Hedging in advanced volatility models: An insight into 'The Fundamental Theorem of Derivative Trading'

Michael Viktor Bolvig Olsen, Rasmus Herkild Svensson & Theis Reinhardt Knapmann

Student thesis: Master thesis


The main subject of this master thesis is to study the main results of The Fundamental Theorem of Derivative Trading (FToDT) and application of these in real option data. Let σact and σimp respecti-vely signify the realized and implied volatility. FToDT states that the purchase of an option with a continuously rebalanced delta-hedge will yield an arbitrage opportunity, while observing the inequality σact > σimp, and buying the option if the inequality is true. The realized volatility is however not observable in the market, as such it is up to the trader to model this process as a function of market data and model parameters. The motivation for researching the application of FToDT in real option data, using several more complex volatility models than the constant volatility approach in Black-Scholes, is (i) to profit in arbitrage opportunities and (ii) to measure the empirical validity of the theorem. We consider (ii) to be indispensably important, the affirmation of which may strengthen the theorem as discretely applicable in measure of hedge portfolio profit-&-loss. Our study utilizes a methodology of constructed P&L paths, and with this we found significant profi-tability by usage of EGARCH or Heston volatility models. With the EGARCH model we have found a stricly positive 95% confidence interval for the option-wise P&L results, and thus conclude the exi-stence of an arbitrage opportunity. The Heston model yields the highest expected P&L result, and thus exceeds EGARCH in profitability, although not concluding the same arbitrage opportunity. We also found superior hedging performance by utilizing a minimum-variance strategy based on volatili-ty sensitivity modelling, however this strategy is hardly profitable compared to EGARCH and Heston. The conclusion of this study is highly applicable in option trading, but this result is somewhat thwar-ted by the introduction of financial frictions in the market, such as transaction costs. An investor with intent to hedge an option now faces the non-trivial choice of cost minimizing hedge frequency given such frictions in the market - an aspect we have not focused our attention towards, but certainly consider a viable suggestion for further studies in the matter of option hedges.

EducationsMSc in Business Administration and Mathematical Business Economics, (Graduate Programme) Final Thesis
Publication date2020
Number of pages137