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Questions tagged [proof]

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0
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2answers
44 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 ...
2
votes
1answer
201 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) ...
3
votes
1answer
139 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 $\...
1
vote
1answer
152 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. ...
4
votes
1answer
127 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: $$Var(S_t) = 0.199\sim0.2$$ ...
1
vote
1answer
168 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 ...
-4
votes
1answer
92 views

Derivation of arithmetic variation of a portfolio over multiple periods [closed]

I am very confused on how to derive the attached equation (15). Would someone be kind enough to walk me through the proof?
1
vote
0answers
878 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 \...
5
votes
1answer
419 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."
0
votes
1answer
153 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)
1
vote
2answers
187 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?
1
vote
1answer
513 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 - \...
0
votes
1answer
91 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 ...
1
vote
1answer
169 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 ...
5
votes
1answer
266 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 ...
7
votes
7answers
1k 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 ...
2
votes
2answers
109 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 ...
5
votes
1answer
974 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 ...
5
votes
1answer
788 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 ...
0
votes
0answers
192 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?
1
vote
1answer
704 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 ...
5
votes
1answer
356 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(\...
10
votes
2answers
720 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 ...