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3

This is indeed very strange, and is probably a typo in the paper. It would be correct if $s^2$ is the sum of squares of the last 60 days returns, and $s$ is the square root of that. Then the division by $\sqrt{60}$ would give the daily vol. But if $s$ is the standard deviation, as they claim, then we would be doing the division twice and that would be wrong. ...


-3

Volatility over N periods is roughly proportional to sqrt(n). Using the annual number will give a different result that will make the monthly appear lower. Because volatility is ergodic it doenst just increase linearly over time.


0

As time passes P/L line is getting closer and closer to expiration P/L line. Volatility resists P/L line to get closer to expiration P/L line. You can think about Volatility as an elastic element btw P/L line and expiration P/L line. The higher is Volatility the bigger is the gap btw P/L line and the expiration P/L line. Time and Volatility are opposite ...


0

Isn't that exactly how you recover the option price? You paid the option price $C$, conduct delta-hedging continuously with initial capital $-C$ (zero cash to start with). At the end, the option expires and has payoff $X$, your hedging portfolio should then end up with $-X$. The pnl from you hedging portfolio is $-X-(-C)=C-X$ (in cash). You get the payoff $X$...


0

For the Geometric Average Asian Option in BS, there is an arithmetic formula for the price - in fact, it is possible to price it using a BS vanilla options calculator, if you adjust the parameters slightly - as discussed in this blog post: http://www.quantopia.net/asian-options-iii-geometric-asian/ As the pricing formula is the same, the Greeks can be ...


1

Following the notation in Hull, let $H$ be the barrier level. I list the prices of European-style down-and-out barrier options with continuously observed barrier. If $H\leq K$, then $$c_{di}=S_0e^{-qT}(H/S_0)^{2\lambda}N(y)-Ke^{-rT}(H/S_0)^{2\lambda-2}N(y-\sigma\sqrt{T})$$ and $$c_{do}=c-c_{di}.$$ If $H>K$, then $$c_{do}=S_0N(x_1)e^{-qT}-Ke^{-rT}N(x_1-\...


2

Note that yahoo is posting the forward dividend yield. Other yields, trailing (that you seem to describe) and indicated, are described (Investopedia) here and (Wikipedia) here. See also this, this, and related links on Stack Exchange. In general, for (estimated) discrete dividends $ (D_i)_{1\leq i\leq n} $ at future times $(0<)t_1<\ldots<t_n (\leq T)...


1

With theoretical modeling you just put in the number of days till expiration in years. If time to expiration is less than 1 year, it will just be some decimal. That means you can put in any time to expiration you want. In practice, how you model the decay overnight depends on your own analysis, judgement, modeling, trading strategy etc.. You could do the ...


1

Another way to look at it, is that we have a one-dimensional Brownian motion process driving the market but two risky assets. The market price of risk process (giving the equivalent martingale measure), $\lambda$, must then respect two conditions: $$ \lambda \sigma_1 =\mu_1 -r $$ $$ \lambda \sigma_2 =\mu_2 -r $$ which implies $$\frac{\mu_1-r}{\sigma_1}=\...


2

Here is a simple solution using the equivalence of no arbitrage and the existence of a stochastic discount factor. Let the SDF be $\Lambda(t)$. This evolves as $$\frac{d\Lambda(t)}{\Lambda(t)}=-rdt-\varphi(t) dW(t),$$ where we used the fact that the drift of the SDF is the risk-free rate and that there is only one source of uncertainty. The standard pricing ...


1

(An attempt to answer @noob2's question posed in answer above.) Black-Scholes is homogeneous: $$ xC(S,K) = C(xS,xK) $$ for all $x>0$. This is true even if one triplicates $S$ (new variables but taking the same value as $S$), as the two copies are always divided by $K$. Taking derivative wrt $x$ gives: $$ C(S,K)=S(\partial_1C)(xS,xK) + K(\partial_2C)(xS,xK)...


2

I don't know "what this identity intuitively means" but I can tell you an anecdote about how I encountered it. (This is a true story, though I won't mention the school and the professor). The professor wrote on the blackboard the BS equation (with no dividends): $$C=S N(d_1) - K e^{-rT} N(d_2)$$ and asked: what is Delta, i.e. what is $\frac{dC}{dS}$...


3

For (32), under Black-Scholes model ($r^*$ foreign interest rate, in FX world, or continuous dividend, in equity world), we have Gamma $$\frac{\partial^2 C}{\partial S^2} = \mathrm{e}^{-r^*\tau}\frac{\phi(d)}{S\sigma\sqrt{\tau}} $$ and Dual Gamma $$\frac{\partial^2 C}{\partial K^2} = \mathrm{e}^{-r\tau}\frac{\phi(d -\sigma\sqrt{\tau})}{K\sigma\sqrt{\tau}} $$ ...


3

Yes you are correct: the formula you found is the so-called Black formula. What that showed is that under the Black-Scholes assumption of a constant rate, working under the risk-neutral measure or under the $T$-forward measure is exactly the same. When rates are stochastic, however, you do not know the value of $B_T = e^{\int_0^T{r_t \mathrm{d} t}}$ and to ...


1

This looks similar to a cliquet or "ratchet" option: an option with a strike price which resets occasionally. The Wikipedia definition of a cliquet is a bit too restrictive since one of the most common uses of such options was by Japanese firms which issued warrants and convertible bonds in the 1990s after the implosion of the Japanese real estate ...


1

The bellwether Indices for testing, are NASDAQ, Technology sector, S & P 500 Big 500 capital weighted Stocks, Russell 2000, MID sector stocks and some small stocks. It is better to use the data fro their relative, ETF's eg. QQQ, SPY, IWM. The Dow is covered by the Nasdaq and the S&P 500, it is the 20 biggest stocks on the market and is not useful. ...


0

Short-term power options have very high volatility and kurtosis. If you can find power data from, say, 2001... that would have jumps. However, those data are likely difficult to come by. If you are going to model jumps, you might do better by looking into work by Todorov and Tauchen. They find jumps in both the underlier and in volatility itself. Their work ...


3

KeSchn and I pointed out in the comments that this it is not possible to represent all stock dynamics using the Generalized Black Scholes model. For example, there can be jumps at random moments and not just at random moments but also jumps of random size. These jumps can affect either $\mu_t$ or $\sigma_t$. Models with too many sources of randomness are not ...


4

SPY pays dividends ~1.8%, and the expiry is ~3y (as of date was 2018, 2021 expiry), so the it looks like there is a discount Assuming $0 time value $$OptionValue=Intrinsic Value+Time Value $$ $$OptionValue= (S-K)-Dividend$$ $$OptionValue=267x(1-0.02)-267x1.8\%x3=\\\$247$$


1

I suspect that the reference value is, well, only for reference and not the real price. Maybe the price of SPY when the call contract is bought is lower


1

As the other answer says, expected PnL does not depend on hedging portfolio, so you can hedge with whatever vol, expected PnL is the same. In this particular case, you can simply observe that in the paper, the PnL for the case of hedging with actual vol has the gamma term, and the other two terms combine to form a brownian motion under the risk neutral ...


1

CME (as of now) also publishes it. Folder: ftp://ftp.cmegroup.com/irs/ file name: irs_close_quotes_OISUSD_YYYYMMDD.csv, e.g. 20200717 CURVE_NAME,TENOR,RATE USD LIBOR-OIS DISCOUNT CURVE,2 Years,0.0033500000 USD LIBOR-OIS DISCOUNT CURVE,3 Years,0.2200000000 USD LIBOR-OIS DISCOUNT CURVE,5 Years,0.2212830000 USD LIBOR-OIS DISCOUNT CURVE,10 Years,0.2172810000 ...


2

Let $C=(S-K)^+$ and $P=(K-S)^+$. Then it is clear, for any positive integers $i$ and $j$, \begin{align*} C^i P^j = 0. \end{align*} Consequently, for any positive integer $n$, \begin{align*} (C+P)^n = C^n + P^n. \end{align*} Your conclusion now follows immediately.


0

Volatility surface explains the variance evenly spread across the duration, starting from effective time when it is published. For 8 hours expiry, referring to ON volatility is enough. Next, calculate the duration by taking the difference of the expiry of ON volatility and the effective time of the surface. Usually surface will be expiring at GMT14 and 15 on ...


0

I'm not too sure if I interpret the question correctly, but I am inclined to say that as the call is long gamma, 'jumps' (second order moves) would always result in higher value in the delta hedged portfolio, and therefore should be built into the price of the option. So the call should be more expensive if it has jumps.


3

Following this answer, let $\mathbb Q$ be the probability measure associated to the risk-free bank account as numeraire and $\mathbb Q^1$ the probability measure associated to the stock as numeraire. You know that the standard equation $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t^\mathbb{Q}$ can be written as $\mathrm{d}S_t=(r+\sigma^2)S_t\mathrm{d}...


3

Heavier tails, or a higher probability of extreme outlier values, meaning the investor is more likely to experience extreme events (e.g. tail losses). EDIT: @noob2, valid point, see this article on modelling and forecasting the kurtosis and returns distribution of financial markets: irrational fractional Brownian motion model approach


-1

For positivty, I think it will also depend on the boundary condition. The idea is briefly stated as following. Let say we have a boundary condition $f(S_T,T) = min\{S_T - K,0\}$, $f(b,0) = 0$ and $f(a,t) = 0$. By Feynman-Kac Theorem, the solution of the original PDE can be written as: $f = e^{-rT}\mathbb{E}[min\{S_T-k,0\}]$ , where $dS_t = rS_tdt + \sigma ...


0

An easy way to check if you've made a mistake during a longish calculation like your derivation of the skew for a straddle is to numerically evaluate the original integral. It would be a one-liner in Mathematica.


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