Analytical pricing of single barrier options under local volatility models

Funahashi, Hideharu; Kijima, Masaaki

doi:10.1080/14697688.2015.1101483

“…It was already noted, e.g. Funahashi and Kijima (2016), that the approach of Carr et al Carr and Nadtochiy (2011).…”

Section: Remarks On Alternative Ideamentioning

confidence: 98%

“…In this case, the European payoff of a hedging instrument is not a vanilla type in general, leading to the approximations with vanilla options in practice. Funahashi and Kijima (2016) considered the problem of static hedging under the symmetrized volatility model, but the stringent assumptions on the volatility function in this paper or Carr et al (1998) are inconsistent with market behaviors such as the leverage effect or the implied volatility skew. It is discussed later in the paper that the static hedge solution in Carr et al (1998) can be represented as a solution to the integral equation based approach.…”

Section: Introductionmentioning

confidence: 99%

Integral Equation Based Approach for Static Options Replication

Kim

¹

,

Lim

²

2018

SSRN Journal

1

0

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This study provides a systematic and unified approach for constructing exact and static replications for exotic options, using the theory of integral equations. In particular, we focus on barrier-type options including standard, double and sequential barriers. Our primary approach to static options replication is the DEK method proposed by Derman et al. (1995). However, our solution approach is novel in the sense that we study its continuous-time version using integral equations. We prove the existence and uniqueness of hedge weights under certain conditions. Further, if the underlying dynamics is time-homogeneous, then hedge weights can be explicitly found via Laplace transforms. Based on our framework, we propose an improved version of the DEK method. This method is applicable under general Markovian diffusion with killing.

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“…It was already noted, e.g. Funahashi and Kijima (2016), that the approach of Carr et al (1998) is not extendable even to the CEV model. On the other hand, Proposition 1 provides more flexibility when it comes to model selection.…”

Section: Remarks On Alternative Ideamentioning

confidence: 99%

“…In this case, the European payoff of a hedging instrument is not a vanilla type in general, leading to the approximations with vanilla options in practice. Funahashi and Kijima (2016) considered the problem of static hedging under the symmetrized volatility model, but the stringent assumptions on the volatility function in this paper or Carr et al (1998) are inconsistent with market behaviors such as the leverage effect or the implied volatility skew. It is discussed later in the paper that the static hedge solution in Carr et al (1998) can be represented as a solution to the integral equation based approach.…”

Section: Introductionmentioning

confidence: 99%

Static replication of barrier-type options via integral equations

Kim

¹

,

Lim

²

2020

Quantitative Finance

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This study provides a systematic and unified approach for constructing exact and static replications for exotic options, using the theory of integral equations. In particular, we focus on barrier-type options including standard, double and sequential barriers. Our primary approach to static options replication is the DEK method proposed by Derman et al. (1995). However, our solution approach is novel in the sense that we study its continuous-time version using integral equations. We prove the existence and uniqueness of hedge weights under certain conditions. Further, if the underlying dynamics is time-homogeneous, then hedge weights can be explicitly found via Laplace transforms. Based on our framework, we propose an improved version of the DEK method. This method is applicable under general Markovian diffusion with killing.

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“…Merton () derives analytical formulas for single‐barrier options under the BS model. Funahashi and Kijima () provide approximation formulas for single‐barrier options under LV models whose volatility depends on stock price but not time. There are no approximation formulas for double‐barrier options under such models.…”

Section: Introductionmentioning

confidence: 99%

Efficient trinomial trees for local‐volatility models in pricing double‐barrier options

Lok

¹

,

Lyuu

²

2019

Journal of Futures Markets

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A local‐volatility (LV) model captures the volatility smile while retaining the preference freedom of the Black–Scholes model. Past attempts to construct a smile‐consistent tree for the LV surface do not guarantee validity. This paper presents an efficient and valid smile‐consistent tree for the LV model. The only assumption is that the LV surface be upper‐ and lower‐bounded. With this tree, double‐barrier options can be priced with fast convergence even in the presence of volatility smile. This is confirmed numerically. An implied tree is also presented. It recovers the LV surface reasonably well.

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Analytical pricing of single barrier options under local volatility models

Cited by 5 publications

References 20 publications

Integral Equation Based Approach for Static Options Replication

Integral Equation Based Approach for Static Options Replication

Static replication of barrier-type options via integral equations

Efficient trinomial trees for local‐volatility models in pricing double‐barrier options

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