This is called the smile curve and is illustrated in Figure 9 More typically

This is called the smile curve and is illustrated in

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This is called the smile curve, and is illustrated in Figure 9. More typically since then, it is downward sloping at- and around-the-money, and then flattens or sometimes curves upwards for far out-of-the-money strikes ( K >> S t ). An 65
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70 80 90 100 110 120 130 0.225 0.23 0.235 0.24 0.245 0.25 0.255 Strike Price Implied volatility Figure 9: Smile Curve. example from S&P 500 index options is shown in Figure 10. Interpretation of the Smile Curve Such a smile curve, or smirk, tells us that there is a premium charged for out-of-the-money put or in-the-money call options (low K ) above their Black-Scholes price using the at-the- money implied volatility. The smile effect is just a popular condensation of the fact that straight Black-Scholes is not sufficient in today’s derivatives market. Implied volatility, at first a seemingly artificial quantity, turns out to be an enormously useful synoptic variable expressing the extent of departure from Black-Scholes. At this stage, it is clear that the specification of a constant volatility is significantly inaccurate, first from statistical studies of stock price history that indicate the random character of volatility (measured as the standard deviation of returns), and second from the nonflat implied volatility surface I ( K, T ) that tells us how the market is actually pricing derivatives contrary to the Black-Scholes theory. Term Structure of Implied Volatilities There is also a pronounced term structure effect in the implied volatility surface (that is, the surface of market implied volatilities as a function of strike and expiration date). Term struc- ture refers to the variation with respect to the expiration, or term, of the option contracts, 66
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600 800 1000 1200 1400 1600 1800 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Strike Price K Implied Volatility S&P 500 Option Implied Volatilities on 1 June 2007, 204 days from maturity Figure 10: S&P 500 Implied Volatility Curve. The vertical line is at-the-money. or, similarly, with the time-to-maturity. Figure 11 shows typical S&P 500 skews (as a function of moneyness K/S t ) for various maturities. Clearly the skews become steeper for shorter maturities: the departure from Black-Scholes is strongest over short horizons like one or two months, when the market seems to be ascribing a significantly larger probability of a big downward move in the stock price than is implied from historical volatility estimates. Figure 12 shows the enormously increased implied volatilities in the midst of the major market turbulence in Fall 2008. 67
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0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Moneyness=lK/X Implied Volatility Implied Volatilities on 1 June 2007 1 mo 2 mo 3 mo 6 mo 9 mo 12 mo 18 mo Figure 11: Term structure effect: S&P 500 skews for different maturities on 1 June, 2007. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 0.5 1 1.5 2 2.5 3 Log-Moneyness=log(K/X) Implied Volatility Implied Volatilities on 10 Oct. 2008 6 days 41 days 69 days 97 days 160 days 251 days 342 days 433 days Figure 12: S&P 500 skews for different maturities on 10 October, 2008.
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  • Fall '11
  • COULON
  • Stochastic volatility, Smile Curve

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