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Gamma is the sensitivity of the delta with respect to infinitesimal changes in the price of the underlying asset (in whatever unit your underlying is nominated, typically dollar, pounds, euros, ...). So, it is not a percentage change. Instead, the percentage change (option elasticity) equals $\Delta\frac{S}{V}$. This quantity, for instance, gives you the ...


5

With a long time to maturity, your options have a low theta because their time value decays quite slowly. If there are many months to go, the passage of one day does not change the exercise probabilities too much, whereas short life options with only a few days left have a much higher time value decay. Hence, the larger the time to maturity, the lower theta. ...


5

Let $\mathbb{Q}$ be the risk-neutral probability measure which uses the risk-free bank account $(B_t)$ as numeraire. In general, $\mathrm{d}B_t=r_tB_t\mathrm{d}t$. In the Black-Scholes setting, $r_t\equiv r$, we have $B_t=e^{rt}$. The stock measure $\mathbb{Q}_S$ uses the compounded stock price $S_te^{qt}$ as numeraire and is defined via the Radon Nikodym ...


4

Generally the Black Scholes equation is used to refer to the Black Scholes PDE (PD equation). And the formula refers to the analytical formula, usually cover both call and put versions. The extension of the BS to the square or power of S is frequently covered in the textbooks and tests; however, it could be tricky in an interview situation. When using ...


4

Approximating implied volatility of European options can be done in a few ways--this is just one. Below is a python implementation that uses Newton Raphson. You can use the implied_volatility function to find the approximate implied volatility. You can then check it by plugging the output from that back into the option_price function. import numpy as np ...


4

The traders or practitioners’ gamma concept tries to capture the same issue. It is defined as S times gamma divided by 100: $\Gamma_P=\frac{S\, \Gamma}{100}$ Please see page 29 of this document: https://mathfinance.com/wp-content/uploads/2017/06/FXOptionsStructuredProducts2e-Extract.pdf


3

Your derivation is right and as Alex said, the only difference between the equations is how you measure time. There are two possibilities Time going forward, $t\in[0,T]$, Time going backwards, $\tau\in[T,0]$, i.e. $\tau=T-t$. And clearly, after the change of variables, $\frac{\partial V}{\partial t}=-\frac{\partial V}{\partial\tau}$. Amongst others, this ...


3

If the stock prices falls "gently" and the option remains in the money, your delta will converge to 1 and you will have to buy stocks: the gains on your long stock positions will be lower, but the payoff of the option you wrote will be lower too. If the stock prices falls more sharply and gets closer to the strike, there are higher chances that the option ...


3

Shreve is a bit naughty here but, of course, he is right. When you have the risk-neutral measure $\mathbb{Q}$ or $\tilde{\mathbb{P}}$, you can price derivatives as discounted expectation by the very definition of the risk-neutral measure (better called: equivalent martingale measure). So indeed, once you have $\tilde{\mathbb{P}}$, you can price derivatives ...


3

First, you'd rather simulate $\log(X)$ rather than $X$; thus, there is no level dependency in your discretisation scheme, making it more accurate. $$Z_t = \log(S_t)$$ $$dZ_t = \left(r - \frac{\sigma^2}{2}\right)dt + \sigma dW_t$$ You can even run one single time step, and the distribution of your final price will still be as accurate! Second, the different ...


2

Under the Black-Merton-Scholes model, and also for a limited class of stochastic volatility models (when volatility is not correlated to the asset), the following relationship holds: $$ C(S,K) = \frac{K}{S} P(S, S^2/K) $$ This relationship is called put-call symmetry. Here is a short introduction: PCS Also, note that under geometric Brownian motion, $1+x$...


2

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 ...


2

Your code looks fine and it is encouraging that both MC simulations yield similar results. Please look at this simplified code for the analytical part of the Monte Carlo simulation. As you know, $$S_T=S_0\exp\left(\left(r-\frac{1}{2}\sigma^2\right)T+\sigma W_T\right).$$ A call is path-independent, so there is no need to simulate the entire path. I guess you ...


2

For Risk-neutral Pricing to “work”, you need assumptions where risk elimination by trading financial instruments is possible : no counterparty risk, no transaction costs, continuous trading, continuous asset paths. If such assumptions are not fulfilled (which is the case in real markets ; however, for large banks they are sufficiently near from reality), ...


1

Yes. You should use that function to calculate the implied volatility - market convention is to always quote implied volatility using the Black-Scholes model. Traders may execute a trade simply by agreeing a level of implied volatility combined with the use of the corresponding Bloomberg option pricing page. Someone once said, "it is the wrong number in ...


1

When you here implied volatility in finance, it usually means Black volatility or Bachelier volatility. In your case, since you have the prices from Heston, you can use Black-scholes to get the implied volatility. In that scenario, don't think about Black-scholes as a model, but as a translator to better understand the option price. For option traders, ...


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