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1

Both integrand are different. One includes $\phi(u-i)$ and the other one simply $\phi(u)$. As one expects, in the Black-Scholes model, $\Pi_1$ and $\Pi_2$ collaps to $\Phi(d_1)$ and $\Phi(d_2)$. Note firstly that if $X\sim N(\mu,\sigma^2)$, then \begin{align*} \phi_X(u) &= e^{iu\mu-\frac{1}{2}\sigma^2u^2}, \\ \phi_X(u-1) &=\phi_X(u) e^{\mu+\frac{1}{...


0

For Java, combining @vjond answer (as the initial starting estimate for implied Volatility), with a basic Cumulative Density Function for Normal distribution (CDN), and the Black Scholes Model applied to Newton Raphson, below is a basic Implied Volatility calculation for Java : For the NR code (C), plz refer http://finance.bi.no/~bernt/gcc_prog/recipes/...


-1

You're replicating the value of the long call. As with your small move example, other things equal, if the underlying doesn't move, your position loses money (as an actual call would) due to theta. It works exactly as it should, the position itself just loses money as a result.


2

Let $F$ be a claim (an option), then in the Black-Scholes model and assuming zero interest rates the SDE for the claim is $$ dF = \frac{\sigma S}{F} F_S F dW $$ where the subscript $S$ denotes the partial derivative with respect to $S$. So the instantaneous volatility of $F$ is $$ \frac{\sigma S}{F} F_S $$ The dollar gamma is equal to $K^2 C_{KK}$, where $...


1

This won't transform a version of the heat equation that can be solved analytically. The extra term results in the time-integral of Geometric Brownian Motion, which has no known analytical transform. The sum (or equally, the arithmetic average) of lognormally distributed variables is believe to result in a Bessel process that converges to an inverse gamma ...


1

The use of forwards is just another method to look at the underlying. The Black-Scholes options model utilizes Spot and handles the carry as an interest rate in the model. On the other hand the Black Model uses forwards instead. Since the forward price would take into account the carry, both models should yield the same result if one is accounting for all ...


1

Your question is using some terminology incorrectly. Forward volatility refers to the volatility realized from t1 to t2 given that it's currently t0 and t0 < t1 < t2. What you are talking about is whether the moneyness of an option is expressed in relative to the spot or relative to the forward. Which parametrization to pick is a choice; as long as ...


2

Insurers do use derivative pricing models such as Black-Scholes to price the sort of guarantees you describe. As far as I know, this used to be known as the "replication method" in the industry jargon, and it allows insurers to price guarantees in a market-consistent manner, hence enabling them to efficiently hedge them with traded instruments. In particular,...


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