TY - BOOK
T1 - Essays on Quantitative Finance
AU - Jönsson, Martin
PY - 2016
Y1 - 2016
N2 - This thesis consists of ve research papers written during the period March 2014 - April
2016. The papers can be read independently and their abstracts are:
1. European Option Pricing with Stochastic Volatility Models under Pa-
rameter Uncertainty. We consider stochastic volatility models under parameter
uncertainty and investigate how model derived prices of European options are affected.
We let the pricing parameters evolve dynamically in time within a specied
region, and formalise the problem as a control problem where the control acts on
the parameters to maximise/minimise the option value. Through a dual representation
with backward stochastic dierential equations, we obtain explicit equations for
Heston's model and investigate several numerical solutions thereof. In an empirical
study, we apply our results to market data from the S&P 500 index where the model
is estimated to historical asset prices. We nd that the conservative model-prices
cover 98% of the considered market-prices for a set of European call options.
2. The Fundamental Theorem of Derivative Trading. When estimated volatilities
are not in perfect agreement with reality, delta hedged option portfolios will
incur a non-zero prot-and-loss over time. However, there is a surprisingly simple
formula for the resulting hedge error, which has been known since the late '90s. We
call this The Fundamental Theorem of Derivative Trading. This paper is a survey
with twists of that result. We prove a more general version of it and discuss various
extensions and applications, from incorporating a multi-dimensional jump framework
to deriving the Dupire-Gyongy-Derman-Kani formula. We also consider its
practical consequences both in simulation experiments and on empirical data thus
demonstrating the benets of hedging with implied volatility.
3. Risk Minimization in Electricity Markets. This paper analyses risk management
of xed price, unspecied consumption contracts in energy markets. We model
the joint dynamics of spot-price and consumption of electricity, study expected loss
minimisation for dierent loss functions and derive optimal static hedge strategies
based on forward contracts. These strategies are tested empirically on 2012-2014
Nordic market data and compared to a simpler hedge strategy which is widely employed
by the industry. Results show that our suggested hedge outperforms the
commonly used with a higher reward-to-risk ratio, which can be exploited to release
a premium from the contract. The realised cumulative prot-and-loss from our suggested
hedge is greater for almost every single one-month period of the considered
data, whilst the hourly realised payout results in a 66% out-performance probability.
4. Stochastic Volatility for Utility Maximisers. Using martingale methods we derive
bequest optimising portfolio weights for a rational investor who trades in a bondstock-
derivative economy characterised by a generic stochastic volatility model. For
illustrative purposes we then proceed to analyse the specic case of the Heston
economy, which admits explicit expressions for plain vanilla Europeans options. By
calibrating the model to market data, we nd that the demand for derivatives is
primarily driven by the myopic hedge component. Furthermore, upon deploying our
optimal strategy on real market prices, we nd only a very modest improvement
in portfolio wealth over the corresponding strategy which only trades in bonds and
stocks.
5. Volatility is Log-Normal, But not for the Reason you Think. Stochastic
volatility models have increased enormously in popularity since their introduction
in the late eighties. Not the least for hedging and option pricing purposes since
they do well in tting the implied volatility surface. In fact, their pricing ability is
often the reason for advocating such a model, whilst their ability to capture the
underlying dynamics is loosely motivated. In this paper we test for what is a good
model of volatility based on the latter perspective: we brie
y review three wellknown
stochastic volatility models, and concentrate on the instantaneous variance
in Heston's model, a log-normal model and in the 3-over-2 model. Since volatility
is a non-observable process, we employ the technique of realized volatility to obtain
variance measurements and from these we form a goodness-of-t analysis based on
the concept of uniform residuals. To assess the model-classication ability of our
analysis, we perform a Monte Carlo study. We then apply the methodology in an
empirical study, where our results show that the log-normal model
AB - This thesis consists of ve research papers written during the period March 2014 - April
2016. The papers can be read independently and their abstracts are:
1. European Option Pricing with Stochastic Volatility Models under Pa-
rameter Uncertainty. We consider stochastic volatility models under parameter
uncertainty and investigate how model derived prices of European options are affected.
We let the pricing parameters evolve dynamically in time within a specied
region, and formalise the problem as a control problem where the control acts on
the parameters to maximise/minimise the option value. Through a dual representation
with backward stochastic dierential equations, we obtain explicit equations for
Heston's model and investigate several numerical solutions thereof. In an empirical
study, we apply our results to market data from the S&P 500 index where the model
is estimated to historical asset prices. We nd that the conservative model-prices
cover 98% of the considered market-prices for a set of European call options.
2. The Fundamental Theorem of Derivative Trading. When estimated volatilities
are not in perfect agreement with reality, delta hedged option portfolios will
incur a non-zero prot-and-loss over time. However, there is a surprisingly simple
formula for the resulting hedge error, which has been known since the late '90s. We
call this The Fundamental Theorem of Derivative Trading. This paper is a survey
with twists of that result. We prove a more general version of it and discuss various
extensions and applications, from incorporating a multi-dimensional jump framework
to deriving the Dupire-Gyongy-Derman-Kani formula. We also consider its
practical consequences both in simulation experiments and on empirical data thus
demonstrating the benets of hedging with implied volatility.
3. Risk Minimization in Electricity Markets. This paper analyses risk management
of xed price, unspecied consumption contracts in energy markets. We model
the joint dynamics of spot-price and consumption of electricity, study expected loss
minimisation for dierent loss functions and derive optimal static hedge strategies
based on forward contracts. These strategies are tested empirically on 2012-2014
Nordic market data and compared to a simpler hedge strategy which is widely employed
by the industry. Results show that our suggested hedge outperforms the
commonly used with a higher reward-to-risk ratio, which can be exploited to release
a premium from the contract. The realised cumulative prot-and-loss from our suggested
hedge is greater for almost every single one-month period of the considered
data, whilst the hourly realised payout results in a 66% out-performance probability.
4. Stochastic Volatility for Utility Maximisers. Using martingale methods we derive
bequest optimising portfolio weights for a rational investor who trades in a bondstock-
derivative economy characterised by a generic stochastic volatility model. For
illustrative purposes we then proceed to analyse the specic case of the Heston
economy, which admits explicit expressions for plain vanilla Europeans options. By
calibrating the model to market data, we nd that the demand for derivatives is
primarily driven by the myopic hedge component. Furthermore, upon deploying our
optimal strategy on real market prices, we nd only a very modest improvement
in portfolio wealth over the corresponding strategy which only trades in bonds and
stocks.
5. Volatility is Log-Normal, But not for the Reason you Think. Stochastic
volatility models have increased enormously in popularity since their introduction
in the late eighties. Not the least for hedging and option pricing purposes since
they do well in tting the implied volatility surface. In fact, their pricing ability is
often the reason for advocating such a model, whilst their ability to capture the
underlying dynamics is loosely motivated. In this paper we test for what is a good
model of volatility based on the latter perspective: we brie
y review three wellknown
stochastic volatility models, and concentrate on the instantaneous variance
in Heston's model, a log-normal model and in the 3-over-2 model. Since volatility
is a non-observable process, we employ the technique of realized volatility to obtain
variance measurements and from these we form a goodness-of-t analysis based on
the concept of uniform residuals. To assess the model-classication ability of our
analysis, we perform a Monte Carlo study. We then apply the methodology in an
empirical study, where our results show that the log-normal model
UR - http://rex.kb.dk/KGL:KGL:KGL01009261865
M3 - Ph.D. thesis
BT - Essays on Quantitative Finance
PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
ER -