Questions tagged [proof]
The proof tag has no usage guidance.
51
questions
11
votes
2
answers
776
views
Is it possible to understand financial theory without mathematics?
I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a ...
- 337
8
votes
7
answers
2k
views
Proof that no trading system always wins
I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how the price of a futures moves. In a context where one can go long or ...
- 189
7
votes
1
answer
133
views
Show that $\frac{\partial c(t))}{\partial \sigma^2 }>0 \text{ if and only if } S(t)<Xe^{-r(r+\frac{1}{2} \sigma^2 )(T-t)}.$
Statement: if $c(t)$ is the price of the digital cash-or-nothing call option, then direct calculation (under Black-Scholes assumptions) shows that
$$\frac{\partial c(t))}{\partial \sigma^2 }>0 ...
- 840
6
votes
1
answer
491
views
Heat/Diffusion Equation
I am working on a problem where I have successfully reduced a version of Black Scholes to the Heat Equation and then shown the solution to be:
$$u(x,t)=\frac{1}{2\sqrt{t\pi}}\int_{-\infty}^\infty{f(\...
- 119
6
votes
1
answer
250
views
Parametric Stochastic Integral
I need help.
Defining the parametric stochastic integral
$$
F_t = \int_t^T\xi(t,s)g(s)ds
$$
$\\\\$
with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
- 61
5
votes
2
answers
2k
views
Relationship between Vega and Gamma in Black-Scholes model
my question is the following one: I don't manage to prove that, in Black-Scholes model, single-signed Gamma options have values that are monotonic in the volatility. I am looking for an exhaustive and ...
5
votes
1
answer
1k
views
Proof of arbitrage-free implied volatility surface in relation to local volatility surfaces
I'm looking for proof of the following statement:
"The existence of an arbitrage-free implied volatility surface is
equivalent to the existence of a well-defined local volatility
surface."
- 2,081
5
votes
1
answer
638
views
Prove arbitrage opportunity
The continuously compounded interest rate is $r$. The current price of the underlying asset is $S(0)$ and the forward price with delivery time in 1 year is $F(0,1)$. Short selling of the stock ...
- 432
5
votes
1
answer
1k
views
Proving that Absence of Arbitrage does not imply law of one price
I am trying to prove that the Absence of arbitrage statement (AOA) does not necessarily imply the law of one price (LOP). For the definitions of these concepts I am using Cochrane's book "Asset ...
- 163
5
votes
1
answer
2k
views
How to prove the "Law of one price" theorem?
There are two subparts to Fundamental Asset Pricing theorem.
The Law Of One Price (LOOP thereafter) holds if and only if there exists a state price vector.
In a market in which the LOOP holds, the ...
- 51
4
votes
1
answer
617
views
European Call price for an asset with mean reverting (Vasicek model) dynamics
Let's look at a stock with a mean reverting price dynamics:
$$dS_t = a(S-S_0)dt + \sigma dW_t$$
If we let $\sigma=0.25$ and $a=-0.5$ then the variance of this process is: $$Var(S_t) = 0.199\sim0.2$$
...
- 127
4
votes
1
answer
365
views
Expected Shortfall monotonicity
I have to show monotonicity for a more general case than the expected shortfall.
I have to show that
$E(X|X \geq a) \geq E(X|X \geq b), \forall a,b \in \mathbb{R}$ so that $a\geq b$ and $F_X(a-)<1$....
- 181
4
votes
1
answer
486
views
Ito isometry and the covariance of an Ito process
Let $(B_t)_{t \geq 0}$ et $(W_t)_{t \geq 0}$ be two independent Brownian motions and let $f: \mathbb{R} \rightarrow \mathbb{R}$ a deterministic function of time. We define the following process:
\...
- 2,436
3
votes
1
answer
2k
views
Proof behind solution for theta in Hull-White with time-dependent volatility and mean reversion?
I'm studying the following paper on Hull-White model calibration:
Hull-White paper
In this paper they study the general form of the HW model with time-dependent mean reversion and volatility:
$$dr(t) ...
- 75
3
votes
2
answers
135
views
Sums of random variables and independence
I'm having troubles with this proof:
Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean and unit standard deviation. For $(a_0, a_1, ..., a_r)$ a sequence of $r$ real numbers ...
- 676
3
votes
1
answer
125
views
Justification for substituting "Itô differentials"
I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
- 437
3
votes
0
answers
55
views
Martingale positive price process
I hope you can help me with this problem.
In my lecture notes, my professor stated that for a state price deflator $\phi\in L_{n+1}^2(P, F)$ (F being a filtration) and a strictly positive price ...
- 181
2
votes
2
answers
1k
views
Proof of Feller condition for CIR square root process. Any reference?
Could you please give me some reference for the proof of the so-called Feller condition as to a stochastic differential equation of the form:
$$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}dB_t\tag{1}$$
with $\...
2
votes
1
answer
358
views
Proof for ATM delta with Local col
I am looking at a time-homogeneous local volatility model where
ATM implied volatility equals ATM local volatility: $\sigma_{imp}(S_0)=\sigma_{local}(S_0)$
ATM IV Skew = half of LV slope
In general $\...
- 21
2
votes
1
answer
154
views
Proof about discounted zero coupon bond
Hey guys I am having trouble finishing this proof:
Proposition 5.1 Under the above assumptions, the process $r$ satisfies under $\mathbb{Q}$
$$
d r(t)=\left(b(t)+\sigma(t) \gamma(t)^{\top}\right) d t+\...
- 383
2
votes
0
answers
76
views
Filipovic: Where is it used that the world is deterministic
In this text (Damir Filipovic, Term-Structure Models, Springer, 2009) $P(t,T)$ denotes the price of a zero-coupon bond at time $t$ with maturity $T$. I cannot see where the proof uses the ...
- 548
2
votes
0
answers
188
views
Black-Litterman proof with P=I and Omega=tau*Sigma
Elsewhere on this site (link), Richard notes that
\begin{equation}
\Pi_{BL} = \frac{1}{2} \Pi + \frac{1}{2}Q,
\end{equation}
so long as we set $ P = I $ (where $I$ is the identity matrix) and $\Omega ...
- 21
1
vote
1
answer
2k
views
Deriving Delta Hedge error in the B-S setup (part 2)
In this paper paper page 16-19 by Davis
and this discussion
derivation of the hedging error in a black scholes setup,
the derivation of the delta hedging error in the Black Scholes model is discussed.
...
- 1,627
1
vote
1
answer
116
views
Proving Scaled Random Walk Approaches Normal Distribution
I'm reading Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve and I don't understand how he went from the equation on the left to the middle one. If it helps, this section is ...
- 113
1
vote
1
answer
2k
views
Trading over a Ornstein/AR process
For a OU/AR(1) process is there anyway to analytically calculated most probable period of time the process is likely to diverge from the average, before turning to converge.
Basically I am looking ...
- 233
1
vote
1
answer
288
views
Variance of Random Walk with Drift
For Gaussian random variables $\xi_t$ with mean $\mu_t$ and standard deviation $\sigma$, consider the random walk with initial condition $P_0=100$, such that
\begin{equation}
P_t=P_{t-1}(1+\xi_t).
\...
- 128
1
vote
1
answer
446
views
Prove Subadditivity - Entropic Value at Risk
Any insight in how to prove the following risk measure is subadditive?
$\rho_{1-\alpha}(X) = \inf_{z>0}\{z^{-1}\ln(\frac{E[e^{zX}]}{\alpha})\}$, with $\alpha \in ]0,1]$
I want to prove it is a ...
- 51
1
vote
1
answer
386
views
Spot-Forward Relationship - Proof
Does anyone know of a decent proof for the spot-forward relationship of a currency? I've been looking on Google for hours and I'm not getting anywhere. My lecture notes are useless in that they don't ...
- 33
1
vote
1
answer
80
views
Coefficients of univariate regressions equal to those in the multivariate regression
I am currently running fixed effect regressions with multiple dummy variables. These dummy variables are created by a grid of '1' '0':
...
- 154
1
vote
1
answer
186
views
Showing BM $W(s)$ is independent of $W(t)-W(s)$ [closed]
Consider $0\leq s<t$ where $t,s$ represent time index.
I want to show a Brownian motion $W(s)$ is independent of $W(t)-W(s)$.
Specifically, show that $E[W(s)(W(t)-W(s))]=0$
Proof:
Writing $W(s)$...
- 323
1
vote
2
answers
404
views
How can I prove that the sum of two log-normal variable is not log-normal?
I am looking for an analytical proof, that the sum of two log normal random variables is not log-normal. Couldn't find it anywhere, does somebody know where to find it or know how to do it?
- 177
1
vote
1
answer
1k
views
Proving that the $\Delta$ of a call on a futures contract under the B-S model is $N(d_1)$
The author of my textbook says that the $\Delta$ of a call on a futures contract is $N(d_1)$ and not $e^{-rT}N(d_1)$. I wasn't convinced, so I tried to prove this.
Let $F = F_{0, T}(S) = S_0e^{(r - \...
- 339
1
vote
1
answer
474
views
arbitrage proof question
prove the condition $D<R<U$ is equivalent to the absence of arbitrage:
R = risk free investment rate of return. U and D are returns corresponding to the upward/downward price movements of a ...
- 432
1
vote
1
answer
287
views
Proof of Calendar-Spread-Inequality
The Calendar-Spread-Inequality compares the prices of two European Call Options on the same underlying non-dividend-paying stock, but with different maturities $T_1<T_2$. Denote the value of a call ...
- 133
1
vote
0
answers
148
views
HJM Model proofs
I am looking for a source that possibly has the proofs for the material in the first paper on the HJM model Heath, David, et al. “Bond Pricing and the Term Structure of Interest Rates: A New ...
- 153
1
vote
0
answers
2k
views
How to derive the Greek theta from Black-Scholes solution formula?
Which are the steps to compute the theta greek from the BS solution:
$$c(t, x) = xN(d_+(T-t,x)) - K e ^{-r(T-t)}N(d_-(T-t,x))$$
with:
$$ d_\pm (T-t, x) = \dfrac{1}{\sigma \sqrt{T-t}} \left[ \ln \...
- 111
0
votes
1
answer
61
views
Show that the following result holds true for the variance of the return of a portfolio of shares
Start with a portfolio $p$ of $n$ shares, each with weight $x_i = \dfrac{1}{n}$ (for $i$ ranging from $1$ to $n$, discretely). Its return is given by:
$$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\...
0
votes
1
answer
88
views
Proof verification : risk free rate [closed]
I want to prove that
$$r_t = \theta + (r_0 -\theta)e^{-kt}$$
satisfies $$dr_t = k(\theta-r_t)dt, \ r(0) = r_0$$
I have
\begin{split}\frac{1}{\theta - r_t} dr_t = kdt \Rightarrow & \int_0^t \frac{1}...
- 113
0
votes
1
answer
199
views
Prove that a determinant in markowitz method derivation is greater than zero
I want to prove that the following determinant, that appears in the markowitz method of portfolio allocation is greater than zero.
($\mu$ is the vector of returns and $\sum$ is the covariance matrix)
- 853
0
votes
1
answer
939
views
how to derive the cost of carry formula
Can anyone explain why the cost of carry formula looks like this:
$$F_0 = S_0 \cdot e^{(c-y)T}$$
,where $S_0$ equals the spot price when $T=0$, i.e. today. $c$ denotes the cost of carry and $y$ the ...
- 41
0
votes
1
answer
117
views
Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded
How can I prove that the function
$$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$
is bounded/unbounded?
$\sigma_i\left(t\right)$ is the ...
- 353
0
votes
0
answers
47
views
No arbitrage argument for the price process of a forward contract
I was reading the book Stochastic Calculus for Finance II by Shreve and I read the proof that the forward price for the underlying $S$ at time $t$ with maturity $T$ is given by
$$
For_S(t,T) = \frac{S(...
- 195
0
votes
0
answers
84
views
Price of financial assets at $t=0$ in Black-Scholes framework
Given the share price equation
$$
dS_t=rS_tdt+\sigma S_tdW_t
$$
working in the framework of Black-Scholes model, find the price at $t=0$ of the following two financial assets:
(a) The asset pays at $t=...
- 101
0
votes
0
answers
128
views
Understanding arbitrage, defined as a series of cash flows
I'm currently catching up on material presented in the edX-MIT course Foundations of Mondern Finance 1, in which they present a definition of arbitrage that doesn't quite make sense to me. Informally, ...
- 101
0
votes
0
answers
71
views
Can we proof the boundary condition for the Black Scholes derived from a replicating Portfolio?
So for Black Scholes we know that the PDE is the follwing: ${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}=rV-rS{\frac {\partial V}{\partial S}}...
- 28
0
votes
0
answers
278
views
CVA formula proof
I'm struggling to prove the CVA formula in this paper.
Equation (3) is the result of computing the expectation of formula (1). Could you please show me how to prove that?
- 9
-1
votes
1
answer
1k
views
Survival probabilities starting from CDS spreads
How is that possible to get survival probabilities starting from CDS spread?
Could you please provide me with a demonstration?
What is more, is that true that CDS Zero type is necessary so as to get ...
-3
votes
2
answers
242
views
Proof: Brownian Motion Path Continious with Probability One [closed]
How can one show that the paths of the standard Wiener process are continuous in $T$ with
probability one? Can we just proof it with the assumption of independence ? Thank You in advance!
- 28
-3
votes
1
answer
121
views
Proof that adding some quantity of stocks in a portfolio of option does not change the portfolio Gamma
I would like to proof mathematically and intuitively that adding some quantity of underlying to a portfolio of option does not change the overall gamma.
Can you help me?
- 111
-3
votes
1
answer
105
views
Can someone prove (or disprove) this assertion about the normal distribution? [closed]
Let $X$ be distributed as a $Normal (\mu, \sigma^2)$. Then for a fixed $\mu$ it is always the case that:
\begin{equation}
\frac{90th quantile-10th quantile}{\sigma}=constant \quad \forall \sigma>0
...
- 175