# Improving Convergence of Binomial Schemes and the Edgeworth Expansion

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## Abstract

**:**

## 1. Introduction: Convergence of Binomial Trees

## 2. Asymptotic Analysis of Binomial Trees: Distributional Fit

**Remark**

**1.**

**Corollary**

**1.**

**Proof.**

## 3. Improving the Convergence Behavior

#### 3.1. Existing Methods in the Literature

#### 3.2. The 3/2-Optimal Model

**Proposition**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

#### CRR vs. RB

- The initial asset price ${s}_{0}$ is fixed to 100,
- the value x is uniformly distributed between 50 and 150,
- the riskless interest rate r is uniformly distributed between zero and $0.2$,
- the volatility σ is uniformly distributed between $0.1$ and $0.8$,
- the maturity T is chosen uniformly between zero and one years with probability $0.75$ and between one and five years with probability $0.25$.

## 4. Expansions for Barrier Option Prices

#### 4.1. Binomial Trees for Barrier Options

**Lemma**

**1.**

**Proposition**

**2.**

**Remark**

**6.**

#### 4.2. Numerical Results

**Remark**

**7.**

#### CRR vs. RB

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. The Edgeworth Expansion

**Theorem**

**A1.**

**Lemma**

**A1.**

**Lemma**

**A2.**

**Proof.**

## Appendix B. Proof of Proposition 1

**Proposition**

**B1.**

**Remark**

**B1.**

## Appendix C. Proof of Proposition 2

**Proposition**

**C1.**

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**Figure 5.**CRR vs. RB: up-and-in barrier put, $T=1$, $r=0.1$, $\sigma =0.25$, ${s}_{0}=100$, $K=110$, $B=120$.

**Figure 6.**CRR vs. 1-opt.: Up-and-in barrier put, $T=1$, $r=0.1$, $\sigma =0.25$, ${s}_{0}=100$, $K=105$, $B=120$.

**Figure 7.**RB with extrapolation: up-and-in barrier put with $T=1$, $r=0.1$, $\sigma =0.25$, ${s}_{0}=100$, $K=110$, $B=120$.

Parameters | n | CRR Tree | RB Extrapolation | 1-Optimal | BS Value |
---|---|---|---|---|---|

$T=1$ | 100 | 1.0370950 | 1.2728844 | 1.3071811 | 1.3714613 |

${s}_{0}=100$ | 200 | 1.1428755 | 1.3064705 | 1.3528020 | |

$K=110$, $B=120$ | 500 | 1.2210427 | 1.3667218 | 1.3668495 | |

$\sigma =0.25$ | 1000 | 1.2248525 | 1.3608690 | 1.3671287 | |

$r=0.1$ | 2000 | 1.3285299 | 1.3731534 | 1.3713814 | |

4000 | 1.3018025 | 1.3696310 | 1.3710375 |

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Bock, A.; Korn, R.
Improving Convergence of Binomial Schemes and the Edgeworth Expansion. *Risks* **2016**, *4*, 15.
https://doi.org/10.3390/risks4020015

**AMA Style**

Bock A, Korn R.
Improving Convergence of Binomial Schemes and the Edgeworth Expansion. *Risks*. 2016; 4(2):15.
https://doi.org/10.3390/risks4020015

**Chicago/Turabian Style**

Bock, Alona, and Ralf Korn.
2016. "Improving Convergence of Binomial Schemes and the Edgeworth Expansion" *Risks* 4, no. 2: 15.
https://doi.org/10.3390/risks4020015