Computational Methods for Quantitative Finance

Hilber, Norbert; Reichmann, Oleg; Schwab, Christoph; Winter, Christoph

doi:10.1007/978-3-642-35401-4

Cited by 51 publications

(54 citation statements)

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“…By classical results, see [8,15,17,39,45,47,48] for instance, the FD approximations of this equation converge to the solution w of Equation (75), something that we have noticed very clearly in our implementation. The solution w of Equation (75) is, however, different from the solution C SA of Equation (1) for v = h and with κ = 0.…”

Section: Cut-off Errorssupporting

confidence: 55%

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

2014

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We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for β = 1, namely ∂τ u = σ 2 1 2 (∂ 2The terms u j are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of u that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" τ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of ν, similar to Hagan's formula, but including also the mean reverting term. We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility. Contents32 5.2. The SABR model (with general β) 34 References 35

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Section: Cut-off Errorssupporting

confidence: 55%

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

2014

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“…The discrete LCP problem may be solved using the projected successive over relaxation (PSOR) procedure proposed by Cryer (1971). See also Hackbusch (1994);Hilber et al (2013). To solve the LCP using PSOR, we discretize v on a regular numerical grid using a suitable discrete scheme, leading to the discrete LCP…”

Section: Numerical Solutionsmentioning

confidence: 99%

“…The PSOR procedure we adopt is outlined in Hilber et al (2013), Table 5.1, which is reproduced as Algorithm 3, where 0 < ω < 2 is the relaxation parameter, and k = 1, ..., K is the spatial index for z.…”

Section: Numerical Solutionsmentioning

confidence: 99%

Optimal Investment-Consumption-Insurance with Durable and Perishable Consumption Goods in a Jump Diffusion Market

Sun

Perera²,

Shevchenko³

2019

SSRN Journal

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We investigate an optimal investment-consumption and optimal level of insurance on durable consumption goods with a positive loading in a continuous-time economy. We assume that the economic agent invests in the financial market and in durable as well as perishable consumption goods to derive utilities from consumption over time in a jumpdiffusion market. Assuming that the financial assets and durable consumption goods can be traded without transaction costs, we provide a semi-explicit solution for the optimal insurance coverage for durable goods and financial asset. With transaction costs for trading the durable good proportional to the total value of the durable good, we formulate the agent's optimization problem as a combined stochastic and impulse control problem, with an implicit intervention value function. We solve this problem numerically using stopping time iteration, and analyze the numerical results using illustrative examples.

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“…To be able to react instantaneously to market movements, complexity reduction techniques are of special interest. Both European and American options can be based on a parameter-dependent partial differential equation in time and asset price, see, e.g., [2,34] and the references therein. However due to the higher flexibility of exercising American options compared to European options, the mathematical model for an American model has to be enriched by a suitable inequality constraint, reflecting the arbitrage-free principle.…”

Section: Introductionmentioning

confidence: 99%

Complexity reduction for calibration to American options

Burkovska¹,

Glau²,

Mahlstedt³

et al. 2019

JCF

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American put options are among the most frequently traded single stock options, and their calibration is computationally challenging since no closed-form expression is available. Due to the higher flexibility in comparison to European options, the mathematical model involves additional constraints, and a variational inequality is obtained. We use the Heston stochastic volatility model to describe the price of a single stock option. In order to speed up the calibration process, we apply two model reduction strategies. Firstly, a reduced basis method (RBM) is used to define a suitable low-dimensional basis for the numerical approximation of the parameter-dependent partial differential equation (µPDE) model. By doing so the computational complexity for solving the µPDE is drastically reduced, and applications of standard minimization algorithms for the calibration are significantly faster than working with a high-dimensional finite element basis. Secondly, so-called de-Americanization strategies are applied. Here, the main idea is to reformulate the calibration problem for American options as a problem for European options and to exploit closed-form solutions. Both reduction techniques are systematically compared and tested for both synthetic and market data sets.

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Computational Methods for Quantitative Finance

Cited by 51 publications

References 0 publications

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

Optimal Investment-Consumption-Insurance with Durable and Perishable Consumption Goods in a Jump Diffusion Market

Complexity reduction for calibration to American options

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