ASSET MANAGEMENT

LECTURE VIII

Derivatives markets and derivatives in portfolio management Introduction to contracts

• Definition of a derivative: an agreement between two parties which has a value determined by the price of something else (underlying). • Derivative securities include 1. Swaps 2. Options and forwards 3. Futures

• Different uses: – Risk management – Speculation – Reduce transaction costs – Regulatory

• End users of derivative products include corporates, investment managers, individual investors. Derivatives may be looked at from different perspectives: – End users – Intermediaries/market makers – Regulators and researchers 2 1. The

A swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices. A swap provides a means to a stream of risky payments. Swap contracts are traded OTC.

The most common type of swap is the swap (plain vanilla swaps exchange fixed payments for variable payments semi-annually and the contract is settled for differences). The notional of the swap is the amount on which the interest payments are based and the life of the swap is called swap term or swap tenor. • Let’s make a sample: the company A has a debt with a fixed interest rate of 7% and left maturity of 5 years; the company V has a debt with same maturity but floating interest rate equal to 6 months (same amount). • The company A forecast falling interest rates. The company V forecast rising interest rates. • The (IRS) allows to meet the opposite needs. The company A will pay to the company V the 6 months EURIBOR and will receive from the company rate a 7%. The value of the swap has given from the difference between fixed and floating interest rates (euribor): S=FLR-FXR where 7% Euribor

S = swap Euribor Com. A Com. V FLR = floating rate

FXR = fixed rate 7%

3 1. The swap and different rating

• Let’s consider two companies with different rating: • The company A (with rating A) that can borrow at 8% fixed interest rate or floating rate Euribor (5 year maturity); • The company B (with rating B) that can borrow at 10% fixed interest rate or floating rate Euribor +0,5% (5 year maturity). Is it possible for both to reduce the cost of funding?

8% EURIBOR + 0,5

EURIBOR – 0,75

Com. A Com. B 8%

(EURIBOR) (10%)

EURIBOR – 0,75 9 + 0,25

4 1. The interest rate swap

Introduction to derivative contracts, forward and options

• Forward and futures contracts are agreements between two parties: the buyer agrees to purchase an asset from the seller at a specific date at a price agreed to now. Options offer the buyer the right without obligation to buy or sell at a fixed price up to or on a specific date.

• basics – Option to buy is a . Option to sell is a . Option premium – sum paid for the option – Option styles: European (at maturity), American, Bermudan – The price is the price the call buyer will pay to-or the put buyer will receive from-the option seller if the option is exercised – Intrinsic value represents the value that the buyer could extract from the option if the holder exercised it immediately – The time premium component is simply the difference between the whole option premium and the intrinsic component

• Options trade both in over-the-counter markets and on exchanges. The OTC market is much larger than the exchange traded segment

6 The call option

There are 5 different scenarios: A) P < 108, it is not convenient exercise the CALL loosing the premium (OUT OF THE MONEY); B) P = 108, it is indifferent either exercise or not the CALL (AT THE MONEY); C) 108 < P < 110 , exercising the CALL it is possible to reduce the cost of the premium (IN THE MONEY); D) P = 108 + 2, it is convenient exercise the CALL and the P&L isequal to 0 (IN THE MONEY); E) P > 108 + 2, the profit is nominally unlimited (IN THE MONEY).

P&L

30

20

10 Underlying price

0 -2 30 40 50 60 70 80 90 100 110 120 130 140 150 -10

7 A summary of payoff graphs for forward and option contracts

• A long forward contracts pays the difference between the terminal price of the underlying asset and a certain strike level. The opposite is true for short forward positions. Long and short forward contracts just pay this difference every day, not at maturity

• Long calls pay the difference between the terminal price of the underlying asset and a certain strike level, if positive. The opposite is true for short call options

• Long puts pay the difference between a certain strike level and the terminal price of the underlying asset, if positive. The opposite is true for short put options

References: BROWN & REILLY, Chapter XXI; Mc Donald, Derivatives Markets, Pearson

8 Management,Asset lecture VI Option strategies: bull spreads and collars

• An option spread is a position consisting of only calls or only puts, in which some options are purchased and some written. A is a position, in which you buy a call and sell an otherwise identical call with a higher : it is a bet that the price of the underlying asset will increase

• A is the purchase of a put option and the sale of a call option with a higher strike price, with both options having the same underlying asset and having the same date. Example: purchased zero- cost collar

9 Other option spreads and non-directional strategies

• A is a position in which one sells a call and buys an otherwise identical call with a higher strike price. Clearly, it is a bearish position, profiting when asset prices fall

• A is accomplished by using options to create a synthetic long forward at one price and a synthetic short forward at a different price. A box spread is a means of borrowing or lending money: It has no stock price risk

• A is constructed by buying m calls at one strike and selling n calls at a different strike, with all options having the same time to maturity and same underlying asset

• Options can be used to create positions that a non-directional with respect to the underlying asset. Examples are , strangles and butterflies

• Who would use non-directional positions? Investors who do not care whether the stock goes up or down, but only how much it moves, i.e., who speculate on (or diversify, using volatility as an additional asset class )

10 A summary of payoff graphs for forward and option contracts

• A is obtained by buying a call and a put with the same strike price and time to expiration. A straddle is a bet that volatility will be high relative to the market’s assessment

• A is created by buying an out-of-the-money call and put with the same time to expiration. A strangle can be used to reduce the high premium cost, associated with a straddle. Write a straddle + add a strangle =

11 A summary of payoff graphs for various option strategies

• Bull spreads and purchased collars are synthetic long and short positions in the underlying asset obtained by using options. The collar can cost nothing if the option strikes are chosen correctly

• Long straddles and long strangles profit when the movement in the underlying securities is large. The opposite is true for written strangles and written straddles

• Long butterfly spreads profit when the market moves sideways, i.e. when the price moves are moderate. Long ratio spreads profit for small downside moves. Short positions in these strategies yield the opposite result

12 Options parity: the put call parity

1. Buy a share = Asset 1 2. Buy a put at an exercise price X and expiration T = Asset 2 3. Sell a call at an exercise price X and expiration T = Asset 3

Assumptions: put and call are european and no dividends for the underlying Portfolio: Asset 1 = Si = 110 Asset 2 = Pi = 2 Asset 3 = - Ci = 4 Net investment is: Si + Pi + Ci Scenarios:

A) T+1, S = 90 B) T+1, S = 120 • Asset 1 = 90 • Asset 1 = 120 • Asset 2 = 100 – 90 • Asset 2 = 0 • Asset 3 = 0 • Asset 3 = - (120 -100)

100 100

13 Options parity: the put call parity

In either case the investor will sell the stock at time T for the fixed price X... It means that in To the Portfolio is equal to a T-Bill, wich is the face value X discounted using the risk free rate:

So + Po - Co = X/(1 +RFR)^T

1) LONG STOCK + LONG PUT + SHORT CALL = LONG T-BILL

After 6 months

108 = 110/(1 +RFR)0,5 RFR =(110/108)^2 – 1 = 1,037 – 1 = 3,7%

2) Short stock + short put + long call = Short T-Bill ...

In efficient markets there are links among stocks, bonds, put and call options

14 Options parity relationships

• Using the payoff graphs we just introduced you find that (long call) + (short put) = (long forward) and that (long underlying) + (long put) + (short call) = (long risk free security)

• Instead of buying a stock, take a long position in a to buy a stock. Supplement this transaction by purchasing a put option and selling a call option, each with the same exercise price and expiration date. This reduces the net initial investment compared to purchasing the stock in the spot market

• The difference between put and call prices must equal the discounted difference between the common exercise price and the contract price of the forward agreement, otherwise arbitrage opportunities would exist

• A risk-free portfolio could be created using three risky securities: a stock (for example), a put option on the stock and a call option on the stock. With the risk-free security as the fourth security, any one of the four may be replaced with combinations of the other three

• The owners of derivative instruments do not participate directly in payments to holders of the underlying instrument. If the amounts and payment dates are known when puts and calls are written those are adjusted into the option prices. (long stock) + (long put) + (short call) = (long risk free security) + (present value of payments)

15 Summary of the outcome of various investment strategies

References: BROWN & REILLY, Chapter XXI; Mc Donald, Derivatives Markets, Pearson

16 Management,Asset lecture VI Futures…

Futures

Futures are agreements negotiated in exchange markets. A future is an obligation between two parties to buys or sell an underlying at a certain date under particular conditions. • COMMODITY FUTURE (oil, gold) • FINANCIAL FUTURE or INTEREST RATE FUTURE (euribor, ) • (euro/dollar)

Exchange markets:

Chicago Board of Trade (CBOT) that was the first in march 1851, the NYFE, the LIFFE (London International Financial Futures Exchange).

CLEARING HOUSE CLEARING MEMBERS INITIAL (1%-2%)

P&L= (FP-IP) x tick x n of contracts

18 Futures

Futures contracts differ from Forward contracts in the following ways: • Futures trade on organized exchanges while Forwards are private contracts and typically do not trade • Futures are standardized. Forwards are customized contracts that satisfy the specific needs of the parties involved • A clearing house is the counterparty to all futures contracts while Forwards are contractd with the originating counterparty and therefore have credit risk (counterparty) • The government regulates futures markets. Forwards are usually unregulated and do not trade on organized markets

A few examples of futures contracts:

1. futures are time deposits denominated in US dollars and held at banks outside the United States. Buying the contract is equivalent to lending money while selling the contract is equivalent to borrowing. They are based on 90-day and face value of one contract is $1m. One tick price change = $25

2. Stock index futures Based on the level of an equity index. The most famous one is on the S&P500 and trades in Chicago. The value of one contract is x250 the level of the index, therefore the one tick move = $250

19 Elementary applications of derivatives to portfolio management

• Restructuring asset portfolios with futures contracts ─ Shorting futures contracts or buying to increase exposure ─ Tactical asset allocation to time general market movements instead of company-specific trends ─ Hedge position with payoffs that are negatively correlated with existing exposure ─ Converts beta of portfolio to zero, making a synthetic risk-free security, affecting portfolio beta

• Protecting portfolio value with put options ─ Purchasing protective puts ─ Keep from committing to sell if price rises ─ Asymmetric hedge ─ Portfolio insurance

• Portable alpha funds extract alpha using futures hedging. To get the total return simply add the desired quantity of market exposure via derivatives

References: BROWN & REILLY, Chapter XXI; Mc Donald, Derivatives Markets, Pearson

20 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Covered-write and buy-write strategies involve holding a stock and selling a call on the stock. ─ It provides the means to monetize future upside potential with higher income today ─ popularity due to demand for yield given the low interest rate environment ─ for a given time horizon, an investor wants to pick-up extra yield and believes that the upside potential of a stock is limited to certain point but the long term story is still valid ─ an investor who has a neutral, slightly bullish or slightly bearish market outlook but nevertheless wants to have some cushion against the downside ─ the income generated from selling calls is more attractive than the income derived from interest rate instruments

• While the rationale for writing calls is pretty simple, the specifics (such as strikes and maturities) are perhaps more murky ─ Investors typically write calls with maturities of less than three months ─ In terms of strikes, covered writes are typically out-of-the-money (OTM) strikes, anywhere from 100% to 105% of the current share prices ─ the strikes written should conform to the investor’s active view

21 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Covered-write and buy-write strategies involve holding a stock and selling a call on the stock

─ Buy 100 shares of Disney Stock at 30.50. Sell Disney Oct 08 30 strike calls for 2.10 ─ Call has 1.60 of time value ─ Call is 0.50 In the money 30.50-30.00=0.50 ─ Call has 0.50 of intrinsic value [0.50(30.50-30.00)] ─ Cost of trade 30.50-2.10=28.40 Potential profit if assigned=(2.10-0.50)/(30.50-2.10)=5.6% ─ Trade runs 82 days 25.0% Annualized (for comparison purposes only) ─ Stock can fall up to 28.40 6.8% and still break even

22 Management,Asset lecture VI A currency option

Plain vanilla A company has sold goods and receives Euro forward (90 days) spot + forward = 1,2950 + 0,0079 = 1,3029

Buys an option Put Euro/Call USD Strike 1,2850 Importo USD 1.000.000 (Euros 778.210) Premio 0,0255 USD pips where the premium in Euros is -15.323,83 The profile of risk is: If Euro/USD < 1,2850 à the company buys USD 1.000.000 at 1,2850 Spot 1,2950 20.000 P&L Spot 15.000 Forward 10.000 Strike Put Euro/Call USD 5.000

-

-5.000

-10.000

-15.000

-20.000 Euro/USD 1,2329 1,2529 1,2729 1,2850 1,2900 1,3029 23 1,3150 1,3329 1,3529 1,3729 Graduation of strategies

+ • IRS • BUY CAP Group A • BUY FLOOR • Z.C.C. Z.C.C.ASIMMETRIC • FRA

• PREMIUM BASIS WITH CAP Group B • QUANTO SWAP WITH CAP • BASIS CAP

• IRS WITH BARRIER • TWO STRIKE CAP Group C • DIGITAL CORRIDOR • BASIS • BASIS SPREAD

• Group D • DOUBLE BASIS

Grade of hedging hedging of Grade • RANGE ACCRUAL • GAP REVERSAL • PREMIUM BASIS Group E • QUANTO SWAP Speculative • DIGITAL BASIS • EXTRA SWAP

• SELL CAP Group F - • SELL FLOOR • SELL DIGIFLOOR Graduation of interest rate derivative strategies

Group A • CILINDRIC BUY CALL & PUT + • BUY CALL • BUY PUT

Group B • FORWARD EXTRA • PARTECIPATED

• CANAL PLUS Group C • SINGLE KO • BUY CALL KI • BUYU PUT KI

• DOUBLE KO Group D • BUY CALL KO • BUY PUT KO

• CALL CON CAP

Grade of hedging hedging of Grade • PUT CON CAP Group E • CALL CON CAP KI • CALL CON CAP KO Speculative • SELL CALL • SELL PUT Group F • SELL CALL KI - • SELL PUT KO • SELL PUT KI Elementary applications of derivatives to portfolio management

• For the purpose of identifying the most attractive call writing or buy-write candidates: ─ Limited upside potential: We select stocks with limited upside potential or where (price target – price)/ price < 5% (it depends on mkt conditions) ─ Stocks given a rating of neutral by buy side analysts ─ greater than the country and sector averages

3M 3M 3M Ups Price Target 3M 3M Ret ATM Call Poten Vol/ Vol/ vol Prem tial CAvg SAvg

26 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Bull-put and Bear-Call Credit spreads: advantages of Credit spreads ─ Can be bearish or bullish ─ Versatile Strategy; cutting off a piece of risk pie ─ can be short term or longer term strategy

• Essential Dangers... When not to do a Credit Spread! ─ Stocks that swing wildly ─ Earnings periods; movement expected ─ Insufficient Strikes

• How does a Credit Spread compare to a ? – Smaller piece of the risk pie – Smaller credit in dollars – Higher percent return – Larger risk of total loss

27 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• A bull put credit spread on RIMM (126.90$) ─ 110/105 bull put credit spread on RIMM ─ Sell the Sep 110 put option for 2.12, Buy the Sep 105 put option for 1.35 ─ Net Credit is 0.77 (2.12-1.35) ─ At risk is 110-105=5.00-0.77=4.23 ─ Return rate is 0.77/4.23=18.2%, 13.3% downside protection (126.90-110)/126.90 ─ 37 days =179% annualized

• A Bear Call Credit Spread on RIMM ─ 150/155 bear call credit spread on RIMM ─ Sell the Sep 150 call for 1.15, Buy the Sep 155 call for 0.76 ─ Net Credit 0.42 ─ 0.42/(5.00-0.42)=9.2% ─ Upside protection 18.2%

28 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

29 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• What if we do both ...at the same time…

─ 110/105 bull put credit spread on RIMM for 18.2% ─ 150/155 bear call credit spread on RIMM for 9.2% ─ You can get more than 27.4% (18.2+9.2=27.4%) return ─ as long as the stock stays between 110 and 150 ─ you would make the 0.42 +0.77=$1.19 ─ 1.19/3.81=31.2%

30 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

31 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

32 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Stock replacement or cash extraction strategies involve selling an existing stock position and replacing it by buying a call. The advantage of stock replacement is that it provides a leveraged exposure to the upside while limiting the maximum risk to the premium paid ─ Given large recent price appreciation in many cases, taking profits off the table may be a defensive measure to alleviate concerns about the possibility of a pullback ─ While there is definitely risk on the upside, upside potential on the stock remains, and keeping upside exposure is therefore an attractive proposition ─ Low volatility means that options are generally relatively inexpensive (comparing to 2008 and 1H 2009) ─ Given the low level of volatility, calls will benefit from increases in implied volatility, and delta-hedged positions would benefit from the long gamma • For the purpose of identifying the most attractive cash extraction candidates: ─ Large recent price increases: a screening for all stocks with a 3-month return greater than 10% ─ Significant upside potential: stocks with upside potential of greater than 10% ─ Low implied volatility levels: Stocks with implied volatility levels below the sector average and below the country average

33 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• For the purpose of identifying the most attractive cash extraction candidates:

─ Large recent price increases: a screening for all stocks with a 3-month return greater than 10% ─ Significant upside potential: stocks with upside potential of greater than 10% ─ Low Implied volatility relative to country and sector averages

3M 3M 3M Ups Price Target 3M 3M Ret ATM Call Poten Vol/ Vol/ vol Prem tial CAvg SAvg

34 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Combined with a long stock position, a protective-put strategy involves the purchase of a put option to insure against downside risk. Synthetically, a has the same pay-off profile as a long call. The difference is that with a protective put, the investor with the long stock position is entitled to the dividends and the voting rights ─ If an investor is concerned about downside risks for a certain fixed period but believes in the long-term prospects of the stock ─ when the implied volatility is low the cost of the protection is relatively low • For the purpose of identifying the most attractive protective put candidates: ─ Large recent price increases: a screening for all stocks with a 3-month return greater than 10% ─ Significant long term upside potential: stocks with a strong buy rating ─ Current three-month implied volatility in the bottom half relative to history 3M 3M 3M Put Ups Price Target 3M ATM Ret ATM Prem Potenti d rank vol al

35 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• A collar involves the purchase of an OTM put and the sale of a covered call. A collared position exchanges upside potential for downside protection ─ Typically, the sale of a call is used as the means to reduce the cost of put protection ─ the call is often structured to provide enough of a premium to completely offset the cost of the put; a zero-cost collar ─ A collar confines the possible future distribution of returns

• The ideal environment in which to implement a collar strategy is as follows: ─ If an investor is concerned about the downside risk of a particular stock and the cost of protection is too high, he may exchange some upside for that protection. ─ The cost of protection is not high enough to eliminate all the upside potential of the stock.

36 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

37 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

38 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• For the purpose of identifying the most attractive collar candidates: ─ Large recent price increases: a screening for all stocks with a 3-month return greater than 10% ─ Limited upside potential (i.e. 20%) ─ Skew level is low relative to the sector

3M 3M 3M 3M Ups Price Target Skew Ret ATM Call Put Poten Sec vol Prem Prem tial Avg

39 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Is volatility an asset class? ─ How does volatility compare to traditional asset classes? ─ What is the utility of investment into volatility from a portfolio management perspective? ─ What kind of the models which can be used to forecast volatility? ─ Which financial instruments can be used to invest in volatility?

• Using long time series we compare volatility to some traditional asset classes ─ Equity volatility has a negative correlation to equities in the long term and due to large increases of volatility in times of crashes it can be bought into the funds as insurance against losses on equity investments over long time periods ─ In the short-term, one should use volatility forecast models to implement active return enhancement strategies (easier said than done) ─ Equity volatility has extremely high volatility itself. Be sure that volatility has a significantly negative relationship with the rest of the portfolio in order to compensate for its own high volatility contribution.

40 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

Frequency distribution of daily Frequency distribution of S&P Frequency distribution S&P 500 changes in oil prices 500 3-m volatility daily changes daily returns

41 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Risk reduction potential depends on correlation levels ─ Volatility is crucial for managing a derivatives portfolio however, we explore what the utility is for an asset manager to invest in volatility ─ From a long-term perspective, volatility does not offer price appreciation as a reason for investing (average monthly return was -0.3% in the long-term) ─ We believe that investors should consider volatility to be a valuable asset class because of diversification benefits it potentially offers to a traditional portfolio mix ─ As long as correlation between two assets does not equal one, then theoretically combining them together can reduce risk in a mixed portfolio ─ Expectation of levels of correlation between volatility and other asset classes are crucial in this context. Correlation between volatility and levels of the index can, and does, change significantly over time and there were times when we measured a positive correlation between volatility and equity returns

42 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

Returns, variances and correlations estimates

Recommended asset weights for standard deviation = 8%

43 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

Risk-Return frontiers (grey bonds and equity only, The performance the funds would have if could blue bonds, equity and volatility, green all assets) perfectly foresee the future 250 days

44 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

Risk-Return frontiers (grey bonds and equity only, blue bonds, equity and The performance the funds would have if could volatility, green all assets, yellow all assets long-only) perfectly foresee the future 250 days before

45 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Are there models to forecast volatility? ─ Most statisticians would agree that volatility in short term is relatively well forecastable ─ The presence of heteroscedasticity in equity returns is one of the important deviations of the assumptions in Black-and-Scholes derivatives formula ─ Changing volatility has a great impact on investment banks risk management and risk evaluation and is at the core of activities on financial markets ─ Therefore, much effort was invested in development of volatility forecasting models. Efforts in the development of GARCH models were distinguished with an award of 2003 Nobel price to Robert F. Engle and Clive W.J. Granger ─ GARCH is, so far, the most accepted model for volatility forecasting in the market, but is mainly applied to short-term time-frames ─ Characteristics of volatility are clustering and mean-reversion, meaning that if volatility is relatively low it will very probably remain low in the near future, and if volatility increased (declined) a lot on a given day, it will most likely bounce back soon ─ While GARCH gives us the best-known prediction of volatility range so far, it holds only as long as volatility does not change its regime, and in a different regime volatility behaves quite differently ─ Volatility does return to its mean according to GARCH, but when the regime switches occur the mean changes quite significantly. Therefore, most of the research in this area are now focused on forecasting regime changes in volatility patterns

46 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

The Chicago Board Options Exchange SPX Volatility Index reflects a market estimate of future volatility, based on the weighted average of the implied volatilities for a wide range of strikes

47 Management,Asset lecture VI Elementary applications of derivatives to portfolio management

• Which financial instruments can be used to invest in volatility ? ─ A volatility trade can be constructed from plain vanilla options in the market directly, using volatility index futures or alternatively by buying/selling volatility swaps ─ Combining plain vanilla options in order to achieve exposure to volatility only, without a directional price movement bet is the oldest way to trade volatility ─ Its advantages are that one can construct volatility trades using relatively liquid exchange-traded instruments. The disadvantage is that the neutrality to price movements disappears as soon as the price of the underlying moves requiring an expensive hedging of such combined position ─ Variance and volatility swaps are forward OTC contracts in which two parties agree to exchange cash flows during a specified period of time ─ While a exchanges realized variance against a fixed pay-off calculated in units of variance, payoff = units x (σ2r - σ2x), while exchanges realized volatility against a fixed payoff, payoff = units x ( σr - σx) ─ The payoff of a variance swap is nonlinear in volatility. However, variance swaps are easier to hedge, are more liquid and are provided for single stocks as for the indices as well. The main difference to the VIX (or VDAX) future is variance and volatility swaps are related to the realized volatility on the market and the futures to implied

48 Management,Asset lecture VI