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1) A straigthforward application is to price any complex payoff at maturity using this. By that I mean a payoff that is such that the price of the option is $$P = e^{-r(T-t)}E[f(S_T)]$$ Which you can then calculate by integrating $f(S_T)$ w.r.t. to your density. One of the challenges though is to have a proper marks and inter/extrapolation for the ...

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In effect, you are wondering whether to price this option on risk-free probability distributions (B-S drift $r_f$), or real-world ones (B-S drift $\mu$, however calibrated) One cannot short the mutual fund, so the argument for using risk-free is weakened. But, there are various economic equilibrium arguments why using it may still be OK. If you use the ...

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Behavioral Finance is a wide topic, which I believe is still today underestimated by many financial professionals. How can it be used by quants? Well, in portfolio optimization it can be used "as an overlay" in the form of constraints where the optimal portfolio can not be too different from the current portfolio, because clients have behavioral biases ...

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Look the first answer of this thread: How to derive the implied probability distribution from B-S volatilities? Also many papers in Dupire volatility have your formula derivation. For example, look at (10) in http://www.javaquant.net/papers/DupireLocalVolatility.pdf

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Try the VG model by Madan, Carr & Chang.

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To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi$$ where $f(\phi)$ is the characteristic function of the standard normal distribution: $$... 2 the answer for calculating the prices can be found here - see chapter: Black–Scholes valuation ;) The put-call parity in that case is pretty straight forward: P=Se^{-qT}-C. Using the results presented on the Wikipedia page in the aforementioned section this can be proved as follows P=Se^{-qT}-C =Se^{-qT}-Se^{-qT}\Phi(d_1) ... 0 First let's recapitulate: The market is free of arbitrage if (and only if) there exists a martingale measure; The market is complete if and only if the martingale measure is unique; In an arbitrage-free market, not necessarily complete, the price of any attainable claim is uniquely given, either by the value of the associated replicating strategy, or by ... 1 Hint The future world has 4 states: (0.5,0.5), (2,0.5), (0.5,2), (2,2). You have 4 instruments - cash, each stock, and an option they are both \2 which is traded. Take x,y,z,w of each and match the portfolio to the price of the option in each market state. You get 4 equations and 4 unknowns, solve, and supposedly you get a unique solution, which ... 2 There is no difference in information, though the fitting algorithm may increase in complexity. First note that in practice you never have an entire curve or surface of prices C(K,T) of any kind of option. You only have a finite number of observations and even those typically have a bid and an offer. I would therefore argue that the correct picture of ... 1 At long maturities, the real problem tends more to be model error than volatility estimation: over that kind of time period most companies undergo significant capital structure changes, for which there are very few models. 2 Assume the stock pays no dividends before 100 dollars is hit. Interest rates can be arbitrary. Buy 1/100 of a share for 75 cents. Hold until 100 is hit then sell. The payoff of 1 dollar is replicated for an upfront cost of 75 cents. The arbitrage-free value of the option is 75 cents. 2 The consensus nowadays is that stable distributions are not a well fit, although they do possess heavy tails. In particular Cauchy has too fat tails. The reasons for this are disparate, however the first that comes to mind is that empirically longer horizons show a decrease in tail thickness, approaching normality for 1-year returns (although this has been ... 1 I would put it differently. Modelling variance in an additive way (an OU process is in some regard additive) is more natural than e.g. a gemetric Brownian motion model (which on the other hand does not model mean reversion). Volatility as it is a square-root is by no means additive. Let (B_t)_{t \ge 0} be Brownian motion then we have$$ VAR(B_t) = t = ...

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The second equation where you would be using variance instead of standard deviation won't provide "meaningful" paths. The reason is: variance has no meaning/interpretation in space. If you consider a normal distribution of stock returns the standard deviation is actually a number that tells you the difference between the expected value and some quantile. ...

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This is perhaps not a concrete solution to your problem but the space in the comments is limited :) In your setupt you are not actually pricing an option on a basket but on a dynamically allocated portfolio. Thus conventional pricing and hedging approaches won't apply. Also you are underestimating porfolio optimization algarithms. To find an optimal ...

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I would define the weights $w_1,\ldots,w_n$ as whatever number you want and the basket given by $$B_t = \sum_{i=1}^n \frac{w_i}{W}S_t^{(i)}\ , \qquad W = \sum_{i=1}^nw_i$$ so the weights always sum to one. This doesn't make much sense, however, because you are changing the product, not a market variable. This meaning that when the weights change, the ...

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The Price of an American option may contain information on the expected behaviour of it's holder. When might he/she exercise the option ? Contrary to European options that don't. Thus when you are primarily interested in "reconstructing" the transition density - I would stick with the European-Option-Prices. If however you were to price path dependant ...

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No the discounting factor that you use for backward induction won't change. (confer here Chapter IV) This is only seems confusiong due to the mathematical formulation. Introducing continuous dividends basically adjusts your stock price (down) by discoutning the divididend (for it is paid out and thus dicreases the stock value). Your "risk-free" stock value ...

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you get a volatility skew by imposing a neumann-like barrier if market makers think a stock won't surpass a certain threshold, a skew is inevitable if one were to match the pricing under a barrier with the BS formula https://en.wikipedia.org/wiki/User:Barrieroption/sandbox

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