臺灣博碩士論文加值系統

在這篇論文中，我們推導一個對股價報酬的變異數交換做定價公式，
並且與Carr and Wu(2004)此篇論文中的定價公式在相同的架構之下做比較。
此外，我們也在Duan, Ritchken and Sun (2004)所提出假設股價報酬跟變異數都產生內容的
NGARCH(1,1) 不連續模型之下，推導出對變異數交換的定價公式。
我們也發現，跳躍現象的發生在變異數跟報酬均不連續的情況之下，
對變異數交換的價格有明顯的影響。

In this paper we developed a model for valuing variance swaps with jumps in the returns of underlying asset.
We compare our simulation results with those of Carr and Wu (2004) model under the same framework.
We find that our model value of variance swap contracts are very close to those of Carr and Wu model.
We then applied Duan, Ritchken and Sun (2004) GARCH jump framework which analyzes which jumps could happen in both asset return and volatility
to develop a more general model for valuing variance swaps.
From the simulation results, we find that both jumps in return and volatility will significantly affect the values of variance swaps.

Contents
1 Introduction 1
2 Variance swaps 3
2.1 How to price variance when the underlying asset price is continuous . 3
2.2 Hedging for Variance Swaps . . . . . . . . . . . . . . . . . . . . . . . 6
3 Pricing variance swaps when underlying stock return processes are
discontinued 6
3.1 The mixed processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Hedging for Variance Swaps under Mixed-Jump Process . . . . . . . . 10
3.5 The Comparison of our Model and Carr-Wu (2004) model . . . . . . 10
4 Pricing Variance Swaps When the Underlying Stocks Return Processes
Follow GARCH Jump Model 12
4.1 A NGARCH(1,1) Jump Model . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Conclusion 18
References 20
A The proof of equations (26)-(27) 22
B The proof of equation (40) 23
C The proof of equation (41)-(42) 24

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