Exercises 10.1 SVI parametrization Gatheral [89] presents the following \Stochastic Volatility Inspired" (SVI)

parametrization of the implied volatility

BS(; k)2 = a + b



(k ???? m) +

p

(k ???? m)2 + 2



where the ve parameters a; b; ;  and m depend on the maturity  . This parametrization agrees with the Lee moment formula, i.e., the large-strike behavior is at most linear in the moneyness k.

1: Prove that the no-arbitrage condition @2C(;K)

@K2 > 0 is preserved when k >> 1 if and only jb(1 + )j  2