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We know the market model is arbitrage free if and only if there exists a martingale measure $Q$, also the Binomial Model is free of arbitrage if and only if $d\le 1+R\le u\,\,$ (Arbitrage Theory in Continuous Time).It is easy to calculate the martingale probabilities.This Condition is equivalent to saying that $1 + R$ is a convex combination of $u$ and $d$ , ...


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well solve for the value of $q$ that makes the value of the stock divided by the bond be a martingale. You will find that only one value does so. It is the one you posted. If you then define the discounted value of an option to be its expectation of the discounted pay-off, its discounted value is a martingale. So the discounted value of everything is a ...


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If you can get your hands on a physical copy of by Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (Springer Finance) by S. Shreve, I strongly encourage it. If you can't, here is an electronic document which adresses exactly the same issues in the first chapters (and a little bit more in the following).


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I assume this is a plot of option value versus price of the underlying. The only case where it ought to be symmetric is if the pdf of the underlying is symmetric eg normally distributed. I'm guessing your chart assumes a lognormal underlying, which is a non symmetric pdf, so the graph is non symmetric.


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This formula is used for replication of certain payoffs, for example, the log-payoff in Variance replication using options. The value of $\kappa$ can be set to any number, for example, $\kappa=E(S_T)$. This is a decomposition of the payoff, which is not a valuation of the payoff itself, and then further valuation is still needed. For example, based on the ...


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Your valuation is NOT for the knock-out option that you have specified. Let \begin{align*} \tau = \inf\{t \mid 0 \le t \le T, S_t \ge L\}. \end{align*} Here, we set the infimum of an empty set to $\infty$. Then, the payoff of the knock-out option is of the form \begin{align*} (S_T-K)^+ 1_{\tau = \infty}. \end{align*} Under the Black-Scholes setting, this ...


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By definition, the payoff of a log-contract of maturity $T$ writes $$ \phi(S_T) = \ln\left(\frac{S_T}{S_0}\right) $$ Let $\Pi_t$ denote the $t$-value of such a contingent claim. We are interested in the price at $t=0$, best known as the option premium. Theory tells us that the latter premium can be computed as $$ \Pi_0 = e^{-rT} E^{\mathbb{Q}} \left[ ...


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Time $T$ boundary condition is correct $u(T,x)=(x-K_1)^+-(x-K_2)^+$. Time $x\to 0$ boundary condition is known and is equal to $0$. Time $x\to\infty$ boundary condition is also known and is correct $\lim_{x\to\infty}u(t,x)=(K_2-K_1)e^{-r(T-t)}.$ You need to be precise if you want your boundary be "absorbing" or "reflecting".


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Yes, your broker could have used one or combination of many factors: estimated volatility surface from historical returns of your target index, historical returns of similar indexes, implied volatility of similar indexes, existing inventory,etc. Check out these two approaches to deriving surfaces from returns starting slide 14


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Please read "Volatility's Impact On Market Returns" at http://www.investopedia.com/articles/financial-theory/08/volatility.asp. It is important to remember that VIX is a volatility index comprised of options and not stocks. It predicts volatility of future prices. It is not a measure of the present stock market. For a more thorough understanding see the ...


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This Quandl Page provides you the informations you need: a lot of programming languages and other tools are linked to Quandl.


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Thank you for your answer @MarkJoshi. I followed you advice and achieved in deriving the approximation formula. However, I can not fully understand why the fact that Black's formula is linear in $\sigma$ for ATM strikes causes the Rebonato approximation only to be accurate for ATM strikes and not OTM and ITM strikes. I would be grateful if somebody can ...


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Local vol model gives a "too shallow" forward skew. Derivatives of which the price are depending on the forward skew will be mispriced. If i remember correctly, Hagan's paper


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In order to define option price we should follow Black Scholes construction to construct riskless portfolio at t then to state that instantaneous rate of return of this portfolio equal risk free rate r ( t ) where r is a random on [ t , t + dt ] interval. We actually then arrive at the problem which could not be embedded in BS pricing world.


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it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$ \frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A, $$ with $A$ the annuity is remarkably good. One way of deducing these formulas is to do an asymptotic/Taylor expansion about $\sigma=0.$


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When you say 'overprice' I assume you mean model price > market price. In my experience this is true for all reasonable models. It's due to excessive supply of the Bermudan structure in the market.



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