5 research outputs found
Small-Time Asymptotics of Option Prices and First Absolute Moments
We study the leading term in the small-time asymptotics of at-the-money call
option prices when the stock price process follows a general martingale.
This is equivalent to studying the first centered absolute moment of . We
show that if has a continuous part, the leading term is of order
in time and depends only on the initial value of the volatility.
Furthermore, the term is linear in if and only if is of finite
variation. The leading terms for pure-jump processes with infinite variation
are between these two cases; we obtain their exact form for stable-like small
jumps. To derive these results, we use a natural approximation of so that
calculations are necessary only for the class of L\'evy processes.Comment: 22 pages; forthcoming in 'Journal of Applied Probability
General smile asymptotics with bounded maturity
We provide explicit conditions on the distribution of risk-neutral
log-returns which yield sharp asymptotic estimates on the implied volatility
smile. We allow for a variety of asymptotic regimes, including both small
maturity (with arbitrary strike) and extreme strike (with arbitrary bounded
maturity), extending previous work of Benaim and Friz [Math. Finance 19 (2009),
1-12]. We present applications to popular models, including Carr-Wu finite
moment logstable model, Merton's jump diffusion model and Heston's model.Comment: 35 pages, 2 figures. To appear on SIAM Journal on Financial
Mathematic