All Questions
Tagged with finance partial-differential-equations
55
questions
4
votes
1
answer
73
views
Do inequalities in boundary conditions imply inequalities in solutions to a PDE?
I'm working on an unassessed course problem to show the solution $p$ to a PDE satisfies an inequality. I think I can show that of the 2 boundary conditions of the PDE, $p$ is equal to another solution ...
- 2,281
2
votes
0
answers
328
views
Are the Black-Scholes equation models still considered accurate predictors for asset pricing? [closed]
Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could.
I was talking ...
- 2,531
0
votes
0
answers
59
views
Approximation and Transformations for PDE
Here is the Link of the paper for more informations.
There are two questions I really got stucked,
(i) The equation (20) says that $\frac{1}{K(1-K)}$ is aprroximated by Taylor's series around $0.5$
$\...
- 33
1
vote
1
answer
193
views
On the transformation of Black-Scholes equation in to a dimensionless system
Let me show the supposition extracted from S.-P. Zhu et al. / Computers and Mathematics with Applications 64 (2012),
Consider the system of coupled Black-Scholes equations (Regime-switching) for the ...
- 2,184
1
vote
1
answer
114
views
Differentiating the risk-neutral price of a European call
(Black-Scholes formula) The risk-neutral price of a European call is $$C_t = S_tN(d_1) - e^{r\tau}KN(d_2)$$ where $$d_1 = \frac{log(\frac{S_t}{K}) + (r + \frac{1}{2}\sigma^2)}{\sigma\sqrt{\tau}}$$ and ...
- 77
1
vote
0
answers
212
views
Forward price change of variables in Black-Scholes Model
Suppose that $V(S, t)$ satisfies the Black-Scholes PDE:
$$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma ^2 S^2 \frac{\partial ^2V}{\partial S^2} + (r-q)S \frac{\partial V}{\partial S} - rV = 0, \...
- 167
2
votes
0
answers
75
views
On the existence and uniqueness of solution for a regime-switching Black-Scholes problem (coupled parabolic problems)
I am currently working on a regime-switching Black-Scholes model and am having trouble determining the existence and uniqueness of a solution for the problem. Specifically, I am interested in finding ...
- 775
0
votes
1
answer
122
views
On the time value asymptotic behaviour of call option in the generalized Black-Scholes model
In the context of generalised Black-Scholes models,
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2(S, t) S^2 \frac{\partial^2 V}{\partial S^2}+(r(t)-D(t)) S \frac{\partial V}{\partial S}-r(t) V=...
- 2,184
2
votes
1
answer
155
views
Prove that $V(t)=e^{-r(T-t)} \mathbb E\left[S_{t}\right]$ satisfies the Black–Scholes PDE
Let us consider the geometric Brownian motion:
$$
d S_{t}=\mu S_{t} d t +\sigma S_{t} d B_{t}
$$
where $\mu$ is the drift, $\sigma \in \mathbb{R}^{+}$ is the volatility and $B_{t}$ is the Wiener ...
- 7,880
3
votes
1
answer
198
views
Technical difficulties with degenerate PDEs
Crossposted at Quantitative Finance SE
I have seen lot of discussions in this Math. S.E. platform about 'degenerate partial differential equations'. But I still unclear about the 'technical ...
- 775
2
votes
1
answer
211
views
Black Scholes PDE in forward log space
In BS world, we have the stock process in log space $dS_t=(r-\frac{1}{2}\sigma^2)dt+\sigma dW$. Let's say we want to price $f(t,x)=\mathbb{E}_{t,x}[h(S(T)]$. Using Feynman-Kac, we get
\begin{equation}
...
- 43
4
votes
0
answers
122
views
Solve this PDE without guessing and checking
I am trying to solve the Hull-White PDE. Let $\theta : \mathbb{R} \to \mathbb{R}$ be a known deterministic function and $\lambda$, $\eta \in \mathbb{R}$. The PDE is then given by
$$f_t(t, r) + \Big(\...
- 75
2
votes
1
answer
679
views
$(d_2/\sigma) = (1-d_1)$ in Black-Scholes model
I'm trying to derive Vanna from the Black-Scholes Model (BSM) equation, but had a hook up on one of the manipulations in the formula.
Vanna, also referred to as DvegaDspot and DdeltaDvol, is a second ...
0
votes
0
answers
158
views
Trying to Solve the Black Scholes PDE with the Green's Function
I have finished the transformation into the Heat Equation. And I am now at the point of establishing the initial conditions. The article I read said the $\max(S-K,0)$ is now the initial condition ...
- 141
1
vote
1
answer
119
views
Asymptotic behavior of European options [closed]
In some of the numerical work on Black-Scholes generalized models, the boundary conditions on the truncated domain taken from the asymptotic behaviors of European call options, which are given by
$$\...
- 2,184