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2

VIX almost always only spikes when SPX goes down as @Jan Stuller also mentions in a comment. Insofar the question is a bit counterfactual. I frequently use twin axis in the charts that follow. The position of the label corresponds to the axes the ticker belongs to. These are essentially two question in real world scenarios. 1 ) VIX and Vega: VIX up, IVOL up,...


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Vega is the option's price sensitivity to the volatility (i.e. IV). In the graph below, vega is shown to be a strictly positive function in volatility, which means that at any point in the graph (i.e. for any value of IV, irrespective of whether the option is OTM, ATM or ITM), the option price: will increase in value if IV goes up (because Vega is positive) ...


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Your product is sensitive to the price of the put you will recieve conditional on the spot being at a certain place at some point in the future - aka the forward skew. As you increase vol of vol, while still being calibrated to the vanillas (so keeping the square of volatility constant), due to the fact that the put option in the wings is convex in the ...


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An actual option with an independent existence cannot have a negative price. But we are talking here about 'embedded options' that are part of another security (in this case a USTR bond) and cannot be separated from their parent. Their price is not quoted in the marketplace but is found by a calculation. The problem is that this calculation comes up with a ...


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For an option with delayed cash settlement, expiry time $T$ and settlement time $T_p(\geq T)$, paying $(S_T-K)^+$ at $T_p$, the present value of this payment at $T$ is: $$ E_T\left[\beta_T \beta_{T_p}^{-1} (S_T-K)^+ \right] = P(T,T_p)(S_T-K)^+,$$ with $\beta_t = \exp \left(\int_0^t r_u du \right) $, $r$ risk-free interest rate, $P$ associated zero-coupon ...


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In the paper "Optimal Delta Hedging for Options" link the author shows that the minimum variance delta is a function of change in implied volatility. If you use equation on page 9 i.e. $E[\Delta \sigma ]=(\frac{a+b \delta_{bs} + c \delta^2_{bs}}{\sqrt T}) \frac{\Delta S}{S}$ you will get what you want. The parameters a, b, c are fit with OLS ...


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Just to be clear we are talking about an option that pays $max(0,S_1-K)$ paid at time $t=2$. Then the only difference between this and a standard option is the extra discounting from $ t=1$ to $t=2$ . So the price $P$ must satisfy $$ P=BS/(1+r)$$ where BS is the regular Black Scholes price and $r$ is the forward risk free rate from $t=1$ to $t=2$. The ...


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The key here is to observe that the volatility at the time the option is written is not exactly equal to the volatility that the markets actually experience during the option's lifetime. The seller will price the option according to her best estimate of future volatility over its lifetime, but will always prove to have been too high or too low. More ...


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I'm going to go through a coded example to show how you might attempt this, using the python port of the QuantLib library. It will all seem a little mechanical, but hopefully it is instructive. There is quite a bit of setup code required (specifying the interest rates, spot etc., and also a utility function that I use for plotting surfaces), I've pushed this ...


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The answer is that $S_t$ is a random variable which has realizations that can be solved for using a monte carlo or numerical methods. By solving for this value many times (which is what a monte carlo does for instance) you can find a distribution of prices of the underlying at a given time. This is because the process is random so each solve should be ...


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(My attempt is based on $f(S_t)$, $0\leq t \leq \tau$, being a local martingale.) We first note that: $$\tau = \min \{\tau_1, \tau_2 \} $$ where $$\tau_1 = \inf \{t\geq 0 | S_t =1 \}, \; \; \tau_2 = \inf \{t\geq 0 | S_t =2 \}$$ and $$ \{ S_\tau = 2\} = \{ \tau = \tau_2 \} = \{\tau_2 < \tau_1 \}. $$ Aslo, we note that the standard Lipschitz conditions ...


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The generalised Fourier transform $\hat{P}(z)$ of the payoff of a put option with $P(x)=\max\{e^k-e^x,0\}$ is \begin{align*} \hat{P}(z) &= \int_{-\infty}^\infty e^{izx} \left( e^k - e^x \right)^+ \mathrm{d}x \\ &= \int_{-\infty}^k \left( e^ke^{izx}-e^{i(z-i)x} \right)\mathrm{d}x \\ &= \left[ e^k\frac{e^{izx}}{iz} -\frac{e^{i(z-i)x}}{i(z-i)} \...


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Delta is not the probability of finishing in the money as suggested in another answer, N(d2) is. The foot note mentions this. See Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model by Lars Tyge Nielsen for a detailed explanation. If time and vol is low, $d1 \approx d2$ and delta will be closer to the risk-adjusted ...


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(Just what I think is the right start) The pricing PDE comes out of the dynamics of a self-financing portfolio, $\Pi$, hedged against the movements of stock, $S$, its volatility, $v$, and interest rate $r$. With $V$ the target path-independent derivative, $U$ vanilla European option (different from $V$), and $P$ zero-coupon bond, the portfolio would be: $$ \...


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Duffie et al. (2000) show how to obtain the characteristic function of the log asset price in a fairly general affine jump diffusion model. Among others this includes the Black-Scholes (1973) model, the Heston (1993) model, the Bates (1996) model, the Merton (1976) model and the Kou (2002) model. This case also allows you to add stochastic interest rates. ...


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You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and simulate from these using a uniform draw. That's the brute force approach. The variance gamma process is typically represented as a difference of gamma processes or a ...


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Since you write under BS model, it is a tautology in this model. Assume you use Black, which you can without loss of generality, as one can easily be transformed into the other (FX is easier to show with covered interest rate parity in my opinion). Looking at $$d1 = ( log(F/K) + 0.5*σ^2*t ) / (σ*sqrt(t))$$ $$ d2 = d1 - σ*sqrt(t)$$ it is not immediately ...


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Regarding the second question, if we assume that $A$ is lognormal (shifted lognormal) and imply its two (three) moments from respective spot rate components under the appropriate probability measure, then we have its probability density function $f_A$ which allows us to compute the expectation of the payoff the usual way: $$ E\left[\left(\frac{1}{K}-\frac{1}{...


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