New answers tagged implied-volatility
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Given 3 days of stock prices, you could calculate two days of return and hence calculate the annualized historical volatility which would be the close estimate of the implied volatility. However, more data points would give a better picture of the true implied vol of this stock.
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If i understand this correctly, you want to be able to infer a future volatility surface, given the current simulation parameters you have.
What you're essentially trying to do it include the modelling of forward vol/skew in your MC.
Getting the forward vol surface vaguely correct is quite important to price some types of derivative - i.e. anything that has ...
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Let $$\ln\left(S_T/S_t\right) $$
have mean $\mu_\tau$ and standard deviation $\sigma_\tau$, where $\tau=T-t$, and density of its standardized form
$$ X= \frac{\ln(S_T/S_t)-\mu_\tau}{\sigma_\tau} $$
approximated by Gram-Charlier expansion
$$ f_X(x) = \phi(x) - \gamma_{1\tau} \frac{1}{3!} D^3 \phi(x) + \gamma_{2\tau} \frac{1}{4!} D^4 \phi(x), $$
with $\phi$ ...
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I will assume that you have a set of "sheets" like a market maker. I.e different calendar terms of theoretical prices that your model has spat out.
Lets say:
July Straddle = 100
October Straddle = 150
If someone sells the July-Oct Put Calendar, selling Oct, Buying July for 20. If the theoretical value on our sheets is 22. Then that means we have ...
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Here is an answer from a statistical angle:
Your hedged VIX corresponds to a regression residual.
If you estimate the regression by OLS, the residual will by construction be uncorrelated with stock returns.
The two plots look similar because of a constant trend in stock prices that does not affect the correlation between returns and the VIX.
If you do not ...
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The local volatility graph tomorrow doesn't change, unless the implied volatility surface tomorrow is not the same as today. LV takes the implied vol surf today as input, and outputs a instantaneous volatility function of spot and time, which can price vanilla options today exactly the same as the market prices.
In local volatility world, you assume the ...
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The approximation I mentioned earlier is that in order to price an Asian option with strike K and maturity T on an asset with spot price S0, one should use the implied volatility at the modified strike K'=S0*(K/S0)^(6/5) and the same maturity T. This assumes an asset with a flat forward curve, like a futures contract.
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The theorem you want to use is Breeden Litzenberg which says that the density $\phi_{T}$ of your underlying is given by $\phi_{T}(K) = \frac{1}{B(0,T)}\frac{\partial^{2} C}{\partial K^{2}}$ where C is your call price with maturity T and strike K ( you can obtain rthe price with BS formula as implied volatility is given )
From this theorem, deriving your ...
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Further notes:
One shouldn't build an implied volatility surface just from call prices or just
from put prices. One should build it from liquid instrument quotes and, if necessary,
some less liquid ones. Some markets, like FX option one, quote package prices
(butterfly, risk reversal, ATM straddles).
Deciding how to parameterize the implied volatility ...
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In practice, things are actually quite different and a bit more subtle. You really need to differentiate between the underlying being an index or e.g. a single stock. I will try to provide some insight:
Index options are, in general, of European type. The market quotes prices for calls and puts and you can back out the implied vols via the usual BS formula. ...
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just to add to the other answers, the smile is essentially theoretical, in practice since the 87 crash, investors value more downside protection and the demand is higher for out of the money puts, leading to a volatility skew/smirk.
also bear in mind that these effects are more pronounced closer to the maturity, at inception the IV curve is essentially flat.
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Call and a put of the same strike have the same I.V, in theory.
The ONLY reason for this to differ is the limits to arbitrage on call put parity. Now this is a static strategy that has no rebalancing - so the only problem here is transaction costs in buying/shorting the stock. So if you have reason to believe that this strategy is difficult to implement, ...
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Hint:
$$f(x) = g(h(x),k(x)) $$
$$ f'(x) = \partial_1 g (h(x),k(x)) h'(x) + \partial_2 g (h(x),k(x)) k'(x)$$
(ignore red herrings: $t$, $T$, $S_t$ and $r$; focus on $K$ only)
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