w****i
发帖数: 143 | 1 Could anyone tell me why implied volality has a smile effect? I cannot
understand the interview book explanation | N******r
发帖数: 642 | 2 because people like to overpay downside risk than upside, esp in equity
markets. | z****i
发帖数: 406 | 3 转自wilmott
mghiggins
Senior Member
Posts: 313
Joined: Nov 2001
Sat Jan 04, 03 01:54 PM
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Qualitative explanation for why there is a vol smile in stochastic
volatility models from a market-making perspective:
First, let's step back and look at the standard Black-Scholes world. Why do
options have value over intrinsic value? Because they have positive Gamma.
As a market-making trader, you buy an option and Delta-hedge it - now, the
value of your portfolio as a function of the underlying asset price is
parabolic, and you make money whichever way the asset price moves. Nice! You
're willing to pay for that position, and the more volatile the asset price
is (so the higher you expect to shoot up the sides of the parabolic payoff),
the more you're willing to pay for the option.
Now let's imagine that volatility is stochastic (but we'll ignore the
possibility of jumps in the asset price).
A few definitions first (these aren't standard terms, but I like them better
than other labels I've heard before, like Vanna - yeck, that sounds like
bad Indian food):
Vega: sensitivity of my portfolio value to moves in volatility. d(Value)/d(
Vol). All vanilla options have positive Vega.
Vega Gamma: sensitivity of Vega to moves in vol. d(Vega)/d(Vol) or d^2(Value
)/d(Vol)^2. Out-of-the-money options, both high- and low-strike, have
positive Vega Gamma; ATM options have roughly zero Vega Gamma.
Vega DSpot: sensitivity of Vega to moves in asset price (spot). d(Vega)/d(
Spot) or d^2(Vega)/d(Vol)/d(Spot). High-strike options have positive Vega
DSpot; low-strike options have negative Vega DSpot; ATM options have roughly
zero Vega DSpot.
Why there's a smile:
Imagine you buy an option with positive Vega Gamma (any OTM option, either
high- or low-strike). You hedge the outright Vega by selling an ATM option (
which has roughly zero Vega Gamma). Now you're got a parabolic payoff vs
volatility, much like you did vs asset price in the Gamma example at the top
. So, whichever way vol moves, you make money. Woo hoo! You're willing to
pay up for that portfolio, above the standard zero-stochastic-vol (Black-
Scholes) value. The more volatile vol is, the more you're willing to pay.
So this means that, if volatility is stochastic, you tend to be willing to
pay more than the Black-Scholes value for OTM options, but not for ATM
options (since they have no Vega Gamma). This means that implied
volatilities will be higher for OTM options than ATM options, which is the
smile.
Why there's a skew:
Imagine you buy an option with positive Vega DSpot (a high-strike option)
and hedge the outright Vega with an ATM option. Also imagine there's a
positive correlation between moves in spot (the asset price) and vol. If
spot moves up, your Vega turns positive (since d(Vega)/d(Spot) is positive),
but because there's a positive correlation between moves in spot and vol,
you expect vol to go up too. Vol is going up when Vega is positive, so you
make money. If spot goes down your Vega turns negative, but you expect vol
to drop as well because of the positive correlation. So you make money there
too. Whichever way spot moves you make money - and again you're willing to
pay up for that portfolio. The more positive the correlation is, and the
more volatile vol is, the more you're willing to pay.
In this example, if the spot/vol correlation had been negative, you'd lose
money whichever way spot moved, and you would have to be paid (relative to
the zero-correlation price) to take on the portfolio. Similarly if Vega
DSpot were negative (low-strike options) and correlation were positive.
So this drives the skew: if the spot/vol correlation is positive, you're
willing to pay more for high-strike options and less for low-strike options
that the zero-correlation price. This gives a positive skew. If the spot/vol
correlation is negative, the reverse is true, and the correlation leads to
a negative skew.
This is all a bit approximate, since the discussion has been focussed on the
instantaneous Vega Gamma and Vega DSpot - and to get an accurate price for
a derivative you have to include the global behaviour. However, the local
behaviour often drives the pricing because the spot is most likely to stay
near where it started. | EM
发帖数: 715 | 4 users tend to buy otm calls while producers tend to buy otm puts, so it is a
smile
【在 w****i 的大作中提到】
: Could anyone tell me why implied volality has a smile effect? I cannot
: understand the interview book explanation
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