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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,...


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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. ...


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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{\...


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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.


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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 ...


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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 ...


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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.


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If you consider delta, gamma and vega as three variables, and you are able to construct a portfolio with any values, i.e. with three degrees of freedom: $$ [\delta, \gamma, \theta]$$ And you have a space of products which allow you to construct a hedge for any such delta and vega then you must have at least these two degrees of freedom (in some basis): $$ ...


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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.


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


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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 ...


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Breezing through the referenced paper, the point of it seems to be to develop Ito calculus for non-anticipative functionals $$ F(t, X_t),$$ where $X_t := \left\{X(u)\mid 0\leq u\leq t \right\}$ and $\left(X(u)\right)_{u\geq 0}$ is a stochastic process. For example, for $$ F(t,X_t)=t^{-1}\int_0^t X(u)du.$$ I think that, in that introductive paragraph, the ...


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OK, here is what I think. (But you should ask for advice from others in this forum or elsewhere). You computed $\frac{dC}{dK}$ (the dual delta) by a discrete approximation. The result is negative and this is correct (it is negative for a Call and Positive for a Put). In the case of a European Call it is given by the formula $-e^{-r T}N(d_2)$. (See here for ...


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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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In the simplest case, he books the structured product as a sale of the call at 5, he then enters the market and buys a call at 3, so he pockets 2 on the trade.


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If a structured product trader can directly hedge in the market, he will usually do it. Here your example is a little too simplified, many structured products have features that cannot be easily hedged in the market because they are path dependant (barriers), or illiquid (typically a 5 years 60% put on a single stock). Let's take your product, but imagine ...


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There are a few reasons for this: You can use grid methods for simple trades like barriers. But when you have trades which are not only dependent on the terminal distribution i.e. have payoffs which are path dependent like for instance fadeout, they can be only properly priced by an MC model because you would need to model vol of vol as well. When there are ...


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The simple explanation is that in the absence of calendar spread arbitrage, we should observe monotonic option prices with respect to maturity. And option prices are monotonic with respect to increase in volatility. Let $(X_t)_{t \geq 0}$ be a martingale, $L>0$ and $0\leq t_1, t_2$, then we have $$E[(X_{t_{2}} - L)^{+}] \geq E[(X_{t_{1}}-L)^{+}]$$ for ...


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Note that Boyle (1988) introduces $\lambda$ because the CRR parameterisation $u=e^{\sigma\sqrt{h}}$ yielded negative probabilities (and probabilities above one) for reasonable parameter values. Instead, he uses $u=e^{\lambda\sigma\sqrt{h}}$, where $\lambda>1$ and $h=\frac{T}{n}$ is the length of one time step. If you perform the limits, $p_u\to0$ and $...


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In a CRR binomial model, it would seem that the path-wise minimum is a function of the total number of down moves along that singular path. For example, let us fix $N=3$, resulting in $2^N=8$ possible paths that arrive at four $N+1=4$ possible states at maturity. Along each path, it suffices to note that the minimum along that path is simply defined by the ...


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Yes, the VIX took a sharp downfall on 2020/03/02, from 40.11 to 33.42 (-6.69). But that is not what the 2020/04/15 Put options are based on, they are based on the 2020/04/15 VIX Futures (VIJ20), these went from 23.025 on 2020/02/28 to 23.325 on 2020/03/02 an increase of 0.3. The Vix options are based on the futures, not the spot Vix value.


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Maybe implied vol on the VIX fell. An option can lose value even if the underlying goes in its direction if implied volatility falls enough to outweigh the directional effect.


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It is due the number of timestamps in your case. Actually, as the ZC rate is zero, the price of European and american options should be the same. EDIT PROOF: You know that for american options (see proof in pages 4,5 HERE): $S_T-K\leqslant C-P \leqslant S_T-Ke^{-rT}$ When the risk free rate is zero, you get that the call put parity remains valid $S_T-K= ...


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The formula above is usually the price for a year-on-year inflation indexed caplet, so the $\psi_i$ will be the day count fraction over periods $[T_{i-1},T_i]$ where these $i$'s index the year not the inflation index month. Therefore the $\psi_i$ should be close to $1.0$ since the day count will always be for successive years. You could use this formula for ...


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The are implied parameters. You basically do a parametric dimension reduction by implying them across a range of observed prices, checking the errors, and then you might interpolate or even cautiously extrapolate. In reality, desks will have their own spreads but you can generate a volatility surface as a baseline from these parameters. If I remember ...


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