This is a presentation on volatility, tail hedging, and alternative investments given at the Asian Insurance Forum.

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“The classic Black-Scholes option pricing model assumes that returns follow Brownian motion, but return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. Time-changed Levy processes can simultaneously address these three issues. We show that our framework encompasses almost all of the models proposed in the option pricing literature, and it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.

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“In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the difficulties associated with hedging these products. Naive delta-hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and out-of-the-money strangles at each volatility observation.”

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“Despite the controversial discussion about ETNs focused on volatility, these instruments have caught a lot of interest from investors. However, it’s difficult to judge whether it’s mainly retail investors with a buy and hold approach or traders using these instruments for short term trading/hedging or even sophisticated investors like hedge funds who invest/trade these instruments.

“In my view the biggest critique seems to be the fact that both ETNs – iPath S&P 500 VIX Short-Term Futures ETN (VXX) and the iPath S&P 500 VIX Mid-Term Futures ETN (VXZ) – are constantly loosing value. I will present a quick study which clearly shows that these ETNs are viable hedging instruments even for a traditional buy and hold investment approach.”

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“In this paper we develop strategies for pricing and hedging options on realized variance and
volatility. Our strategies have the following features.

• Readily available inputs: We can use vanilla options as pricing benchmarks and as hedging
  instruments. If variance or volatility swaps are available, then we use them as well. We do
  not need other inputs (such as parameters of the instantaneous volatility dynamics).
• Comprehensive and readily computable outputs: We derive explicit and readily computable
  formulas for prices and hedge ratios for variance and volatility options, applicable at all times
  in the term of the option (not just inception).
• Accuracy and robustness: We test our pricing and hedging strategies under skew-generating
  volatility dynamics. Our discrete hedging simulations at a one-year horizon show mean absolute
  hedging errors under 10%, and in some cases under 5%.
• Easy modification to price and hedge options on implied volatility (VIX).

 
“Specifically, we price and hedge realized variance and volatility options using variance and volatility
swaps. When necessary, we in turn synthesize volatility swaps from vanilla options by the Carr-Lee
methodology; and variance swaps from vanilla options by the standard log-contract methodology.”

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“The Volatility Exchange (VolX) is considering launching contracts on the volatility of metals, rates and stock indices, its chief executive told FOi.

“The exchange has a patented methodology which calculates realised volatility over a specific time period. It uses closing prices of an asset over a defined period of one, three or twelve months to calculate the asset’s volatility over that period. It contrasts with the VIX methodology, which uses options to calculate implied volatility. Implied volatility is based on perceived volatility and realised volatility is actual volatility.

“The products are similar to volatility swaps and variance swaps, which are…”

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“Assessing risk is an important part of investing. One commonly
used measure of risk is volatility, which measures
the variability of a security’s return through time. If
Security A and Security B have the same expected return
but Security B has greater variability of return, Security B
is more volatile than Security A. Given an equal return
most investor’s would prefer a security with less volatility,
which means that investors expect a higher return on an
investment when it carries a higher level of volatility.
This paper takes a close look at the basics of volatility, discusses
why it matters in relation to portfolio management,
and suggests some methods for managing and controlling
the impact of volatility. In highly volatile markets,
heightened emotions can lead to clouded judgment.
Controlling the amount of volatility within your portfolio
can allow for more prudent decisions.”

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“We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Levy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Levy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Levy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula as a special case of our framework. In the presence of negatively- skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Levy processes to equity options data. Finally we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly-quoted 2 multiple undervalues the continuously-sampled variance, then it undervalues furthermore the discretely-sampled variance.”

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“The advantage of knowing about risks is that we can change our behavior to
avoid them. Of course, it is easily observed that to avoid all risks would be impossible;
it might entail no flying, no driving, no walking, eating and drinking
only healthy foods and never being touched by sunshine. Even a bath could
be dangerous. I could not receive this prize if I sought to avoid all risks. There
are some risks we choose to take because the benefits from taking them exceed
the possible costs. Optimal behavior takes risks that are worthwhile. This
is the central paradigm of finance; we must take risks to achieve rewards but
not all risks are equally rewarded. Both the risks and the rewards are in the future,
so it is the expectation of loss that is balanced against the expectation of
reward. Thus we optimize our behavior, and in particular our portfolio, to
maximize rewards and minimize risks.

“This simple concept has a long history in economics and in Nobel citations.
Markowitz (1952) and Tobin (1958) associated risk with the variance in
the value of a portfolio. From the avoidance of risk they derived optimizing
portfolio and banking behavior. Sharpe (1964) developed the implications
when all investors follow the same objectives with the same information. This
theory is called the Capital Asset Pricing Model or CAPM, and shows that
there is a natural relation between expected returns and variance.”

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“Since the Chicago Board Options Exchange (CBOE) introduced futures and, subsequently, options on its Volatility Index, or VIX, traders have asked why the contracts don’t necessarily track the underlying in the same way other equity futures track their indexes. Others may wonder why the put-call parity is violated for VIX options. Then, there are the options that trade underwater, the vastly different implied volatilities for each expiration cycle and the question of arbitrage between S&P 500 derivatives and VIX contracts.

“Thankfully, all of these questions can be answered with theoretical research on VIX futures and options pricing and, along the way, can offer guidance to some practical applications of these products. Our findings also apply to recently launched VSTOXX index futures and options listed on Eurex.”

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