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Expected volatility in the underlying price over the life of the option is a major component of the BSM option pricing model. When you calculate the volatility based on the current market price, you're figuring out what the market thinks the volatility would be, that's why it's called implied volatility. So to answer your question, you can either assume a ...

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$E_0[Y_{\lambda,t}] = 1\,\, \forall t$, hence $Y_t$ is a martingale. Hint: Look at the arithmetic moments section of this wiki page on lognormal distribution

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We define the process $Y_t=Y(t,S_t)$ as follows: $$Y_t=\left(\frac{S_t}{S_0}\right)^\lambda \exp\left\{-\left(r\lambda-\lambda(1-\lambda)\frac{\sigma^2}{2}\right)t\right\}$$ Let: $$\alpha=\lambda\left(r-(1-\lambda)\frac{\sigma^2}{2}\right)$$ Then by Itô's Lemma: \text{d}Y_t=-\alpha Y_t\text{d}t+\frac{\lambda}{S_t}Y_t\text{d}S_t+\frac{1}{2}\frac{\lambda(\... 1 Under the condition r=\frac{\sigma^2}{2}, it is true that S_t = S_0e^{\sigma W_t}. Since \begin{align*} E\Big( S_T - \min_{0 \le t \le T} S_t\Big) = E\big( S_T\big) - E\Big(\min_{0 \le t \le T} S_t\Big), \end{align*} what you need is the expectation E\big(\min_{0 \le t \le T} S_t\big). Note that \begin{align*} \min_{0 \le t \le T} S_t = S_0e^{\sigma \... 4 I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ... 2 The standard BS spot delta is a quantity in % of foreign currency (CCY1). The actual hedge quantity must be adjusted if the premium is paid in CCY1. FX options are special in that regards since for a stock option you wouldn't pay in shares typically. Example: we are short EURUSD with 1MM EUR notional, the option premium received is 73669 EUR. Say delta is ... 1 The market will quote Call and Put options prices within a bid-ask spread. In order to imply the volatility, one may choose to use the bid, the ask, or the mid. Although the mid is a better idea in general, there is no right choice. The point is that there is always a spread in the implied volatility. Now, the Put-Call parity only holds within the a spread. ... 4 Only constrained to be <1 in the simplified Black-Scholes setting with zero cost of carry on the underlying. In the more realistic and common setting where the cost of carry of the underlying is higher than the discounting rate, then it is entirely possible for a call to have a delta > 1. This is the case because your future costs are proportional to ... 3 The delta of a European call is: \Delta(call)=N(d_1) $$where N is the cumulative probability function which return value between 0 and 1. Therefore, for a traditional option, your \Delta cannot be greater than 1. 4 Intuitively, in a (log)-space homogenous diffusion model$$ S_t \propto S_0, \forall t \geq 0 $$such that implied volatilities will only depend on the moneyness level and not on the absolute spot level, which is precisely the definition of sticky delta. Mathematically, consider a (log)-space homogeneous diffusion model (be it stochastic or not)$$ \frac{...

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