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The two formulations seem to be exactly the same. If I take the equations from the first method: $w_{D1}*\Delta_{D1} + w_{D2}*\Delta_{D2} = -2$ $w_{D1}*\Gamma_{D1} + w_{D2}*\Gamma_{D2} = -3$ And substitute for delta and gamma of the two options: $-w_{D1}+ 5 w_{D2}= -2$ $2w_{D1} -2w_{D2} = -3$ which after shifting the constants to the left becomes ...


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You should review the difference between implied volatility and realised volatility. Historical volatility is the realised volatility that happen in the past but an option price will have to be determined by the view of the market about volatility in the future for a given period. Normally you do the inverse, ie the option price is given by the market but ...


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The Heston model can have that property. If you make the correlation negative between the Brownian motions in the $dS_{t}$ process and the $d\nu_{t}$ process you imply that price is negatively correlated with variance.


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Yeah this is often called Spot-Vol correlation and is well known. Most people take this into account. I think if you just google spot-vol correlation you will come up with many example/models.


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Derivatives pricing models are not predictive. They simply extract information about the market’s expectations embedded in the prices of market instruments to which they are calibrated. This information can then be used to price other more complex derivatives. Calibration is important for hedging purposes. Suppose you ask your model for the hedging ratio of ...


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If drift is driven by earnings retention policies then the value of an option does depend on drift! I'm 99% on the following reasoning and would welcome input from others to tighten this up. Consider what happens with Save Co., a hypothetical company that owns a pile of cash sitting in a savings account earning 1% APY. Suppose Save Co. is required to ...


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The digital option pays $H$ at time $T$ if $S_T \geq K$ , so its option time at time $t$ is given by $$V_t=E_t\left[e^{-r(T-t)}H 1_{\{S_T \geq K\}}\right]=e^{-r(T-t)}H* P_t(S_T \geq K)$$ The model used is Black-model, that $$dS_t=rS_tdt+\sigma dW_t$$ or $$S_T=S_te^{\left(r-\frac12 \sigma^2\right)(T-t)+\sigma (W_T-W_t)}{}$$ Calculate $ P_t(S_T \geq K)$ ...


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Recall that the price of your contract is \begin{align*} V_t = e^{-r(T-t)} \mathbb{E}^\mathbb{Q} [H1_{\{S_T>K\}}|\mathcal{F}_t] \end{align*} because your option always pays $H$ if $S_T>K$. Next, \begin{align*} V_t &=He^{-r(T-t)} \mathbb{E}^\mathbb{Q} [1_{\{S_T>K\}}|\mathcal{F}_t] \\ &= He^{-r(T-t)} \mathbb{Q} [{\{S_T>K\}}|\mathcal{F}_t] \\...


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$N\left(d_2\right)$ is the risk-neutral probability that the spot is greater than the strike at maturity, therefore the RN probability that you get your payoff.


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There seems to be a consensus that the distribution of daily stock returns lies somewhere between the Gaussian Distribution and the Cauchy Distribution[1][2]. While the former will fail to encompass high volatility events, the latter typically exaggerates their occurence (tails are too fat). I recently implemented a script for calculating european option ...


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Assume the put option with strike 45 is worth 8 and the put option with 40 is trading at 5. For the bull spread, you sell the 45 strike option and buy the 40 strike option. So your payoff and profit will look as follows (profit=payoff+net premium): Instead if you sell 10 options of strike 45 and buy 10 options of the other strike, profit graph will just ...


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In general you have don't have an exact solution for the non linear equation - so you have to use numerical methods. Have you seen this Non linear option pricing book. It covers the topics that you mentioned. People generally do compare the prices produced by non traditional models to the traditional ones - the purpose could be to benchmark the model ...


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