<p>In this paper we investigate pricing of variance swaps contracts. The</p><p>literature is mostly dedicated to the pricing using replication with</p><p>portfolio of vanilla options. In some papers the valuation with stochastic</p><p>volatility models is discussed as well. Stochastic volatility is becoming</p><p>more and more interesting to the investors. Therefore we decided to</p><p>perform valuation with the Heston stochastic volatility model, as well</p><p>as by using replication strategy.</p><p>The thesis was done at SunGard Front Arena, so for testing the replica-</p><p>tion strategy Front Arena software was used. For calibration and testing</p><p>of the Heston model we used MatLab.</p>

In this paper we investigate pricing of variance swaps contracts. The literature is mostly dedicated to the pricing using replication with portfolio of vanilla options. In some papers the valuation with stochastic volatility models is discussed as well. Stochastic volatility is becoming more and more interesting to the investors. Therefore we decided to perform valuation with the Heston stochastic volatility model, as well as by using replication strategy. The thesis was done at SunGard Front Arena, so for testing the replica- tion strategy Front Arena software was used. For calibration and testing of the Heston model we used MatLab.

In this dissertation, the price of variance swaps under stochastic volatility models based on the work done by Barndorff-Nielsen and Shepard (2001) and Heston (1993) is discussed. The choice of these models is as a result of properties they possess which position them as an improvement to the traditional Black-Scholes (1973) model. Furthermore, the popularity of these models in literature makes them particularly attractive. A lot of work has been done in the area of pricing variance swaps since their inception in the late 1990’s. The growth in the number of variance contracts written came as a result of investors’ increasing need to be hedged against exposure to future variance fluctuations. The task at the core of this dissertation is to derive closed or semi-closed form expressions of the fair price of variance swaps under the two stochastic models. Although various researchers have shown that stochastic models produce close to market results, it is more desirable to obtain the fair price of variance derivatives using models under which no assumptions about the dynamics of the underlying asset are made. This is the work of a useful analytical formula derived by Demeterfi, Derman, Kamal and Zou (1999) in which the price of variance swaps is hedged through a finite portfolio of European call and put options of different strike prices. This scheme is practically explored in an example. Lastly, conclusions on pricing using each of the methodologies are given. / Dissertation (MSc)--University of Pretoria, 2019. / Mathematics and Applied Mathematics / MSc (Financial Engineering) / Unrestricted

In this thesis we develop and test a new method for interpolating and extrapolating prices of European options. The theoretical base originates from the local variance gamma model developed by Carr (2008), in which the local volatility model by Dupire (1994) is combined with the variance gamma model by Madan and Seneta (1990). By solving a simplied version of the Dupire equation under the assumption of a continuous ve parameter di usion term, we derive a parameterization dened for strikes in an interval of arbitrary size. The parameterization produces positive option prices which satisfy both conditions for absence of arbitrage in a one maturity setting, i.e. all adjacent vertical spreads and buttery spreads are priced non-negatively. The method is implemented and tested in the FX-option market. We suggest two sub-models, one with three and one with ve degrees of freedom. By using a least-square approach, we calibrate the two sub-models against 416 Reuters quoted volatility smiles. Both sub-models succeeds in generating prices within the bid-ask spread for all options in the sample. Compared to the three parameter model, the model with ve parameters calibrates more exactly to market quoted mids but has a longer calibration time. The three parameter model calibrates remarkably quickly; in a MATLAB implementation using a Levenberg-Marquardt algorithm the average calibration time is approximately 1 ms. Both sub-models produce volatility smiles which are C2 and well-behaving. Further, we suggest a technique allowing for arbitrage-free interpolation of calibrated option price functions in the maturity dimension. The interpolation is performed in parameter space, where every set of parameters uniquely determines an option price function. Furthermore, we produce sucient conditions to ensure absence of calendar spread arbitrage when calibrating the proposed model to several maturities. We use this technique to produce implied volatility surfaces which are suciently smooth, satisfy all conditions for absence of arbitrage and fit market quoted volatility surfaces within the bid-ask spread. In the final chapter we use the results for producing Dupire local volatility surfaces and for pricing variance swaps.

券商和投資大眾越來越了解價格風險管理的重要性，但是對於波動度風險管理工具及其重要性的認知卻較為貧乏。論文探討的即是美歐新興的波動度管理工具：波動度交換契約(volatility swaps)和變異數交換契約(variance swaps)。藉著波動度交換契約，交易者就可以將所暴露的不確定風險轉換為固定的風險。 論文的焦點在於變異數交換契約(variance swaps)公平履約價的訂定。文章中所使用的評價方法是複製法(replictions strategy)，在唯一的假設條件下：股價的變動是連續的，用已知的金融商品複製成新的商品，而複製成本也就是變異數交換契約的公平價格。 在完美的市場中，我們用履約價從零到無限大的選擇權複製變異數交換契約，但是現實的情況下並不允許如此，改用有限範圍的選擇權複製其損益。故再加以討論當假設不成立：股價跳空時，以及用有限範圍履約價對複製策略的影響。 而波動度交換契約(volaility swaps)不管在理論上或是實務上的評價、避險的難度都遠高於變異數交換契約，在第七章節中，引用泰勒展開式和Heston的波動度模型，求得波動度交換契約公平履約價Kvol的評價公式。 一、中文部分： 1.、 寶來金融創新雙月刊 p31-p38 ‘波動性風險可以規避嗎？’ 陳凌鶴、林瑞瑤 2 、國際金融市場泛論與分析 陳松男著 3 、選擇權與期貨：衍生性商品 陳松男著 4 、期貨市場分析 朱浩民著 二、英文部分： 1. Black F, and M Scholes, 1973 ‘The pricing of options and Corporate liabilities” Journal of Political Economy 81, pages 637-659. 2. Carr P ,and D Madan, 1999 “Introducing the covariance swaps” Risk February, pages 47-51 3. Chriss N ,and W Morokoff, 1999 “Market risk for variance swaps” Risk October, pages 55-59 4. Derman E, 1999 “Regimes of volatility” Risk April, pages 55-59 5. Demeterfi K, E Derman, M Kamal and J Zou, 1999 “A guide to variance swaps”Risk June, pages 54-49 6. Dupire B, 1993 ” Model art risk” Risk September, pages 118-120 7. Andreas Grynbichler, Francis A, Longstaff, 1995, “Valuing futures and options on volatility”. Journal of Banking & Finance 20. 8. Carr, P., and D. Madan. “Towards a Theory of Volatility Trading.” In R. Jarrow, ed. Volatility: New Estimation Techniques for Pricing Derivatives. London: Risk Books, 1998, pp. 417-427. 9. Brenner, M., and D. Galai 1989, “New Financial Instruments for Hedging changes in Volatility”, Financial Analysis’s Journal , July-August, pp.61-65. 10 Demeterfi K., E. Derman, M. Kamal and J. Z. Zou, 1999”A guide to Volatility and Variance Swaps.” Journal of Derivatives, summer pp.9-32. 11. Derman, E. and I. Kani. “Riding on a Smile.” Risk. 7, No. 2 (1994), pp.32-39. ─. “Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility.” International Journal of Theoretical and Applied Finance, Vol. 1, No. 1 (1998), pp. 61-110. 12 Neuberger, A. “The Log Contract: A New Instrument to Hedge Volatility.” Journal of Porfolio Management, Winter 1994, pages 74-80. 13 Neuberger, A. 1996. “The Log Contract and Other power Contracts “The Handbook of Exotic Options. Chicago: Irwin Professional Publishing, pages 220-212. 14 Oilver Brockaus and Douplas Long 2000 ‘Volatility Swaps Made Simple' Risk , January, pages 118-120 15 Jim Gatheral “Case studies in Financial course Notes” Spring 2000,Merrill Lynch 16 Bemd Rolfes and Eric Henn ,1999 “A vega nation.” Risk December, pages 26-28 17 Whaley R, 1993 “Derivatives on market volatility :hedging tools long overdue.” Journal of Derivatives, fall, pages 71-84 18 Cheryl L.Sulima , 2001 “Volatility and Variance Swaps” Capital Markets .News, Federal Reserve Bank of Chicago,March ,pages 1-4 19 Nina Mehta “Equity Vol Swaps Grow UP.”, Derivatives Strategy Magazine , July 1999 20 Dean Curnutt “The Art of the Variance swaps ” Derivatives Strategy Magazine , February 2000