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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 ...

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Just to add to @KeSchn's answer: At $T$ the option has price $(S_T - K)_+$. This is non-differentiable at $S_T = K$. Hence the delta is 1 when $S_T > K$, 0 when $S_T <K$ and not defined when $S_T = K$. That is not a problem since the probability that $S_T = K$ is 0 almost surely. The limit behaviour, i.e. when $t \rightarrow T$, is as in KeSchn's ...

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You simply take limits. Recall that in the Black-Scholes world $$d_1=\frac{\ln\left(\frac{S_t}{K}\right)+\left(r-q+\frac{1}{2}\sigma^2\right)(T-t)}{\sigma\sqrt{T-t}}.$$ As $t\to T$, we have $d_1\to\begin{cases} \infty & \text{if } S_t> K \\ 0 & \text{if } S_t=K \\-\infty & \text{if } S_t<K \end{cases}$. Thus, $\Delta=\Phi(d_1)e^{-q(T-t)}... 2 If you're looking to price a European option under a stochastic short rate, you can take a look at the Bakshi, Cao and Chen (1997) paper. Some of their model combine stochastic volatility, jumps in the price process, as well as a stochastic short rate process -- such as what you ask. The convenience of their approach is that all of their models imply that ... 0 Re your first question: Use the implied volatility$\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)$plane and see how many times they cross each other. At a more formal level, you can check if the Jacobian matrix is nonsingular everywhere. Nonsingularity of the Jacobian matrix (i.e., the determinant is not null) is a local argument ... 0 The basic difference is that for calculating the option's price within the classic BS-framework, you mostly use the historical vol (which is extracted from time series with a model). But this is only a theoretical (arbitrage free) price. At an option's exchange, you will see supply and demand meeting each other. Assuming perfect and efficient capital markets,... 2 Your question makes perfect sense; one has to define volatility. Volatility can be used interchangeably for a number of different metrics. Realized volatility - the observed volatility of the underlying asset (and btw, there are many quite different ways of measuring it). Implied volatility - the number you get when you run your option pricer in reverse. ... 2 The benefits of normal/lognormal have been well-described already. The problems with normal/lognormal, especially in the tails, are equally almost-universally known and appreciated. They persist because they make a variety of applied derivative problems easily solvable (and easily adaptable to similar but slightly different problems). Imagine you could ... 1 Under the BSM model, the terminal stock price is assumed to be lognormally distributed, with expected value equal to$\mathrm{E}_t(S_T)=S_0e^{r(T-t)}$. In order to achieve this in your simulation (of your log-normal stock process), you may want to modify your code: c = np.random.normal(r-0.5*sigma**2/365, sigma/np.sqrt(365)) # instead of mean=0 This ... 1 You are right that many models are based on normal distribution or log-normal distribution which are connected. It seems that there is three reasons (of course, maybe more) for using normal distribution: During "good times" price differences and other variables really behave according to a normal distribution You can easy calculate with normally distributed ... 6 The Geometric Random Walk: The Starting Point Let me begin by being a little more specific. The simplest, yet relatively sound model of asset prices that we have is this one: $$ln S(t+1) = \mu - \Psi_{t+1}(-1) + ln S(t) + \epsilon(t+1), \; \epsilon(t+1) | F_t \sim N(0,\sigma^2).$$ where$\Psi_{t+1}(u) := ln E_t \left( \exp( -u ...

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Rebonato called the whole process "the wrong number in the wrong formula to get the right price". We use Black-Scholes much the same way that we look at price-earnings ratio in equities. This is beneficial for traders. First, it translates a fast-moving price into a slow moving valuation metric. Second, it gives us an idea of value. Is this option rich or ...

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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, $$Vega\equiv\frac{\... 1 I believe you are applying the cap formula to value the floor. From the link you sent, try this:$$floorlet = D [(K-F)N(-d) + \sigma \sqrt{T} n(d)] $$Where the d will be the same. 5 1. Let me first reconcile the Black-Scholes pricing formula with the idea of prices being determined by supply-and-demand. Even if it is not explicitly said this way, from an equilibrium perspective, the Black-Scholes formula defines the unique price of risk that is consistent with the absence of arbitrage. In fact, you explicitly use this price when you ... 1 Based on your computation, you can observe that the N’ term is always positive, between 0 and 0.4. As \sigma is always positive, you can focus on the -d_2 term. When d_2 > 0, i.e. call is ITM, delta has a negative sensitivity to volatility ; conversely for OTM call. That is in line with your remark. 1 This is one of those situations that is not practically possible but is possible in theory. For example , the contingent payoff at T=20 is just S_(20). But the world is such that at t=11, the stock is negatively correlated with interest rates to such an extent that the forward price S(11,20) observed at t=11 for the stock at T=20 actually moves in ... 1 Each path is evolved based on the vol and a random number. The higher the vol the more the paths will diverge. Paths will diverge if you increase time as well. The solution is to increase the number of paths as vol or time increases to get a standard deviation of terminal values that you are comfortable with. 2 This is a good question. See my answer to a question here The point is that under Black-Scholes (and also many SV models) not only European prices but also American options prices are homogeneous of degree 1 in strike and spot as the optimal exercise time does not affect the homogeneity property in strike and spot price. Hence also for American options ... 4 You do not really need the dynamics of S_t^2. You can simply apply your standard technique from risk-neutral pricing. The time zero price of a European-style contract with payoff X is given by$$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^{-... 3 As volatility has a great influence on option prices, you'd like to sell options in high volatility environments and purchase options in moments of low volatility. But what is high/low volatility? Implied volatility rank (IVR) and implied volatility percentile (IVP) tell you this. The implied volatility rank is given byIVR=\frac{IV-52Low}{52High-52Low},...

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