+\sigma_2dW_2(t))$$ for $t\in [0,T]$, constants $\mu_i,\sigma_i$, initial values ...
1
vote
0answers
303 views

Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ ...