1. What contributions have the authors mentioned in the paper "Small-time asymptotics for implied volatility under the heston model" ?
The authors apply the Gärtner-Ellis theorem from large deviations theory to the exponential affine closed-form expression for the moment generating function of the log forward price, to show that it satisfies a small-time large deviation principle.. The rate function is computed as Fenchel-Legendre transform, and the authors show that this can actually be computed as a standard Legendre transform, which is a simple numerical root-finding exercise.
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2. What is the first term in Eq 15?
(15)The first term in Eq 15 is non-negative on (p−, p+) because θ 2 cot θ 2 ≤ 1 if θ in (−2π, 2π), which will be the case when p lies in the interval (p−, p+) (by careful checking of the entries in the table in Theorem 1.1).
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3. What is the simplest way to get the lower bound?
Using the put-call parity, the authors can combine Corollaries 2.1 and 2.2 to obtain the following result, which holds for all x ∈ R, x ̸= 0.8 Corollary 2.3t log ( E(St −K)+ − (S0 −K)+ ) = t log ( E(K − ST )+ − (K − S0)+ ) = Λ∗(x) , (28)where x = log KS0 .
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4. What is the Heston stochastic volatility model?
Λ∗ is continuous and strictly increasing when x > 0, and− lim t→0 t logP(Xt − x0 < k) = − lim t→0 t logP(Xt − x0 ≤ k) = inf {x:x≤k}Λ∗ ′ (x) = Λ∗(k) (4)for k < 0, because Λ∗ is continuous and strictly decreasing when x < 0.
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