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Register a new userMarginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations
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Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l leq d$. Global conditions are found which replace the well-known not-in-cutlocus condition known from heat-kernel asymptotics. Our small noise expansion allows for a second order exponential factor. As application, new light is shed on the Takanobu--Watanabe expansion of Brownian motion and Levys stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in a compagnion paper.
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We extend the Bismut-Elworthy-Li formula to non-degenerate jump diffusions and payoff functions depending on the process at multiple future times. In the spirit of Fournie et al [13] and Davis and Johansson [9] this can improve Monte Carlo numerics for stochastic volatility models with jumps. To this end one needs so-called Malliavin weights and we give explicit formulae valid in presence of jumps: (a) In a non-degenerate situation, the extended BEL formula represents possible Malliavin weights as Ito integrals with explicit integrands; (b) in a hypoelliptic setting we review work of Arnaudon and Thalmaier [1] and also find explicit weights, now involving the Malliavin covariance matrix, but still straight-forward to implement. (This is in contrast to recent work by Forster, Lutkebohmert and Teichmann where weights are constructed as anticipating Skorohod integrals.) We give some financial examples covered by (b) but note that most practical cases of poor Monte Carlo performance, Digital Cliquet contracts for instance, can be dealt with by the extended BEL formula and hence without any reliance on Malliavin calculus at all. We then discuss some of the approximations, often ignored in the literature, needed to justify the use of the Malliavin weights in the context of standard jump diffusion models. Finally, as all this is meant to improve numerics, we give some numerical results with focus on Cliquets under the Heston model with jumps.
We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Holder continuous with exponent $deltain (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $deltage 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotanis criterion for martingality of a one-dimensional diffusion in natural scale.
(Renegar, 2016) introduced a novel approach to transforming generic conic optimization problems into unconstrained, uniformly Lipschitz continuous minimization. We introduce radial transformations generalizing these ideas, equipped with an entirely new motivation and development that avoids any reliance on convex cones or functions. Perhaps of greatest practical importance, this facilitates the development of new families of projection-free first-order methods applicable even in the presence of nonconvex objectives and constraint sets. Our generalized construction of this radial transformation uncovers that it is dual (i.e., self-inverse) for a wide range of functions including all concave objectives. This gives a powerful new duality relating optimization problems to their radially dual problem. For a broad class of functions, we characterize continuity, differentiability, and convexity under the radial transformation as well as develop a calculus for it. This radial duality provides a strong foundation for designing projection-free radial optimization algorithms, which is carried out in the second part of this work.
Following Boukai (2021) we present the Generalized Gamma (GG) distribution as a possible RND for modeling European options prices under Hestons (1993) stochastic volatility (SV) model. This distribution is seen as especially useful in situations in which the spots price follows a negatively skewed distribution and hence, Black-Scholes based (i.e. the log-normal distribution) modeling is largely inapt. We apply the GG distribution as RND to modeling current market option data on three large market-index ETFs, namely the SPY, IWM and QQQ as well as on the TLT (an ETF that tracks an index of long term US Treasury bonds). The current option chain of each of the three market-index ETFs shows of a pronounced skew of their volatility `smile which indicates a likely distortion in the Black-Scholes modeling of such option data. Reflective of entirely different market expectations, this distortion appears not to exist in the TLT option data. We provide a thorough modeling of the available option data we have on each ETF (with the October 15, 2021 expiration) based on the GG distribution and compared it to the option pricing and RND modeling obtained directly from a well-calibrated Hestons (1993) SV model (both theoretically and empirically, using Monte-Carlo simulations of the spots price). All three market-index ETFs exhibit negatively skewed distributions which are well-matched with those derived under the GG distribution as RND. The inadequacy of the Black-Scholes modeling in such instances which involve negatively skewed distribution is further illustrated by its impact on the hedging factor, delta, and the immediate implications to the retail trader. In contrast, for the TLT ETF, which exhibits no such distortion to the volatility `smile, the three pricing models (i.e. Hestons, Black-Scholes and Generalized Gamma) appear to yield similar results.
This paper proposes the sample path generation method for the stochastic volatility version of CGMY process. We present the Monte-Carlo method for European and American option pricing with the sample path generation and calibrate model parameters to the American style S&P 100 index options market, using the least square regression method. Moreover, we discuss path-dependent options such as Asian and Barrier options.
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