Question

In the SABR model, suppose that $F_0=5, \beta=0.5, \sigma_0=0.447$ (equivalent to a lognormal volatility of $20 \%$ ), and $T=1$. Show how the volatility smile varies with $\rho$ for (a) $v=0.6$ and (b) $v=1.2$. Consider value of $\rho$ equal to $0.4,0.2,0,-0.2,-0.4$ and values of the strike price equal to $4.0,4.5,5.0,5.5,6.0$.

    In the SABR model, suppose that $F_0=5, \beta=0.5, \sigma_0=0.447$ (equivalent to a lognormal volatility of $20 \%$ ), and $T=1$. Show how the volatility smile varies with $\rho$ for (a) $v=0.6$ and (b) $v=1.2$. Consider value of $\rho$ equal to $0.4,0.2,0,-0.2,-0.4$ and values of the strike price equal to $4.0,4.5,5.0,5.5,6.0$.

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Options, Futures, and Other Derivatives

Options, Futures, and Other Derivatives

John C. Hull 10th Edition

Chapter 27, Problem 23

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Step 1

Step 1: Calculate the implied volatilities for each combination of $\rho$ and strike price using the SABR model formula: $$\sigma(K) = \frac{\alpha}{(F_0K)^{(1-\beta)/2}} \left[1 + \frac{(1-\beta)^2}{24}\log^2\left(\frac{F_0}{K}\right) + Show more…

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In the SABR model, suppose that $F_0=5, \beta=0.5, \sigma_0=0.447$ (equivalent to a lognormal volatility of $20 \%$ ), and $T=1$. Show how the volatility smile varies with $\rho$ for (a) $v=0.6$ and (b) $v=1.2$. Consider value of $\rho$ equal to $0.4,0.2,0,-0.2,-0.4$ and values of the strike price equal to $4.0,4.5,5.0,5.5,6.0$.

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