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4

Some option prices can't be converted to volatility. E.g. A bid for an in-the-money call which is below its intrinsic value. So sometimes NaN is a valid answer. Best way to handle it is to do precursory checks before going down to the search loop.


3

Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these have been relaxed.) In particular, there are two schools of criticism, one is that time homogeneity makes little sense in the real market. In fact, the Recovery ...


3

There are lots of papers online and here are a few I would suggest math.umn riskworx G. Dimitroff, J. de Kock Nowak, Sibetz I you have matlab there is an step step example to calibrate SABR model. Since it uses the financial toolbox of matlab for a few functions I dont think you can replicate it in any other language. There must be C++ code available ...


3

This goes back to the so-called First Fundamental Theorem of Asset Pricing saying that markets are arbitrage free if and only if there exists at least an equivalent risk neutral measure. So the reason why we are using risk neutral measures to price options is because it allows us to represent discounted stock diffusions as martingales and therefore express ...


2

The OTM binary is not increasing with volatility. Simply plot the price as a function of implied volatility. eg take S = 0.99 K = 1 r=0 T =1 and $\sigma$ vary upwards in steps of 0.01 starting at 0. It peaks around 0.04. Differentiating $d_2$ with respect to $\sigma$ makes this behaviour obvious.


2

All option pricing formulas except this one and this one use some sort of historical volatility . I can't see how you can use the Black Sholes framework and not use some sort of historical volatility uses an order book uses geometric shapes and volume


2

If you want to estimate volatility from historical data, the only best linear unbiased estimator (BLUE) is $$\sigma=\sqrt{\frac{1}{T-1}\sum_{i=1}^T (r_i-E(r_i))^2}$$ Any other estimator will hence either be biased or not consistent. Another approach could be to estimate volatility via a GARCH model, which has shown good empirical results in the past. It is ...


2

If both options are out of the money and volatility is zero then both are worth zero. If there is a positive probability that the lower strike option pays off then the inequality is indeed strict.


1

When pricing FX options, the underlying is the spot or forward exchange rate. The foreign currency is analogous to a stock where the owner of the foreign currency receives a "dividend yield" equal to the risk-free rate in the foreign currency.


1

The unit for volatility is the same unit as the random variable. In the case of the index, yes, it is the index points. The units of the variance is the unit of the random variable squared. Standard deviation is the square root of variance and has the same units as the random variable.


1

if you let the implied vol depend on K you get two terms the first is $N(d_2) $ but you get a correction term which is the slope times the vega $$ \frac{\partial C}{\partial \sigma} \frac{\partial \sigma}{\partial K}.$$ (see eg my book)


1

I don't think that it has a name on its own, but you can write $$ (S_T - K + K)\,1_{S_T>K} = (S_T-K)_+ + K\,1_{S_T>K} $$ so it's a 1 call plus K binary calls. Binary are hard to hedge, the payoff looks like _|‾, going sharply from out-of-the-money to in-the-money. The delta changes fast and it's difficult to hedge the position. In practice, you give ...


1

There are two approaches. Price call and put options with various strikes. Plot their BS implied volatilities. Find the slope of the graph. Price a call and digital call with the requisite strike. Compute the implied volatility of the call. Use the fact that $ DC(model) = DC(BS) - skew \times callvega,$ to solve for the skew. (See eg Section 7.7 of my ...


1

For any data that is not strictly tabular and unchanging in schema, you should rule out SQL solutions. Option pricing fits that description in my experience, because high-liquidity stocks, currencies, or bonds, will have a far bigger set of strikes and maturities than lower liquidity instruments. Thus in a relational database you will have to have columns ...



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