# Questions tagged [proof]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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."
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $$Var(S_t) = 0.199\sim0.2$$ ...
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### 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 $F_X(a-)<1$....
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### 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: \...
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### 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 ...
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### 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?
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### 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 ...
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### 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!
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### 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?
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Let $X$ be distributed as a $Normal (\mu, \sigma^2)$. Then for a fixed $\mu$ it is always the case that: \frac{90th quantile-10th quantile}{\sigma}=constant \quad \forall \sigma>0 ...