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Under the Black-Scholes framework, you can calculate the implied volatility, given the option's price, underlying's price, time to maturity and the risk free rate. To calculate the implied volatility you have to use a root finding method, since there is not a closed form of the inverse of the B-S option pricing equation for volatility. In the real world ...


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There is no skew/smile for forward contracts, but there is for options based on it (caps, floors, swaptions, options on futures). Then it would be the simple Black Formula that should be used in theory (using futures price). The mere existence of the smile is an indicator that the model is fundamentally flawed and it is important to apply a correction.


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As Black Scholes model, you can assume a two-sided Truncated Normal Distribution as riskneutral density $f(x)$ (with $\mu=0,\sigma=T-t$) for the returns and then price the option payoff $H$ as usual by: $$V_t^H=e^{-r(T-t)}E(H_T|F_t)=e^{-r(T-t)}\int_{-X}^{X}H(S_te^{r(T-t)+\sigma W_{T-t}})f(W_{T-t})dW_{T-t}$$ The integral must likely be calculated ...


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The condition $$ud=1\text{, or equivalently }u=1/d$$ is necessary to ensure convergence of the Binomial tree's mean $\mu$ and standard deviation $\sigma$ to nonfinite values when $n$ (number of steps) goes to infinity. Cox-Rubinstein-Ross showed in their famous paper, that to achieve this, we must have: $$u=e^{\sigma\sqrt{t/n}}\text{, ...


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These options can be priced by adding an early exercise premium value to the intrinsic value: http://www.statistics.nus.edu.sg/~stalimtw/PDF/lb-float.pdf



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