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Given the weaknesses of these traditional volatility investing methods, we introduce variance swaps, the first product to provide investors with pure volatility exposure.

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Read out Stop Share. You currently have no access to view or download this content. Please log in with your institutional or personal account if you should have access to this content through either of these. Showing a limited preview of this publication:. Webshop not currently available. Implied basket correlation dynamics.

Copy to clipboard. Log in Register. Full Access. Purchase this item. Volume 33 Issue This issue All issues. In a boundary case, accounting for the realized variance risk factor in portfolio decisions can be seen as a promising alternative from a portfolio performance perspective. The authors examine the best return-risk combination through the calculation of the Sharpe ratio.

They also assess another different portfolio strategy: the risk parity approach. For fruitful discussions on earlier drafts, we wish to thank Sofiane Aboura. Thinh Vu provided excellent research assistance. Chevallier, J. Report bugs here. Please share your general feedback. You can join in the discussion by joining the community or logging in here. You can also find out more about Emerald Engage.

Visit emeraldpublishing. Answers to the most commonly asked questions here. Abstract Purpose In asset management, what if clients want to purchase protection from risk factors, under the form of variance risk premia.

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Finance and Risk Engineering. Overview Fingerprint. Abstract It is shown that the absence of call spread, butterfly spread and calendar spread arbitrages is sufficient to exclude all static arbitrages from a set of option price quotes across strikes and maturities on a single underlier. Access to Document Link to publication in Scopus. Link to citation list in Scopus. Finance Research Letters , 2 3 , In: Finance Research Letters , Vol.

Finance 54 , — Hernandez-Hernandez, D. In: Proceedings of the American Control Conference Hodges, S. Futures Mark. Iihan, A. Finance Stoch. Jorion, P. McGraw-Hill, New York Kelly, J. Bell Syst. Ledoit, J. MIT Levy, H. Finance 40 , — MacLean, L. In: Zenios, S. North Holland Merton, R. Bell J.

Musiela, M. US Printing Office, Washington Rouge, R. Finance 10 , — Schied, A. Sircar, R. SIAM J. Control Optim. Thorp, E. Vince, R. Wiley, New York Zhu, Q. Download references. Correspondence to Q.

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Chevallier, J. Report bugs here. Please share your general feedback. You can join in the discussion by joining the community or logging in here. You can also find out more about Emerald Engage. Visit emeraldpublishing. Answers to the most commonly asked questions here. Abstract Purpose In asset management, what if clients want to purchase protection from risk factors, under the form of variance risk premia.

Findings The authors find that optimized portfolios based on variance-covariance matrices stemming from VRP do not consistently outperform the benchmark based on daily returns. Please note you do not have access to teaching notes. You may be able to access teaching notes by logging in via Shibboleth, Open Athens or with your Emerald account. If you think you should have access to this content, click the button to contact our support team. Contact us. One should also bear in mind that variance swaps are not exempt from counterparty risks.

Since the product is OTC, both counterparties face the risk of the other going bankrupt. BNP Paribas : variance swap seller. Another feature of variance swaps is that taking short positions on a variance swap on a single stock may be risky. Whereas the total loss on short positions is unbounded, the maximum loss on long positions is equal to the notional times the variance strike.

This is important as there is no limit to the potential loss of being short on a variance swap. As a result, variance swaps are often capped in such a way that:. The graph below compares the payoffs from capped and uncapped variance swaps. The maximum gain and thus loss for the counterparty is limited and thereby facilitates the hedging of the instrument.

As we will demonstrate in the next section, a variance swap can in theory be replicated by a portfolio of options with a continuum of strike prices. In practice, however, there is no liquidity for options whose strike prices are far from ATM. Traders are therefore unable to completely hedge variance swaps, whereas capped variance swaps do not require a continuum of strike prices. Before examining variance swap pricing in detail, we shall explain the logic behind the mathematical derivation.

Firstly, variance swap pay-off is by definition a function of the variance and is independent of the stock price level. However, the sensitivity of an option to variance depends on the stock price level. The sensitivity of an option to the variance, or variance vega, is centered around the strike price and will thus change daily according to changes in the stock price level. As the graph below shows, the variance vega declines as the stock price moves away from the strike price and is also an increasing function of the strike price.

The goal is therefore to create an options portfolio with a constant variance vega. This can be done by investing in a portfolio of options inversely weighted by the square of their strike prices. The following two graphs display the variance vega of:. The addition of options with strike prices away from ATM flattens variance vega and thus makes the portfolio sensitive to variance but invariant to stock price.

Carr and Madan suggest a formal method for pricing variance swaps that has the advantage of requiring very few assumptions about stock price dynamics. Instead of defining a process for the stock price, Carr and Madan only assume that markets are complete and that trading can take place continuously.

Although this section is rather mathematical it describes one of the greatest results in derivatives research and one that has largely contributed to the growth of the variance swaps market. Carr and Madan further assume that a market exists for futures options of all strikes. In this case, they show that any payoff f FT of the futures price FT can. As Carr and Madan point out, the above terms can be interpreted as:. In the absence of arbitrage, the above breakdown must prevail for initial values.

Therefore, the initial value of the payoff is equal to:. Carr and Madan thus prove that an arbitrary pay-off can be obtained from bond and option prices without making strong assumptions about the stochastic process driving the stock price5. Since the initial cost of achieving this strategy is given by A , the fair forward value of the variance at time 0 should be equal to:.

Variance-swap strike prices may thus be replicated by a continuum of puts and calls inversely weighted by the square of their strike price. Note that the above valuation is model-free since we did not have to state a specific process for the dynamic of the stock price in order to derive the formula.

It is also worth mentioning that the above formula provides a market-based estimator of future realized volatility. Volatility, not variance, is the most commonly used risk measure. First, it is measured in the same unit as stock returns. However, variance is less meaningful despite it being the square of the standard-deviation. As shown above, variance swaps can be replicated with the help of a linear combination of options and a dynamic position in futures.

Replicating volatility is more complex and requires a non-linear combination of derivative instruments. Options traders are therefore linearly exposed to variance and are thus likely to be more interested in variance swaps. The same reasoning holds true for an investor wishing to hedge the volatility exposure of his options portfolio.

However, the strike price of a variance swap is often defined in terms of volatility as it is more economically meaningful. The specific dynamics of volatility are of particular interest and these may be summarized in the following four points:. Furthermore, as opposed to stock returns, volatility tends to revert back towards its mean and usually remains within a high or a low regime for a long period of time. The other empirical characteristic of volatility is that it is usually negatively correlated with the underlying asset return.

The 6-month variance swap strike is calculated using the method suggested by Derman et al. Variance swaps would have traded an average of 1. The positive average comes from the existence of a variance risk premium, which we highlight in the next section. Volatility is usually negatively.

Variance swap strike prices as a forecast of future realized volatility. By definition, the variance swap strike price should be a good proxy of future realized volatility. If this assertion holds true, regressing the variance swap strike price against realized volatility should yield a slope and a constant equal to one and zero, respectively. The same conclusion holds for ATM implied volatility despite the fact that the constant and the slope are closer to zero and one, respectively.

One of the reasons why variance-swap strike prices do not provide a fair proxy of future realized volatility is again the presence of a variance risk premium. The T-stat measures whether this parameter shows a statistically significant difference from zero: it needs to be below The square measures the overall explanatory power of the equation. As already mentioned, sellers of variance swaps bear a bigger risk than buyers as their potential loss is unlimited.

They should therefore be rewarded with a variance risk premium that is reflected in a variance strike price higher than realized variance on average8. Indeed, in a risk-neutral world, this variance risk premium should be equal to zero. A risk-neutral investor should sell a variance swap at his expected realized variance.

In practice, however, the presence of risk aversion makes the variance risk premium positive. As a result, one way to test the effect of risk aversion on the variance risk premium is to check whether the difference between the variance-swap strike price and realized variance is a function of the skew. As shown by the following table, the higher the skew, the higher the variance risk premium.

See also Driessen et al. In this section, we show how variance swaps can be used to take positions on volatility or hedging it. The most obvious use of variance swaps is to bet on the difference between current implied and future realized volatilities. With variance swap strike prices defined by a combination of puts and calls, they may be viewed as a weighted average of implied volatility across strikes. If one expects future realized volatility to be above implied volatility, as measured by the strike price of a variance swap, investors can use a long position in a variance swap to take this view by means of a buy-and-hold strategy, as opposed to option delta-hedging which would require daily monitoring of the delta position.

One way to play an expected rise in volatility term structure is to enter into two different variance swaps with two different maturities. An investor can then take advantage of this expected steepening of volatility term structure by:. This type of strategy. Here, we present two different single-stock volatility strategies, namely volatility pairs and dispersion trading.

Since the volatilities of two stocks within the same sector are usually driven by the same factors, their spread is often mean-reverting: should one implied volatility diverge from the other, it is likely to revert back. Their levels are closely related and the implied volatility spread is reverting towards a mean around zero. Our implied volatility valuation model can also provide some ideas for implied volatility pairs trading. This model helps calculate fair values for the implied volatility of single stocks according to their beta, 5-year CDS, size and stock returns Dispersion trading consists of buying the volatility of an index and selling the volatility of its constituents according to their index weights.

It is defined as11 :. By definition, it changes over time as stock prices change and is equal to:. As correlation tends to revert back towards its long-term mean, so does dispersion. Dispersion trades may be executed by buying a variance swap on the index and selling variance swaps on the constituents according to their weight in the index. Nowadays, dispersion is usually quoted by mentioning the level of volatility of the index, the average weighted spread of volatility with its constituents and the level of correlation.

Given that structured products are by definition sensitive to volatility, variance swaps are natural tools for hedging volatility exposure. A CPPI is a dynamic strategy that can be applied to stocks or other assets.

The following paragraphs describe how CPPI works. Let us suppose that the initial investment is with a 5-year time horizon. The investor first determines a floor that is equal to the value of a 5-year bond. The cushion C0 is equal to the initial investment I0 minus the bond floor B0. In that case, the final value of his investment would be guaranteed against a decline below its initial value.

The higher the multiplier, the greater the participation in an appreciation of the risky asset. The investment in the risky asset is dynamically adjusted in order to ensure a return of at maturity: if the risky asset declines, exposure to the risky asset is consequently reduced. Once the total value of the portfolio reaches the bond value, the CPPI is said to have been cashed-out and all the money is invested in the risky bond as there is no more cushion.

CPPIs may be demonstrated to be negatively related to the volatility of the underlying asset for the following reasons:. Variance swaps can be useful tools. On the other hand, option-based products such as ODBs, are usually positively related to volatility.

On this basis, variance swaps could help structured-product managers to hedge their volatility risks efficiently. Variance swaps are currently used by hedge fund managers to bet on volatility. They can also be used to protect against volatility risk. While numerous studies demonstrate the correlation between hedge fund strategies and equity market returns, few have focused on the effect of volatility.

Convertible arbitrage, for example, benefits from a rise in volatility as higher volatility usually creates more arbitrage opportunities. Conversely, when volatility declines - as it has in the last two years - convertible hedge funds can improve their performance by selling variance swaps. Conversely, hedge funds with strategies negatively correlated with volatility, such as event-driven arbitrage or distressed companies, could yield higher returns in a high volatility market if they buy variance swaps.

As discussed previously, the gamma exposure of variance swaps is insensitive to the level of the underlying asset. In the event the stock price rises or declines, the gamma exposure depends solely on the initial value of the portfolio.

In general, investors focus on the number of portfolio units they manage and not on the initial cash value of their portfolio. Gamma swaps are by definition products that answer to this need. K is the strike and N the notional amount. In continuous-time, the gamma swap payoff is equal to:. Gamma swaps are thus equivalent to variance swaps whose nominal is proportional to the level of the underlying asset.

Pricing gamma swaps is as easy as pricing variance swaps. We have already shown that whereas the variance swap vega should be independent of the stock price, the gamma-swap vega should be a linear function of the stock price. The aim is to create an options portfolio whose vega will be a linear function of the stock price. As the following graph illustrates, an option portfolio using a continuum of strike prices and inversely weighted by their strike prices provides a vega that is linear with respect to the stock price.

As shown in the appendix, the value — or the strike price — of a gamma swap is given by:. This means that if the distribution of stock returns is skewed to the left, gamma swaps minimize the effect of a crash, thereby making it easier for the trader to hedge. In this case, hedging does not require additional caps, unlike variance swaps which need to be capped.

Gamma swap strike prices should thus be lower than variance swap strike prices. The gamma swap strike price would systematically have been slightly lower than the variance swap strike price by 1. The gamma swap payoff is slightly delta-positive, given that it is a function of the performance of the stock since inception.

As we pointed out before, dispersion is calculated by the difference between the realized index volatility and the market-cap weighted sum of the realized volatility of its constituents. The dispersion between time 0 and T is thus described by:. If an investor trades dispersion using variance swaps weighted by the initial weights of the stocks in the index, he faces the risk of a possible change in weights over time until maturity of the variance swap. Gamma swaps, however, offer a more efficient way to trade dispersion.

The payoff is thus equal to the average dispersion over the period [0,T] weighted by index performance. As a result, forward-start variance swaps enable investors to take positions on future volatility without having to enter two different variance swaps. Corridor variance swaps are a variant of variance swaps that only take into account daily stock variations when the stock is in a specific range.

Corridor variance swaps therefore enable bets to be taken on the pattern of the stock. If the stock move sideways and stays within the defined range, K Corr 2 will be high. If the stock moves sharply upward or downward and leaves the range quickly, K Corr 2 will be low. Financial engineering never stops producing new products and up corridor variance swaps are an example of such innovation.

Up corridor variance swaps are a variant of corridor variance swaps and have the following payoff:. Hence, squared returns are counted in if and only if the stock price lies above a predefined level denoted B. Indeed, one can also define a down corridor variance swap whose payoff would be defined by:.

Forward-start variance swaps enable positions to be taken in variance between two future dates. Aggregating a down corridor variance swap and an up corridor variance swap yields the classic variance swap. This particular payoff has several advantages:. Another reason why it is cheaper is that no purchases of expensive OTM puts are required to replicate it. The ATM down corridor variance swap has a threshold level B equal to the initial level of the index, while only returns associated with an index level below the threshold are counted in.

Comparison of 1-year variance swaps with 1-year ATM up and down corridor variance swaps in vol terms. Up conditional variance swaps are a variant of up corridor variant swaps, whose payoff is described by:. The conditional variance swap payoff is such that if the stock price never trades above the threshold B , it will be null whereas the up corridor variance swap payoff would be equal to K 2.

It also enables bets to be taken on very specific volatility behavior. Take the case of an index whose initial value is As a result, timing is less of an issue for an investor who buys a conditional variance swap rather than a corridor variance swap. The goal of this section is not to explain why the smile exists, but how to trade it.

Although theoretical models exist for pricing such products, they are not traded as yet Nevertheless, investors can use other methods based on existing volatility products to take views on the smile. The first one involves buying a variance swap and selling a gamma swap. As shown previously, variance swaps and gamma swaps can be replicated by a portfolio of options comprising a continuum of strike prices, inversely weighted by the square of the strike price and the strike price, respectively.

This means that the spread between the variance swap and gamma swap strike prices should be an increasing function of the skew. The table plots the prices of 1-year variance swaps and gamma swaps for the three different skews. One can see that the spread between the two increases when the skew rises.

Indeed, variance swaps and gamma swaps yield different vega exposures. By being long a variance swap and short a gamma swap, an investor does not get a vega-neutral position.