Heston和SABR模型二者皆是市場上模擬隱含波動度時，常用到的模型，本篇論文詳細介紹二者的推導、參數校準方式以及意義，然後以台指選擇權作為樣本，研究距到期日天數以及履約價對於二者在計算台指選擇權之隱含波動度時，會有甚麼影響，另外，我們對二者做敏感度分析，觀察在上下變動各個參數10%的大小時，計算誤差會有甚麼變化。

研究結果顯示，固定履約價時，愈接近到期日，兩個模型計算誤差的差距會越大，同時，該現象會隨著離價平愈遠而愈明顯，另一方面，固定距到期日天數時，在履約價遠離價平的過程中，計算誤差的差距會突然暴增，爾後變小。不過，最重要的是，SABR的計算誤差都是明顯小於Heston的。在敏感度分析方面，價外買權的部分，Heston的計算誤差因參數變化而有明顯增加，而SABR因參數變畫增加的計算誤差大部分則不超過0.5%。在價外賣權的部分，Heston表現較佳，但並沒有明顯優於SABR，整體而言，Heston的計算誤差對參數變化的敏感性是高於SABR的。

Heston and SABR model are usually used on simulating implied volatility. This paper details derivation, parameter calibration and parameter meaning of both models. Then we use TXO as sample to study how the day number before expiration date and strike price affect the performance when the two models are used to simulate implied volatility. We also observe how the simulation error change when we increase or decrease parameter value 10%.

The study shows that the difference between two models simulation error will increase when the expiration date close and the strike price is fixed. This situation will be more clear when the difference between strike price and price which is at the money increase. On the other hand, the simulation error will suddenly sharply increase and then decrease when the strike price is gradually away from at-the-money price and the day number before expiration date is fixed. The most important thing is SABR model simulation error is significantly smaller than Heston model simulation error. The sensitivity analysis shows that Heston model simulation error of call option which is out-of-the-money has significantly increase when parameter value is changed. However, SABR model simulation error is increased not more than 0.5% when we use the same kind of option. If we consider put option which is out-of-the-money, Heston model has a better performance but has no obviously difference between Heston and SABR model. In general, Heston model simulation error is bigger than SABR model simulation error when we increase or decrease parameter value.

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