Aprendizaje y comportamiento

When it comes to trading variance swaps, an important practical consideration

is the optimal timing of the dynamic replication strategy. Bondarenko [2014]

analyses the impact of non-optimal rebalancing times on the VRP on the S&P 500

index and argues that knowledge about the considerable deviations are relevant

for exchanges, traders and regulators. In fact, financial derivatives have been

developed that exploit risk premia between different monitoring and rebalancing

schemes and those who trade, clear or certify such products have to be aware of

the risks involved. Bondarenko [2014] compares results for the standard squared

log-return characteristic with those for squared simple returns and Neuberger’s

discretisation-invariant variance characteristic.

While one branch of the literature experiments with different definitions of

the realised leg used for defining a swap, more recent studies of variance risk

premia such as Egloff et al. [2010] and Konstantinidi and Skiadopoulos [2014]

employ market quotes (i.e. CBOE Volatility Index (VIX) futures prices) rather

subject to a significant bias, as documented by A¨ıt-Sahalia et al. [2015] and many

others. The empirical relationship between the realised variance of the S&P index

and the VIX index is discussed in Hsu and Murray [2007]. However, no market

quotes are yet available for skewness, kurtosis and higher-moment swap rates.

Again following the methodology of Carr and Wu [2009], Ammann and Buesser

[2013] analyse the VRP in the foreign exchange market. The authors detect a

significant negative premium for intraday realised variance at a low-frequency,

however, the picture becomes blurred when they analyse high-frequency data.

Both the VIX index and the T-Bills – Eurodollar (TED) spread do have an impact

on the VRP. Yet, there are considerable residual premia that are strongly time-

varying. This confirms some main results from Guo [1998] who documents a

significant, time-varying VRP in the foreign exchange market. Since a rise or

drop in the exchange rate can be good news to the one and bad news to the other

market participant, unlike with equities, there is no leverage effect in the foreign

exchange markets. As a result, the VRP can not be explained by the premium

paid for the underlying exchange rate risk. Although Ammann and Buesser [2013]

claim that their methodology is model-free, they implicitly assume continuous

monitoring and a pure diffusion process for the exchange rate by using squared

log-returns for the floating leg of the swap. Also in the foreign exchange market,

Bakshi et al. [2008] develop a stochastic discount factor model for the exchange

rate triangle spanned by the US Dollar, British Pound and Japanese Yen which

takes the variability of return skewness into account. Both the global and the

currency-specific risk premia are stochastic and exhibit individual reactions to the

economic environment. The authors find that negative, country-specific shocks

remain widely ignored.

Broadie et al. [2007] discuss model specification issues for the equity market

based on a large sample of S&P 500 futures and options prices. In particular, the

authors find evidence for jumps in the futures and the volatility process and anal-

yse how these risk factors are priced in the market. They argue that “intuitively,

volatility jumps should induce positive skewness and excess kurtosis in volatility

increments” (p.1454) and propose a statistic for estimating the phenomenon of

jumps in volatility. They conclude that, while introducing price jumps into a

stochastic volatility model always yields significantly higher pricing performance,

there is an interference between jumps in volatility and the risk premium associ-

ated with the volatility of price jump.

When analysing the contribution of jumps to the VRP in the equity market

based on high-frequency data, Bollerslev and Todorov [2011] find that more than

50% of the premium can be associated with tail risk. They further report an

asymmetry between jumps under the physical and jumps under the risk-neutral

measure and attribute the large proportion of downside risk premium to investor

fear of extreme negative market events. A new Investor Fear Index, as opposed to

the VIX index, distinguishes clearly between common variance uncertainty and

investors’ fear. Using a new class of discrete-time models, Christoffersen et al.

[2012] find that the risk premium associated with uncertainty about the jump

intensity has a stronger impact on option prices than the VRP. Their approach

allows for time-varying conditional skewness and kurtosis, which both depend on

the jump intensity. The affine dynamic the authors use for modelling the pricing

of jump fears on the time-varying VRP is also addressed in Todorov [2010], who

shows that investors’ expectations about jumps change considerably after a market

crash.

Although the direct way of getting exposed to variance risk is to trade variance

swaps, a delta-hedged options portfolio is an important benchmark strategy. The

main difference is the directional risk which is not present in the case of a variance

swap investment. Bakshi and Kapadia [2003] compare the VRP with the average

returns of such a hedged position and find that excess returns are less negative

for OTM than for ATM options and more negative in times of financial distress.

Essentially, the gains or losses on the options position depend on the VRP and

the (model-dependent) portfolio vega.