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1

If there is no interest rate, the european and american put prices are the same for every strike. More details can be found in my answer for the question below: Longstaff Schwartz Algrorithm in R


1

If I am not mistaken, as you have already stated you have the long run relationship $$ h\left(1-\beta-\alpha\gamma^2\right)=\omega + \alpha $$ I suggest you impose the following restrictions that should ensure $h_t$ to stay positive: \begin{align} \omega&>0\\ \alpha&>0\\ \beta &>0\\ \beta+\alpha\gamma^2&<1\\ \end{align} I ...


0

I’d say Hull’s “Options, Futures and Other derivatives”, whatever edition is fine.


0

If you have the path likelihood, you can try just writing that function and optimizing it directly. You might have some issues with the variance piece. This looks a lot like parameter inference for SDE, data-assimilation etc. I think if you write a proper likelihood function with priors for all parameters and same via some MCMC or MC (Gibbs) that is ...


0

I'm not sure what you're asking here quite, it seems to me that you are inputting a shorter time to maturity (from one day to one hour) and noticing a decrease in the contract value. Theta, the derivative of the option price with regards to time, is negative for for all options so this will always be the case no matter the time scale. Are you sure you have ...


7

I will be glad to help, but let me first advise you away from working on this topic until you have an academic position. This topic has been poison for me, but I am slogging on anyways. Before you use anything I do, get permission from your academic advisor. I have an unpublished article on options pricing, and I am proposing a new branch of stochastic ...


1

There were many attempts to switch from normal distribution to some other which can describe a market more accurate, i.e. distribution with fat-tails (e.g. Cauchy distribution or broader familly of so-called stable distributions). These distribution allow you to model Black swans. However, as you pointed out, there is a problem with calculation of mean ...


0

Re your first question: Use the implied volatility $\sigma_{imp}(X,\tau)$ for strike $X$ and expiry $\tau$. The option price, and hence the implied volatility, is driven by the options markets. Your option model should first and foremost be able to replicate observed option prices (hence, you plug in implied vols).


0

At an informal level, this is a system of two nonlinear equations in two unknowns, hence you can plot it in the $(r,\sigma)$ plane and see how many times they cross each other. At a more formal level, you can check if the Jacobian matrix is nonsingular everywhere. Nonsingularity of the Jacobian matrix (i.e., the determinant is not null) is a local argument ...


0

The basic difference is that for calculating the option's price within the classic BS-framework, you mostly use the historical vol (which is extracted from time series with a model). But this is only a theoretical (arbitrage free) price. At an option's exchange, you will see supply and demand meeting each other. Assuming perfect and efficient capital markets,...


2

Your question makes perfect sense; one has to define volatility. Volatility can be used interchangeably for a number of different metrics. Realized volatility - the observed volatility of the underlying asset (and btw, there are many quite different ways of measuring it). Implied volatility - the number you get when you run your option pricer in reverse. ...


1

I am not familiar with the deep mathematical intricacies of advanced no-arbitrage theory, an extremely technical subject. However, from reading literature reviews, I suspect this is an historical legacy of the research path that led to the most general versions of no-arbitrage theory. If you consider dividend-paying assets whose dividends are not ...


0

I figured out the source of the problem today and it was indeed the stupid mistake that comes along with working on something like this late at night. In the paper, the repport risk-neutral estime is $\tilde{\mu} = \mu + 1/\eta$. Since $\eta$ is so damn small and negative, I was grossly understating the value of $\mu$... I corrected the mistake directly ...


1

In the following, I am assuming the BS73 model and I assume that "ATM" means $$ S = Xe^{-r\tau} $$ The pricing formula for a European call then becomes $$ \tag{1} O\propto N\left(+\frac{1}{2}\sigma\sqrt{\tau}\right)-N\left(-\frac{1}{2}\sigma\sqrt{\tau}\right) $$ times some scaling factor which is irrelevant for our purpose. Clearly, $$ Vega\equiv\frac{\...


3

Let me venture a guess. If I had to design a system from scratch, I would probably prefer GARCH processes to properly stochastic conditional volatility processes. The fact that one step ahead, the conditional volatility process is known makes filtering both trivial and faster. Moreover, this class of option pricing model affords me all the flexibility of ...


1

For shorting a stock what you would do is to have a margin account with your broker. As an example, in the American jurisdiction, according to Regulation T from the Federal Reserve, you would provide a 150% of the value of your position as initial margin (50% of additional marging). And the daily margining would be done against your margin account (both ...


1

Based on your computation, you can observe that the $N’$ term is always positive, between 0 and 0.4. As $\sigma$ is always positive, you can focus on the $-d_2$ term. When $d_2 > 0$, i.e. call is ITM, delta has a negative sensitivity to volatility ; conversely for OTM call. That is in line with your remark.


0

There are many ways to estimate model parameters. In your case, if you're going to use only option data, I strongly suggest defining your pricing error in the (Black-Scholes-Merton) implied volatility space. Specifically, I would minimise this: \begin{equation} \frac{1}{N} \sum_{i=1}^N \left( IV(C_{it}^\text{model}, \Theta) - IV(C_{it}^\text{observed}) \...


0

Perhaps someone assumed that there are 250 trading days per year for this time series instead of 252.


4

You do not really need the dynamics of $S_t^2$. You can simply apply your standard technique from risk-neutral pricing. The time zero price of a European-style contract with payoff $X$ is given by $$V_0=e^{-rT}\mathbb{E}^\mathbb{Q}[X\mid\mathcal{F}_0].$$ Thus, \begin{align*} V_0 &= e^{-rT}\mathbb{E}^\mathbb{Q}[\mathbb{1}_{\{S_T^2\geq K\}}] \\ &= e^{-...


2

These quotes may not be synchronised. Also if the probability of being below either strike is zero, then why would the price change, both options will be worth zero ? Just an extreme example that shows why not. The sensitivity of the price wrt strike is $N(d_0)e^{-rT}<1$.


3

It is not a contradiction, we are looking at two different phenomena: The Vol Smile is about a comparison on two call options $C_1$ and $C_2$ at a point in time: S is the same for both options (and does not change!), but $C_1$ has strike $K_1$ and $C_2$ has strike $K_2$. To fix ideas let's say $K_2 > K_1$. Then: $$\Delta_2 < \Delta_1$$ and $$IV_2&...


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