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1

Shelly Natenberg has a typo. He means lognormal

1

If the realized vol is $\sigma$ then your stock follows the GBM $$\frac{dS}{S}=r\,dt+\sigma\,dW_t\,.$$ If you price your call $C(t,x)$ with implied vol $s$ and using the Black-Scholes formula it satisfies the PDE $$\tag{1} \partial_t C+\frac{1}{2}s^2x^2\partial^2_xC+x\,r\,\partial_xC-r\,C=0\,.$$ The hedge portfolio consists of $\partial_x C$ amounts of ...

0

It is fairly standard to hedge a sold option as follows: at any time $t$ buy $\alpha(t)=\frac{\partial}{\partial S}c(t,S(t))$ amounts of stock $S(t)\,,$ and invest $\beta(t)=\frac{c(t,S(t))-\alpha(t)S(t)}{B(t)}$ into the money market account $B(t)=e^{rt}$ By definition, the hedge portfolio $X(t)=\alpha(t)S(t)+\beta(t)B(t)$ exactly matches the option value ...

0

The premium on a put option may be estimated very easily using the following equation that I have discovered, which is fairly accurate compared to BS when strike price is on the x-axis and premium is on the y-axis: $2\sigma e^{-\frac{\left|k-S\right|}{4\sigma}}\ +\left|\frac{k-S}{2}\right|+\frac{k-S}{2}$ For a call option, reflect the equation in the y axis ...

3

Here is another solution using Plotly. First of all let me correct a typo in your code def Vanna_(S, K, T, r, sigma): lista = [] d1 = (np.log(S / K) + (r + 1/2 * sigma ** 2) * T) / (sigma * np.sqrt(T)) d2 = d1-sigma*T**(1/2) return (1 / np.sqrt(2 * np.pi) * S * np.exp(-d1 ** 2 * 1/2) * np.sqrt(T))/S * (1- d1/(sigma*np.sqrt(T))) Then let me ...

5

Something like this? from mpl_toolkits import mplot3d from itertools import product S = np.linspace(100,120) vols = np.linspace(0.05,0.7) combs = list(product(S, vols)) values = [Vanna_(underlying, K, T, r, sigma) for underlying, sigma in combs] x, y = np.hsplit(np.array(combs), 2) fig = plt.figure() ax = plt.axes(projection="3d") ax.scatter3D(x,...

0

First, let's go back to basics to answer why theta can be both positive and negative, and why it's referred to as time decay? At it's core, an option's value is composed of two components: intrinsic value, and time value. As time passes, the proportion of the 'time value' gradually decreases until the option is worth exactly its intrinsic value at its ...

4

It’s just the effect of interest. If you are long a deep ITM European put, it is worth the PV of K minus the stock price. But one day later the PV of K has grown a bit. That’s it. It’s the opposite for calls because you have to pay the K, so bringing the date closer costs you money. This is all assuming interest rates are positive.

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