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jayjo
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Joined: March 31st, 2016, 6:46 pm

No-Arb Conditions for Puts and Calls & Transitioning from absolute prices to moneyness space

April 17th, 2018, 12:41 am

I'm reading Fengler's paper on Arb-Free Smoothing and would like some clarification on the weights [$]w_i[$] proposed in the paragraph addressing the financial theory of fitting the spline.


From the perspective of financial theory, one might worry that the sum of squared differences of [$] y_i - g ( u_i ) [$] in (2) may not be the right measure of loss, since an investor is only interested in relative prices. This concern can be addressed by using the underlying asset price as numéraire. By setting [$]w_i = S^{−2}_{t}[$] and switching to a spot moneyness space [$]\tilde u = u/S_t[$], one can conduct the minimization on relative option prices after some obvious adjustments to the no-arbitrage constraints in (19). The resulting curve estimate [$](\mathbf{\tilde g ^\top }, \tilde \gamma ^\top )[$] can be inflated again via [$]g_i(u) = S_t \tilde g_i(\tilde u)[$] and [$]\gamma_i(u) = \tilde \gamma _i(\tilde u)/S_t[$], which yields a natural cubic spline as can be verified from (12). Seemingly this approach comes at the additional cost of a homogeneity assumption. However, as can be observed from (15), in choosing as smoothing parameter [$] \tilde \lambda = \lambda S_{t}^{−3}[$] the program in relative prices is equivalent to the former one in absolute prices (up to the aforementioned scales).


Before putting into the optimization software, I convert all my strikes (knots [$]g_i[$]) to moneyness space (ensuring my [$]h[$] are now correctly spaced). Then my weights [$]w_i = S^{−2}_{t}[$] only impact the [$]\mathbf{y}[$] vector and the matrix [$]\mathbf{B}[$].

The optimization solves without the no-arb constraints - and is a good fit but still has arb opportunities in the IVS (as it should, the constraints have not been added). I believe I'm having a problem with the "obvious" adjustments to the no-arb constraints. Can we be specific about what they are? Also, can we adjust them for puts, as well?

The optimization without the no-arb constraints should require no modifications for put prices (because it is a generic method for fitting a spline to observed data - it's from Green & Silverman - Nonparametric Regression and Generalized Linear Models).
$$ \begin{align} \frac{g_2 - g_1}{h_1} - \frac{h1}{6}\gamma_2 & \geq & -e^{-\int_{t_m}^{T} r_s ds} &  \\
- \frac{g_n - g_{n-1}}{h_{n-1}} - \frac{h_{n-1}}{6}\gamma_{n-1}  & \geq & 0 &  \\
g_1 & \leq & e^{- \int_{t_m}^{T} \delta_s ds} S_t \\
g_1 & \geq  & e^{- \int_{t_m}^{T} \delta_s ds} S_t  - e^{- \int_{t_m}^{T} r_s ds} u_1  
\end{align}$$ 

The original optimization problem also includes [$]g_n \geq 0[$] and [$]\lambda_i \geq 0[$] which I've excluded because they won't change for puts or for moneyness space. What are the changes for puts and the obvious changes for moneyness space?
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