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Analyzing the "True" Delta, Gamma and Vega of European Swaptions in Black-76 and Bachelier Models Using Python
Mälardalen University, School of Education, Culture and Communication.
2019 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Abstract [en]

Option pricing models such as the Black-76, Bachelier and Stochastic Alpha Beta Rho (SABR) models define delta, gamma and vega using the forward rate as the underlying instrument. In terms of options on swaps, this underlying instrument is the forward swap rate, which is the market fixed rate on the swap. This work employs the Black-76 and Bachelier models to determine delta, gamma and vega of a European swaption, but rather uses the swap value as the underlying instrument. We use data in our implementation and the work is done in the python programming language. 

Our results yielded relatively higher absolute delta values than those implied by the conventional Black-76 and Bachelier models. This means that our method yields relatively higher sensitivity of the swaption value to small changes in underlying asset value. It also means higher trading volumes of the swap contract to hedge against small changes in the value of the underlying swap using our method. It was also observed from our gamma values that both our method and the conventional Black-76 and Bachelier models can provide better sensitivities, relative to each other, to big changes in the underlying swap value. This, however, depends on the choice of strike rate. Further, our work produced comparatively lower absolute vega values, hence, lower sensitivity to changing implied volatility. To be able to use volatility values in both the normal and log-normal sense, we converted from normal to log-normal volatilities. This was achieved numerically using the Newton Raphson method implemented in python. 

Changes in swap value and volatility were mimicked using basis point additions and subtractions of certain parameter values, specifically, floating rates, forward rate and volatility values. Each basis point adjustment in the necessary parameter yielded a different delta, gamma or vega value. Since the swap value or volatility can change multiple times within a specific time period, it was observed that there can exist a series of delta, gamma and vega values within a specific period of time.

Place, publisher, year, edition, pages
2019. , p. 73
Keywords [en]
European swaption, delta, gamma, vega, Black-76 model, Bachelier model, swap value, swap forward rate, python.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-44925OAI: oai:DiVA.org:mdh-44925DiVA, id: diva2:1338671
Subject / course
Mathematics/Applied Mathematics
Supervisors
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Available from: 2019-07-25 Created: 2019-07-23 Last updated: 2019-07-25Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
  • ieee
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