# Bayesian Option Pricing Framework with Stochastic Volatility for FX Data

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## Abstract

**:**

## 1. Introduction

## 2. Stochastic Volatility Model and Option Pricing

#### 2.1. Problem Formulation

- It should be a simple model that enables us to apply the standard Bayesian inference via MCMC;
- It should be convenient to transform the model under $\mathbb{P}$ to the model under $\mathbb{Q}$, the risk-neutral probability;
- The resulting $\mathbb{Q}$-process should generate option prices close to market prices.

- (P
_{1}) - Equivalent probability measures: $\mathbb{P}\left(A\right)=0$ iff $\mathbb{Q}\left(A\right)=0$ for any event A;
- (P
_{2}) - Martingale property: ${\mathrm{E}}^{\mathbb{Q}}\left[\left.{S}_{t+1}\right|{\mathcal{F}}_{t}\right]={S}_{t}{e}^{r}$, where r is the one-step continuously compounded interest rate;
- (P
_{3}) - Equivalent local variance: ${\mathrm{V}}^{\mathbb{Q}}\left(\left.{y}_{t+1}\right|{\mathcal{F}}_{t}\right)={\mathrm{V}}^{\mathbb{P}}\left(\left.{y}_{t+1}\right|{\mathcal{F}}_{t}\right),$

**Theorem**

**1.**

**Proof of Theorem**

**1.**

#### 2.2. The SV Model with VG and t Error Distributions

#### 2.3. Bayesian Framework

**θ**and $\mathit{h}$ together can be viewed as an augmented parameter space. In the Bayesian paradigm, a full Bayesian approach for performing Bayesian inference is via the simulation-based MCMC algorithms, which iteratively sample posterior realisations from the joint posterior distribution

**θ**.

#### 2.4. Option Pricing

- Step A: For j = 1:J where J is the number of simulated stock price paths,
- A1:
- Set ${h}_{1}^{\ast}={h}_{N}$;
- A2:
- Set ${S}_{0}^{\ast}={S}_{N}$;
- A3:
- Sample ${\lambda}_{{h}_{1}}^{\ast},...,{\lambda}_{{h}_{T}}^{\ast}$ from $Ga\left(\frac{\widehat{\nu}}{2},\frac{\widehat{\nu}}{2}\right)$;
- A4:
- Sample ${\lambda}_{{y}_{1}}^{\ast},...,{\lambda}_{{y}_{T}}^{\ast}$ from $Ga\left(\frac{\widehat{\alpha}}{2},\frac{\widehat{\alpha}}{2}\right)$;

- Step B: For i = 1:T,
- B1:
- Sample ${y}_{i}^{\ast}\left(j\right)$ from $N\left(r+\widehat{\beta}{e}^{{h}_{i}^{\ast}\left(j\right)/2},{\lambda}_{{y}_{i}\left(j\right)}^{\ast}{e}^{{h}_{i}^{\ast}\left(j\right)}\right)$;
- B2:
- Sample ${h}_{i}^{\ast}\left(j\right)$ from$$N\left(\widehat{\mu}+\widehat{\varphi}\left({h}_{i-1}^{\ast}\left(j\right)-\widehat{\mu}\right),{\left({\lambda}_{{h}_{i}\left(j\right)}^{\ast}\right)}^{-1}{\widehat{\tau}}^{2}\right);$$
- B3:
- Set ${S}_{i}^{\ast}\left(j\right)={S}_{i-1}^{\ast}\left(j\right){e}^{{y}_{i}^{\ast}\left(j\right)}$;

- Step C: By (16), set$$C\cong \frac{{e}^{-rT}}{J}\sum _{j=1}^{J}max\left({S}_{T}^{\ast}\left(j\right)-K,0\right).$$

## 3. Empirical Studies

#### 3.1. FX Market Data

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Definition**

**A1.**

## References

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**Figure 2.**Comparison of volatility smile on the YEN–USD FX rate from the Black-Scholes (BS) model, $VG$-t SV model, and the market.

**Table 1.**Deviance information criterion (DIC) and Bayesian information criterion (BIC) of stochastic volatility (SV) models with different error distributions. N-N: normal-normal; $VG$: variance gamma.

Model | N-N | t-t | t-$VG$ | $VG$-t | $VG$-$VG$ |
---|---|---|---|---|---|

DIC | 383.3 | 379.9 | 378.6 | 353.3 | 360.2 |

BIC | 386.6 | 376.9 | 379.7 | 370.9 | 371.0 |

Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | Sd. |
---|---|---|---|---|---|---|

$-0.0188$ | $-0.0027$ | $-0.0003$ | 0.0002 | 0.0031 | 0.0310 | 0.0052 |

K = 87.94 | K = 89.28 | K = 90.15 | K = 91.37 | K = 92.32 | K = 93.53 | |

Mid | 0.1318 | 0.1281 | 0.1258 | 0.1229 | 0.1213 | 0.1205 |

Bid | 0.1154 | 0.1184 | 0.1185 | 0.1176 | 0.1167 | 0.1163 |

Ask | 0.1483 | 0.1377 | 0.1331 | 0.1283 | 0.1259 | 0.1247 |

K = 93.84 | K = 95.88 | K = 97.24 | K = 98.23 | K = 99.79 | ||

Mid | 0.1219 | 0.1241 | 0.1274 | 0.1301 | 0.1343 | |

Bid | 0.1175 | 0.1189 | 0.1203 | 0.1206 | 0.1182 | |

Ask | 0.1264 | 0.1293 | 0.1345 | 0.1395 | 0.1503 |

Par. | Mean | Sd | 95% CI |
---|---|---|---|

β | 0.078 | 0.064 | ($-0.046$ , 0.203) |

τ | 0.225 | 0.067 | (0.128 , 0.361) |

μ | $-10.420$ | 0.351 | ($-10.950$, $-9.585$) |

ϕ | 0.873 | 0.088 | (0.687 , 0.988) |

ν | 8.176 | 4.155 | (2.531 , 18.114) |

α | 15.622 | 8.582 | (4.290 , 34.868) |

**Table 5.**Results of option prices on the YEN–USD FX rate from the Black-Scholes (BS) model, the SV model, and the market.

Model | K = 87.94 | K = 89.28 | K = 90.15 | K = 91.37 | K = 92.32 | K = 93.53 |

Mkt | 5.728 | 4.480 | 3.711 | 2.712 | 2.040 | 1.335 |

BS | 5.658 | 4.339 | 3.507 | 2.410 | 1.676 | 0.934 |

SV | 5.679 | 4.372 | 3.555 | 2.480 | 1.759 | 1.020 |

Model | K = 94.84 | K = 95.88 | K = 97.24 | K = 98.23 | K = 99.79 | |

Mkt | 0.049 | 0.509 | 0.272 | 0.169 | 0.077 | |

BS | 0.414 | 0.189 | 0.054 | 0.019 | 0.003 | |

SV | 0.497 | 0.259 | 0.104 | 0.055 | 0.022 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Choy, S.T.B.; Wong, H.Y.
Bayesian Option Pricing Framework with Stochastic Volatility for FX Data. *Risks* **2016**, *4*, 51.
https://doi.org/10.3390/risks4040051

**AMA Style**

Wang Y, Choy STB, Wong HY.
Bayesian Option Pricing Framework with Stochastic Volatility for FX Data. *Risks*. 2016; 4(4):51.
https://doi.org/10.3390/risks4040051

**Chicago/Turabian Style**

Wang, Ying, Sai Tsang Boris Choy, and Hoi Ying Wong.
2016. "Bayesian Option Pricing Framework with Stochastic Volatility for FX Data" *Risks* 4, no. 4: 51.
https://doi.org/10.3390/risks4040051