[dresdner klein wort, bossu] introduction to volatility trading and variance swaps

0

Introduction to Volatility Trading and Variance Swaps

EQUITY DERIVATIVES WORKSHOP

University of Chicago — Program on Financial Mathematics

Sebastien BossuEquity Derivatives Structuring — Product Development

24 February 2007

1

1

Disclaimer

This document only reflects the views of the author and not necessarily those of Dresdner Kleinwort research, sales or trading departments.

This document is for research or educational purposes only and is not intended to promote any financial investment or security.

2

2

‘Cash’

Futures Options

Exotics

Introduction

Typical Trading Floor — Instruments

Volatility Trading and Variance Swaps

3

3

Introduction

Typical Trading Floor — Front Office


Trading

Marketing/ Sales

Research

Quants

Economists

Structurers/ Financial

Engineers

4

4


►Key concepts behind Black-Scholes.............................................................

►Black-Scholes in practice ............................................................................

►Managing an option book ...........................................................................

►Trading volatility ..........................................................................................

►P&L path-dependency ................................................................................

►Variance swaps ...........................................................................................


5

12

21

27

39

45

5

Key concepts behind Black-Scholes►Why is Black-Scholes used in practice?►Key strengths►Key limitations

6

6

Key concepts behind Black-Scholes

Why is Black-Scholes used in practice?

Derman: ‘In 1973, Black and Scholes showed that you can manufacture an IBM option by mixing together some shares of IBM stock and cash, much as you can create a fruit salad by mixing together apples and oranges. Of course, options synthesis is somewhat more complex than making a fruit salad, otherwise someone would have discovered it earlier. Whereas a fruit salad's proportions stay fixed over time (50 percent oranges and 50 percent apples, for example), an option's proportions must continually change. [...] The exact recipe you need to follow is generated by the Black-Scholes equation. Its solution, the Black-Scholesformula, tells you the cost of following the recipe. Before Black and Scholes, no one ever guessed that you could manufacture an option out of simpler ingredients, and so there was no way to figure out its fair price.’

My Life as a Quant, John Wiley & Sons, 2004


7

7


Key strengths

►Sensible model for the behavior of stock prices: random walk / log-normal diffusion

►Intuitive parameters: spot price, interest rate, volatility

►Arbitrage argument: dynamic hedging strategy


8

8


Key limitations

►True or False?‘Stock prices follow a random walk / a lognormal diffusion’

►False – Stock prices are determined by supply and demand which are influenced by countless economic factors. If a company announces bankruptcy, its stock price WILL go down with 100% probability.


9

9


Key limitations

►True or False?‘If I buy an option at a higher price than ‘the’ Black-Scholes price, I will lose money.’

►False – Different agents have different uses for options:

►Bets: individual investors, asset managers

►Hedging: corporate investors

►Volatility trading: traders, hedge funds


i.e Black-Scholes is not an arbitrage price in the sense that one loses money when trading at a different level (compared to forward contracts / futures where there is a ‘strong’ (static) arbitrage.)

10

10


Key limitations

►True or False?‘If I buy an option at a price higher than ‘the’ Black-Scholes price and I follow ‘the’Black-Scholes delta-hedging strategy, I will lose money.’

►False but in one case:

►False in practice: discrete hedging, jumps, stochastic volatility may have a positive impact on P&L

►False in theory: even if we assume that the realised stock price process is a log-normal diffusion…

►…unless we also assume that the realised stock price process follows a log-normal diffusion with the same volatility parameter as the one used to price the option


11

11


So what is Black-Scholes?

►NOT a good model for the real behavior of stock prices

►NOT a good model to determine the ‘fair value’ of an option: different agents give different values to the same option

►NOT a good model to arbitrage option prices: too many factors are ignored

►BUT a powerful, simple toy model to:

►Understand which factors influence the price of an option and estimate its manufacturing cost

►Interpret a market quote (more on this in the next section)


12

Black-Scholes in Practice► Implied Volatility

13

13

Black-Scholes in Practice

Input/Output Diagram

Spot Price

Black Scholes

Volatility

Interest & DividendRates

Maturity

Strike Price

Option Price


14

14


Implied Volatility Diagram

Spot Price

Black Scholes

Volatility

Interest & DividendRates

Maturity

Strike Price

Option Price


15

15


Implied Volatility Example: S&P 500 Dec-08 options

Source: Bloomberg. Data as of 18 October 2006.


16

16


Implied Volatility Example: S&P 500 Dec-08 options


Implied volatility

12%

14%

16%

18%

20%

22%

24%

800 1000 1200 1400 1600 1800Source: Dresdner Kleinwort.

17

17


Implied Volatility Smile and Term Structure

►Implied Volatility Smile (or Skew)

►In Black-Scholes the volatility parameter is assumed to remain constantthrough time…

►… in practice every option expiring at time T has a different implied volatilitydepending on its strike K

►Implied Volatility Term Structure

►Also every option struck at level K has a different implied volatility depending on its expiry T.


18

18

331

680

968

1083

1198

1290

1348

1429

1544

1659

1970

2831

17-Nov-0617-Oct-07

16-Oct-11

0%

10%

20%

30%

40%

50%

60%


Implied Volatility Surface

smile

+ term structure =


implied volatilitysurface

ATM Implied volatility

0%2%4%6%8%

10%12%14%16%18%20%

Oct-06 Oct-07 Oct-08 Oct-09 Oct-10 Oct-11

Implied volatility

12%

14%

16%

18%

20%

22%

24%

800 1000 1200 1400 1600 1800

19

19


Reasons for Implied Volatility

►In the early days options often had the same implied volatility. After the October 1987 crash, volatility surfaces appeared. Why?

►Stock prices do not follow the log-normal diffusion postulated by Black-Scholes. Mainly: volatility is itself volatile, jumps can occur! In particular, stock prices and volatility are negatively correlated: when the market goes down, volatility goes up

►Delta-hedging cannot take place continuously, transaction costs can be significant.

►From a fundamental value perspective, implied volatility can be seen as a market adjustment to take into account everything which Black-Scholes does not.

►From a relative value perspective, implied volatility has become the standard measure to compare option prices, in a similar way as yield for bonds.


20

20

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00%

SPX Index linear regression


Reasons for Implied Volatility: Evidence of Equity Skew


daily return

change in S&P 500 implied volatility

21

Managing an Option Book►Greeks►Hedging

22

22

Managing an Option Book

Definition of an Option Market-Maker

►The job of an option market-maker is to provide liquidity to option buyers and sellers while securing her margin (bid-offer). Thus she will try to minimize the impact of market factors on the mark-to-market of her option book to make it as close as possible to a risk-free portfolio.

►Typical factors:

►Change in spot price: small & large

►Change in implied volatility

►Passage of time

►Change in interest rate

►Change in dividends.


23

23


Greeks

►The change in option price f resulting from a change in one factor is named sensitivity or ‘Greek’:

Change in f due to change in dividendsMu

Change in f due to change in interest rateRho

Change in f due to passage of timeTheta

Change in f due to change in implied volatilityVega

Second-order change in f due to (large) change in spot price = Change in Δ due to change in spot priceGamma

Change in f due to (small) change in spot priceDeltaSf∂∂

=Δ

2

2

Sf

∂∂

=Γ

σ∂∂

=fV

tf∂∂

=Θ

rf∂∂

=ρ

qf∂∂

=μ


24

24


Greeks at Book Level

►By linearity of differentiation, the Greeks of an option book are equal to the sum of the individual Greeks multiplied by the positions. For example:

►Book = Long 1,000 Option 1 and Short 500 Option 2

►Book Value = 1,000 x Price 1 – 500 x Price 2 (‘mark-to-market’)

►Book Delta = 1,000 x Delta 1 – 500 x Delta 2

►Etc.


25

25


Hedging

►The standard approach to minimize the impact of market factors on the mark-to-market of an option / a book of options is to offset (‘hedge’) the Greeks with a relevant instrument

►Example: Delta-hedging

►Initial Book Delta = $5,000 per S&P 500 index point

►Delta-hedge = Sell 5,000 units of S&P 500

►Final Book Delta = 0. This means that the book mark-to-market value isimmune to (small) changes in the level of S&P 500.


26

26


Hedging

►To hedge other Greeks than Delta (e.g. Gamma, Vega…) our market-maker must trade other instruments.

►However, it is usually impossible to perfectly hedge all Greeks.

►This implies that the market-maker is left with some risks.

►Her job is to design her option book so as to be left with the risks she is comfortable with (e.g. long Vega if she believes volatility is on the rise etc.).


27

Trading Volatility►Where does volatility appear in Black-Scholes?►Daily option P&L equation►Volatility trading equation

28

28

Trading Volatility

Definition

Taleb: ‘Volatility is best defined as the amount of variability in the returns of a particular asset. [...] Actual volatility is the actual movement experienced by the market. It is often called historical, sometimes historical actual. Implied volatility is the volatility parameter derived from the option prices for a given maturity. Operators use the Black-Scholes-Merton formula (and its derivatives) as a benchmark. It is therefore customary to equate the option prices to their solution using the Black-Scholes-Merton method, even if one believes that it is inappropriate and faulty, rather than try to solve for a more advanced pricing formula.’

Dynamic Hedging: Managing Vanilla and Exotic Options, John Wiley & Sons, 1997


29

29

Trading Volatility

DefinitionHistorical (‘Realised’) Volatility

►Annualized standard deviationof daily stock returns:

where:

►Volatility parameter in Black-Scholesmodel of stock prices (random walk / lognormal diffusion):

Implied Volatility

∑=

−−

=N

ttHistorical rr

N 1

2)(1

252σ

∑=−

==N

tt

t

tt r

Nr

SSr

11

1ln

tImpliedt

t dWdtS

dS σμ +=


30

30

►True or False?‘An option market-maker sold a call at 30% implied volatility and delta-hedgedher position daily until maturity. The realised volatility of the underlying was 27.5%. Her final P&L must be positive.’

►False – The trading P&L on a delta-hedged option position is ‘path-dependent.’

Trading Volatility

Implied vs. Realised


31

31

Trading Volatility

Where does realised volatility appearin Black-Scholes?►Consider the Black-Scholes Partial Differential Equation:

►With Greek notations:

►What is σ? σRealised or σImplied?

tf

SfS

SfrSrf

∂∂

+∂∂

+∂∂

= 2

222

21σ

Θ+Γ×+Δ×= 22

21 SrSrf σ


32

32

Trading Volatility

Where does realised volatility appearin Black-Scholes?

►Remember that Black-Scholes derive ‘the’ price of an option by modelling the behaviour of an option market-maker who follows a delta-hedging strategy.

►In this idealized world there is only one volatility:

σRealised = σImplied = σ

►In reality traders tweak the model through the volatility parameter to make up for the model imperfections. As such they don’t believe in the model, they merely use it.


33

33

Trading Volatility

Daily option P&L equation

►The daily P&L on an option position can be decomposed along the Greeks:

Full Daily P&L = Delta P&L + Gamma P&L + Theta P&L + Vega P&L + Rho P&L + Mu P&L + Other

►Other = high-order sensitivities (e.g. sensitivity of Vega to a change in the spot price…)


Note that Delta P&L, Gamma P&L and Theta P&L correspond to the ‘state variable risks’ modelled in Black-Scholes, while the Vega P&L, Rho P&L, Mu P&L etc. correspond to ‘parametric risks’ which are not modelled in Black-Scholes. A more sophisticated model such as stochastic volatility with jumps would transfer some parametric risks (Vega) to the state variable risks universe, leading to different Greeks than Black-Scholes.

34

34

Trading Volatility


►Assuming constant volatility, zero rates and dividends, and ‘Other’ is negligible, we obtain the reduced daily option P&L equation:

Daily P&L = Delta P&L + Gamma P&L + Theta P&L

= Δ x (ΔS) + ½Γ x (ΔS)2 + Θ x (Δt)

where ΔS is the change in stock price and Δt is one trading day (1/252nd).


35

35

Trading Volatility


►For a delta-hedged option position, we have Δ = 0. Hence:

Daily P&L = ½Γ x (ΔS)2 + Θ x (Δt)

►Typically Gamma and Theta have opposite signs:

►For a long call or put position, Gamma is positive and Theta is negative, i.e. the trader is long shocks/volatility (she makes money as the stock price moves) and short time (she loses money as maturity approaches.)

►For a short call or put position, the situation is reversed.


36

36

Trading Volatility


►Graph of Gamma vs. Theta


p/l at start of day p/l at close of business

profit

loss

S

ΘΘΘ

Γ Γ

37

37

Trading Volatility

Volatility trading equation

►In fact, Theta can be expressed with Gamma through the proxy formula:

►Plugging the proxy into the daily option P&L equation:

22

21

ImpliedS σΓ−≈Θ

[ ]

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ−⎟

⎠⎞

⎜⎝⎛ ΔΓ≈

Δ×−ΔΓ≈

22

2

222

21

)(21&

tSSS

tSSLPDaily

Implied

Implied

σ

σ


38

38

Trading Volatility

Volatility trading equation

►This equation tells us that the daily option P&L on a delta-hedged option position is driven by two factors:

►Dollar Gamma, which has the role of a scaling factor and does not determine the sign of the P&L

►Variance Spread (realised vs. implied), which determines the sign of the P&L

►Thus, a trader who is long dollar gamma will make money if realised variance is higher than implied, break even if they are the same, and lose money if realised is below implied.

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡Δ−⎟

⎠⎞

⎜⎝⎛ ΔΓ≈

22

2

21& t

SSSLPDaily Impliedσ


‘The Second-Most Important Equation’ in finance according to Derman

39

P&L path-dependency►Case study►Path-dependency equation

40

40

P&L path-dependency

Case study

►An option market-maker sold a 1-year call struck at €110 on a stock trading at €100 for an implied volatility of 30%, and delta-hedged her position daily until maturity.

►The realised volatility of the underlying was 27.5%

►2 months before maturity, her P&L was up €100,000

►Yet her final trading P&L is down €60,000

►How did the profits change into losses?


41

41

P&L path-dependency

Case study

►The hard life of an option trader...

0

20

40

60

80

100

120

0 14 28 42 56 70 84 98 112

126

140

154

168

182

196

210

224

238

252

-80,000

-40,000

-

40,000

80,000

120,000

160,000Stock price Cumulative P/L (€)

Strike = 110

Stock price

Cumulative P/L

Trading days


42

42

P&L path-dependency

Case study

►Realised Volatility and Dollar Gamma

40%

18%

29%

0

20

40

60

80

100

120

0 14 28 42 56 70 84 98 112

126

140

154

168

182

196

210

224

238

252

0%

10%

20%

30%

40%

50%

60%Stock price Volatility

Dollar Gamma

Strike = 110

Stock price

50-day realizedvolatility

Trading days


Note that the graph of the dollar gamma actually corresponds to a short position.

43

43

P&L path-dependency

Path-dependency equation

►Summing all daily option trading P&L’s until maturity, we obtain the path-dependency equation:

where γt = ½ x Γ(t-1, St-1) x St-12 is the Dollar Gamma at the beginning of day t and

rt = (St - St-1)/St-1 is the stock return at the end of day t.

►With this expression, we can clearly see that the final P&L is the sum of the daily Variance Spread weighted by the Dollar Gamma. Thus, days when Dollar Gamma is high will tend to dominate the final P&L.

( )∑=

⎥⎦⎤

⎢⎣⎡ Δ−≈

N

tImpliedtt trLPFinal

1

22& σγ


44

44

P&L path-dependency

A path-independent derivative?

►For the final P&L to be path-independent (in the sense of the equation in the previous page), the Dollar Gamma must be constant.

►This defines a new type of derivative, the log-contract:

►Log-contracts are not traded, but they are closely connected to Variance Swaps introduced in the next section.

abSScfSc

Sf

Scct ++=→=

∂∂

→=Γ→= ln22

2

2γ


45

Variance Swaps► Introduction►Hedging & Pricing►Mark-to-Market Valuation

46

46

Variance Swaps

Introduction

►A variance swap is an exotic derivative instrument where one party agrees to receive at maturity the squared realised volatility converted into dollars for a pre-agreed price:

where:

►St is the closing price on day t

►N is the number of trading days between the trade date and the maturity date

►252 is the number of trading days in a year

►Kvar is the variance strike expressed in volatility percentage points (e.g. 30%) and is not to be confused with the strike of a vanilla option which is a stock price level

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛×= ∑

=

−

−

2var

1

returnlogSquared

2

1

ln2521$ KSS

NPayoffSwapVariance

N

t t

t

43421


47

47

Variance Swaps

Introduction

►Variance Buyer: [Dresdner]

►Variance Seller: [‘Sigma LLC’]

►Underlying asset: S&P 500

►Start date: Today

►Maturity date: Today + 1yr

►Strike (Kvar): 30%

Example

►Scenario 1

►Realised volatility 20%

►Payoff = $1 x (0.22 – 0.32)= $-0.05

► Thus, the variance buyer (Dresdner) pays 5 cents to the variance seller (Sigma)

►Scenario 2

►Realised volatility 40%

►Payoff = $1 x (0.42 – 0.32)= $0.07

►Here Dresdner receives 7 cents from Sigma

Payoff Calculation


48

48

Variance Swaps

Introduction

►Note:

►Returns are computed on logarithmic basis rather than arithmetic

►The mean return is not subtracted (‘zero-mean assumption’)

►In practice the log-returns are multiplied by 100 to convert from decimal to percentage point representation, and the variance strike is quoted in volatility points (30 for 30%)

►Often the number of variance swap units is calculated from a notional specified in volatility terms (e.g. $100,000 per volatility point):

var2 KNotionalVegaNotionalVariance×

=


49

49

Variance Swaps

Introduction

►Variance Swaps are actively traded on the major equity indices: S&P 500, Nasdaq, EuroStoxx50, Nikkei...

►Typical bid/offer spread is ½ to 2 volatility points (‘vegas’)Indic. mids SPX NDX SX5E NKYNov-05 15.6 16.7 17.4 17.5Dec-05 15.2 16.6 17.0 17.5Jan-06 15.3 17.4 16.9 17.4Mar-06 15.7 17.9 17.5 17.2Jun-06 16.2 18.9 17.7 17.6Sep-06 16.6 19.8 18.0 17.8Dec-06 16.9 20.4 18.6 18.1Jun-07 17.4 21.3 18.7 18.6Dec-07 17.8 21.9 19.3 19.2Jun-08 18.3 22.3 19.6 19.7Dec-08 18.7 22.6 19.9 20.1Dec-09 19.6 23.1 20.3 20.4


Source: Dresdner Kleinwort. Data as of October 2005.

50

50

Variance Swaps

Hedging & Pricing

►Compare the Variance Swap payoff (1) with the P&L path-dependency equation (2):

►Substantially, the difference between equations (1) and (2) lies in the weighting of the squared daily log-returns:

►Variance Swaps are equally weighted

►The final P&L of a delta-hedged vanilla option position is Dollar-Gamma-weighted

( )

22var

1 1

22

1

252(1) ln

(2) &

Nt

t t

N

t t Impliedt

SVariance Swap Payoff KN S

Final Option P L r tγ σ

= −

=

⎛ ⎞= −⎜ ⎟

⎝ ⎠

⎡ ⎤= − Δ⎢ ⎥⎣ ⎦

∑

∑


51

51

Variance Swaps

Hedging & Pricing

►Thus, a Variance Swap could be hedged by an option with constant Dollar Gamma: the Log-Contract

►However, Log-Contracts do not trade in the market

►Problem: Can we find a combination of vanilla calls and puts with the same maturity as the variance swap such that the aggregate Dollar Gamma is constant?

►Formally: Find quantities (‘weights’) w1Put, w2

Put, ... and w1Call, w2

Call, ... such that:

cwwCallPut N

i

Calli

allCi

N

i

Puti

PutiAggregate =+= ∑∑

== 11

γγγ►Put i struck at Ki

Put

►Call i struck at KiCall


52

52

Variance Swaps

Hedging & Pricing

►Answer: Use weights inversely proportional to the square of strike: wi = 1/Ki2

K = 25

K = 50 K = 75

K = 100 K = 125 K = 150 K = 175 K = 200

Aggregate

0 50 100 150 200 250 300

Dollar Gamma

S

Constant Gamma Region


53

53

Variance Swaps

Hedging & Pricing

►A perfect hedge would require an infinite number of liquid options with strikes forming a continuum [0, ∞)

►The fair strike of a Variance Swap (i.e. the level of Kvar such that the swap has zero initial value) is then given as:

where r is the constant interest rate, T is the maturity, Put(k), Call(k) denote the price of a put or call struck at k% of the underlying asset’s forward price.

⎥⎦⎤

⎢⎣⎡ += ∫∫

∞+

1 2

1

0 2*var )(1)(12 dkkCall

kdkkPut

kTeK

rT


54

54

Variance Swaps

Mark-to-Market Valuation

►Because variance is additive, the mark-to-market valuation of a variance swap position can be linearly decomposed between past (realised) variance and future (implied) variance:

where:

►Realisedt is the realised volatility between the start date 0 and date t (under zero-mean assumption)

►Impliedt is the current fair strike of a (notional) variance swap starting on date t and ending on date T

►Kvar is the strike level agreed on the start date 0

⎥⎦⎤

⎢⎣⎡ −

−+= −− 2

var22)( KImplied

TtTRealized

TteMTM tt

tTrt


55

55

References & Bibliography

►My Life as a Quant, Emanuel Derman, Wiley (2004)

►Dynamic Hedging: Managing Vanilla and Exotic Options, Nassim Taleb, Wiley (1997)

►More Than You Ever Wanted To Know About Volatility Swaps, K. Demeterfi, E. Derman, M. Kamal, J. Zou, Goldman Sachs Quantitative Strategies (1999)

►Self-referencing:

► Introduction to Variance Swaps, Wilmott Magazine (March 2006)

► Just What You Need To Know About Variance Swaps, with E. Strasser, R. Guichard, JPMorgan Equity Derivatives Report (2005)


[dresdner klein wort, bossu] introduction to volatility trading and variance swaps

Documents