Smile risk is often managed using the explicit implied volatility formulas developed for the SABR model [1]. These asymptotic formulas are not exact, and this can lead to arbitrage for low strike options. Here we provide an alternate method for pricing options under the SABR model: We use asymptotic techniques to reduce the SABR model from two dimensions to one dimension. This leads to an effective one-dimensional forward equation for the probability density which has the same asymptotic order of accuracy as the explicit implied volatility formulas. We obtain arbitrage-free option prices by numerically solving this PDE. The implied volatilities obtained from the numerical solutions closely match the explicit implied volatility curves, apart from a boundary layer at very low rates. For very low-rate environments, or for very low strikes, the implied absolute (normal) volatility dips downward, closely matching market observations. We also show how negative rates can be accommodated by replacing the F factor with (F + a) .

The SABR model has two variables, the forward asset price and the local volatility . A singular perturbation analysis has shown that the marginal density Q(T, F) defined by can be found through by solving a one‐dimensional (1D) effective forward equation of the form where and This reduces the valuation of European options to one spatial dimension from two. Recently, similar asymptotic analyses have shown that for all commonly used stochastic volatility models, the marginal density Q(Tex, F) can be obtained through O(ε2) by solving the same 1D effective forward equation. The only differences are in theformulas for the coefficients b and c in terms of each model's fundamental parameters. These stochastic volatility models include the Heston and generalized Heston models, the mean‐reverting SABR (λ‐SABR) and dynamic SABR models, the exponential volatility models, ZABR‐like models, cross‐FX SABR models, and the SABR models for baskets and spreads. Here we analyze the above “universal” effective forward equation, and obtain explicit asymptotic formulas for the implied volatilities of European options for all the above models. These new formulas reduce to the original SABR implied vol formulas under moderate conditions, but are much more accurate in extreme situations.

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Under the New Education Policy, all teacher education programmes will include instruction on how to teach children with specific impairments. The LATEST PROVISION in the NEW School POLICY (NEP) 2020, approved by India's Union Cabinet in July 2020, encourages, and promotes "Barrier-free access to Education for all Children with Disabilities." India's first Education Policy was initially established in 1986 and last revised in 1992. Since then, India needed a change in its entire Education Policy. The New Education Policy describes the much-awaited new reforms that India was looking for. These adaptations are designed with special consideration for children with impairments and those from low-income families. The focus of this new rule's implementation will be on two things. Imparting knowledge and know how to teach faculties on how to teach children with specific disabilities. These nobs are keeping in mind the Right of Persons with Disability Act-2016 to provide barrier-free education to children with disability. The most common barrier to a child's access to pre-school and primary education is disability. Less than 40% of school buildings have ramps, and only around 17% of schools have accessible restrooms. According to the NEP, children with disabilities will be able to participate equally in all aspects of the educational system. This paper focuses the provisions given in the New Education Policy 2020 for the divyangjan to show how education for them must be barrier-free trying to reach them. It is also a step to bring the divyangjan in the mainstream of learning.