New answers tagged options
8
Great question. Let me try provide some insights and thoughts regarding your points and questions raised. It may not be a full answer but hopefully it helps to connect the contents in the paper/book with some trading intuition:
From a theoretical perspective, I don't see any mistake in your thinking regarding skew decay but two questions arise on my end: ...
4
It depends what you exactly call Dupire's formula. If you take the original formula, valid under zero interest rates and dividends (or equivalently, considering undiscounted option prices on the forwards), which reads
$$\sigma_L^2 = 2 \frac{ \frac{\partial C}{\partial T} }{K^2 \frac{\partial^2 C}{\partial K^2}}\,.$$
Then the formula for a put is the same, ...
0
The maximum profit for a collar is the call strike less the collar's cost (at expiration).
The maximum loss is the collar's cost less the put strike (at expiration).
Prior to expiration, the speed at which the profit or loss approaches the maximum depends on the time remaining and is accelerated (or decelerated) by the implied volatility.
If there's a ...
0
You can potentially try multi-level MC to help with the Euler discretization step size and thus reduce the overall work done when increasing the number of paths. Check out https://archive.siam.org/meetings/uq16/giles.pdf
0
For everyone's benefit, as per the answer here on Cross Validated, the distribution should be Logit-normal distribution.
1
Greeks represent rates of price change. Obviously they vary across models, if they didn't, the models would agree on prices across all market scenarios and thus the models have to be the exact same, which is a contradiction.
As far as hedging goes, nobody has the right hedge. For a Greek to be 'true' the model should:
Exactly capture the underlying ...
4
$\frac{1}{S_t}$ is log-normal
If $S_t$ is a geometric Brownian motion, so is $\frac{1}{S_t}$ and indeed any power $S_t^\alpha$. Simply use Itô's Lemma and set $f(t,x)=\frac{1}{x}$,
\begin{align*}
\mathrm{d}f(t,S_t) &= \left(0-\mu S_t\frac{1}{S_t^2}+\frac{1}{2}\sigma^2S_t^2\frac{2}{S_t^3}\right)\mathrm{d}t-\sigma S_t \frac{1}{S_t^2}\mathrm{d}W_t \\
&=-...
1
The shifted exponential of a lognormal distribution, just as the exponential of a lognormal distribution is a known in finance because of zero-coupon bond option. I am not aware of it being otherwise known.
You can use a lognormal assumption on the bond price instead of the yield. This will allow you to use the Black formula. You can approximately fit the ...
8
hope I am not too late to the party.
tl;dr Taleb's paper draws incorrect conclusions from a set of wrong assumptions.
In practice, the movements of the forecast at 538 are very much in line with what can be defined an "arbitrage free prediction" based on a binary option model.
The gist of the article is summarized in its incipit.
A standard ...
2
Call option:
$$\mathbb{P}\left(S_t\geq K\right)=\mathbb{P}\left(S_0e^{(rt-0.5\sigma^2t+\sigma W_t)}\geq K\right)=\\=\mathbb{P}\left(W_t\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma}\right)=\\=\mathbb{P}\left(Z\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma\sqrt{t}}\right)=\mathbb{P}(Z\leq d2)$$
So we have shown the well-...
1
Interpolate after building the surface. Won't your step #2 do this for you, do you really need step #3?
Definitely it changes every day. Look up "sticky strike" and "sticky delta" if you want to see how you can use a vol surface on a previous day as an approximation, if you don't have a fresher one.
3
The option payoff diagram for European Option will be exactly the same.
The intuitive reason that option value changes with volatility is that it changes the probability of winning the jackpot.
Think about it, options provide you downside protection. So, for example, when the stock is volatile, it has a higher probability of getting to a high price. If you ...
1
Have you reviewed the Black Scholes formulae? For a given spot price, volatility is back-calculated based on options premium that investors are willing to pay.
Given constant volatility as you suggest, the option value will decay as it gets close to the expiration. In other words, an option at a strike price equal to spot price, will have an intrinsic value ...
3
I am not sure where you're going, but GARCH models usually have two equations: (1) an equation describing the conditional expectation of the dependent variable and (2) an equation describing the conditional variance of the error term in equation (1) as a process that is perfectly anticipated 1 period ahead.
When you use returns in equation (1), equation (2) ...
6
It depends a little bit what you're trying to do.
If you can statically replicate the payoff of a position at $t=0$,
then putting on that hedge will insulate you from all risk coming
from the contract. Payoff doesn't need to be linear - for example,
you can perfectly replicate a call option using a put option and a
futures contract
If you want to use only ...
0
I will attach you a youtube video on the derivation of the Heston Model by a really intersting channel called quantpie. I think it might be useful.
https://www.youtube.com/watch?v=KncOcHnEA3Q&t=880s
Regards,
-1
In Black and Scholes derivation of their PDE (Journal of Political Economy, Vol. 81 No. 3, 1973, p.642) they tacitly assumed $w_1 = 0$. This resulted in the disappearance of drift from their PDE. Correcting for that ommission, restores drift to its rightful place in the (now more complicated) PDE that results. For more details see my answer to my question "...
1
As Will advised to check, there is actually no arb for both put & call considering the spread to be crossed (particularly for Dec expiry) on both legs, ie hitting Dec expiry put wing at bid and lifting Jan at offer.
1
To add to @ilovevolatility 's answer, in brevity no.
The covariance of a portfolio consisting of two options $O_1$ and $O_2$ on assets $S_1$ and $S_2$ is
$$
Cov=\mathrm{E}_\mathbb{P}\left[\left(O_1(S^{(1)}_t,t)-\mathrm{E}\left[O_1(S^{(1)}_t,t)\right ]\right)\left(O_2(S^{(2)}_t,t)-\mathrm{E}\left[O_2(S^{(2)}_t,t)\right ]\right)\right]
$$
Let's have a look at ...
2
My understanding of a "geared put structure" is that it is a bought ATM put option on a stock, whereby the ATM put-option buyer sells (at the same time) some OTM puts. The number of OTM puts sold is greater than the ATM puts, to make the pay-off function decrease to zero linearly. The structure buyer owns the underlying stock and buys the structure ...
3
Let's work under Black-Scholes, with two correlated GBMs:
$$
dX = \sigma X dW, \quad dY = \nu Y dZ, \quad dWdZ =\rho dt
$$
I've taken interest rate is zero for simplicity, does not influence the covariation anyway.
Suppose $F$ is a claim on $X$ and $G$ is a claim on $Y$. Both satisfy the BS PDE, hence
$$
dF = \left(\frac{\sigma X}{F}\frac{\partial F}{\...
3
1 Not really pricing focused, but have you looked at: Marc Allaire and Marty Kearney. Understanding Leaps: Using the Most Effective Option Strategies for Maximum Advantage. McGraw-Hill 2002.
2 an article on the rho of long-dated commodity options, rather than equity options https://onlinelibrary.wiley.com/doi/abs/10.1002/fut.21954
3 Gurdip Bakshi, Charles ...
0
As regards option prices, most people would use the midpoint of the spread as the fair price. Then, risk free rates would be computed by using the yield on very safe government fixed income instruments like a US Treasury Bond. You can pick two bonds with times to maturity straddling that of your option contract and you interpolate them linearly to get a risk-...
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