d ˆ Z t 70 where ˆ Z t t is a Brownian motion correlated with W Here α is

D ˆ z t 70 where ˆ z t t is a brownian motion

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· · · d ˆ Z t , 70
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where { ˆ Z t , t 0 } is a Brownian motion correlated with W . Here α is called the rate of mean-reversion and m is the long-run mean-level of Y . The drift term pulls Y towards m and consequently, we would expect that σ t is pulled towards f ( m ) (approximately). The second Brownian motion ˆ Z is typically negatively correlated with W and can be written ˆ Z t = ρW t + radicalbig 1 ρ 2 Z t , where Z is a Brownian motion independent of W , and ρ is the correlation coefficient. Density Function with stochastic volatility What effect does stochastic volatility have on the probability density function of the stock price? To answer this question qualitatively, we plot in Figure 13 the density function (estimated from simulation) of a popular stochastic volatility model. 40 60 80 100 120 140 160 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Stock Price in 6 months Figure 13: Density functions for the S&P 500 index distribution six months forward. The Black- Scholes density (dashed curve) uses the constant volatility σ = 0 . 1 , and the stochastic volatility density is generated by simulation. The correlation is ρ = 0 . 2 . The mean growth rate of the stock is μ =0 . 15 . Notice the fatter tails because of the random volatility. In particular the negative corre- lation causes the tails to be asymmetric: the left tail is fatter. Stochastic Volatility implies Smile When the correlation is zero, it is known that stochastic volatility models predict option prices whose implied volatilities smile. This result is an important success of these models, although the fact that one observes smile curves does not necessarily imply that volatility is stochastic. For the correlated case ( ρ negationslash = 0), a similar general result is not known, but numerical simulations for a lot of models display a negative skew for negative correlation and positive 71
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skew for positive correlation. This is a strong indicator that stochastic volatility models are “doing the right thing”. Difficulties New difficulties are associated with the stochastic volatility approach: Volatility is not directly observed. As a consequence, estimation of the parameters of a specific model and the current level of volatility is not straightforward. There is no canonical stochastic volatility model that is generally accepted. We have to deal with an incomplete market which means derivatives cannot be perfectly hedged with just the underlying stock. In addition, a volatility risk premium (the price the market attaches to volatility uncertainty or “crash-o-phobia”) has to be estimated from option prices. 15 Hedging with Greeks As we have seen, stock price volatility appears to be best treated as a random process, rather than constant, as in the Black-Scholes model. However, volatility is not a traded quantity, and so different instruments are needed to hedge against the risk of changing volatility. In practice, derivative instruments (usually vanilla contracts such as put and calls) are used to de-sensitize (or immunize) a portfolio against changes in volatility, and for the effects of infrequent delta hedging.
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  • Fall '11
  • COULON
  • Stochastic volatility, Smile Curve

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