FEAR &GREED VOLATILITY MARKETS. Emanuel Derman. Quantitative Strategies Group Goldman Sachs & Co. Quantitative Strategies. Page 1 of 24. Fear&Greed.

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1 FEAR &GREED IN VOLATILITY MARKETS ~ Emanuel Derman Group Goldman Sachs & Co. Page 1 of 24 Fear&Greed.fm

2 Are There Patterns to Volatility Changes? Since 1987, global index options markets are persistently skewed. How do/should volatilities and the skew change as markets move? Every description of data involves an articulated or unarticulated model. There are at least three models for volatility change: An apocryphal Sticky-Strike Rule, that reflects Greed; An apocryphal Sticky-Delta Rule, that reflects Moderation; A theoretical Implied Tree Model, that reflects Fear. Each rule leads to different predictions for valuing & hedging options. Which works best? And why? Traders daily reports are sometimes unreliable. They focus on liquid at-the-money volatility, a moving target, but they own definite strikes. Therefore, ignore everyone and look at the data through the prism of models. Page 2 of 24 There appear to be several distinct periods ( regimes ) in which different rules seem to hold. Often, S&P 500 implied volatilities seems to oscillate between the Fear Rule and the Greed Rule... Producing Moderation in the long run, but not the short.

3 Contents 1. INTRODUCTION: GLOBAL IMPLIED VOLATILITIES 2. GREED (STICKY STRIKE) 3. MODERATION (STICKY DELTA) 4. FEAR (STICKY IMPLIED TREE) 5. WHAT REALLY HAPPENS: MODEL REGIMES Page 3 of 24

4 PART I PART I INTRODUCTION: GLOBAL INDEX IMPLIED VOLATILITIES INTRODUCTION: GLOBAL INDEX IMPLIED VOLATILITIES Page 4 of 24

5 A Persistent Negative Global Skew A persistent large skew, almost linear, and inconsistent with Black- Scholes. PART I INTRODUCTION: GLOBAL INDEX IMPLIED VOLATILITIES Global Three-Month Volatility Skews Mar D Put Atm 25D Call Nikkei 225 S&P 500 Hang Sendg g FTSE 100 DAX CAC 40 MIB 30 SMI AEX Page 5 of 24 Σ( K ) = Σ atm bk ( S 0 )

6 PART I INTRODUCTION: GLOBAL INDEX IMPLIED VOLATILITIES Page 6 of 24 A Negative Correlation with the Index The S&P 500 index and its at-the-money three-month implied volatility, Sep through Nov Three-Month Implied Volatilities of SPX Options INDEX ATM Note - you don t own at-the-money volatility, you own a fixed strike.

7 PART 50 I INTRODUCTION: 40 GLOBAL 35 INDEX IMPLIED 30 VOLATILITIES 25 Page 7 of 24 Volatility Behavior By Strike Is Complex Three Month Implied Volatilities of SPX Options What s going on here? INDEX ATM 750" 800" 850" 900" 950" 1000" 1050" 1100" 1150" 1200"

8 What s The Future Skew? We know the current skew Σ(K) = Σ atm b(k - S 0 ). Hypothetical Implied Volatility of Three-Month SPX Options Index Strike PART I INTRODUCTION: GLOBAL INDEX IMPLIED VOLATILITIES ? 18? ? 20? ? 22? Page 8 of 24 What will happen when the index moves? What s the S-dependence in Σ(S,K)? Distinguish carefully between Σ(S,K) and Σ atm (S) = Σ(S,S).

9 PART II PART II GREED (STICKY STRIKE) GREED (STICKY STRIKE) Page 9 of 24

10 Complacency or Greed: Sticky Strike Model The simplest & most convenient model for changing the implied volatility of an option as the index moves is not to change it at all. This is the or complacency model, or sticky strike, the closest thing to Black-Scholes. It s also the lazy-trader model. STICKY STRIKE Σ( S, K) Σ( K ) = Σ atm bk ( S 0 ) PART II GREED (STICKY STRIKE) Characteristics Fixed-strike volatility is independent of S. Therefore, because of the negative skew, at-the-money volatility falls with rising S. = BS. In a rising market, you can think of this model as representing Irrational Exuberance or Greed: At-the-money options are the most liquid. When the market rises, at-the-money volatility falls, and you are selling the most liquid options more and more cheaply, as though you need never worry about future index declines. Page 10 of 24

11 How Options Trees Evolve In The Sticky Strike Model Index Strike Known 90 PART II GREED (STICKY STRIKE) Fixed-strike volatility is independent of S. Therefore, because of the negative skew, at-the-money volatility falls with rising S. = BS. Page 11 of 24

12 PART III PART III MODERATION (STICKY DELTA) MODERATION (STICKY DELTA) Page 12 of 24

13 Rational Moderation At-the-money volatility is the rational estimate for the future cost of replicating liquid options issued now. On average, over the long run, at-the-money volatility should be independent of index level. If you have no special expectations about the future, you should keep at-the-money volatility unchanged. Given the negative skew, as the index rises, you need to raise every strike s volatility to keep at-the-money volatility unchanged. PART III Traders refer to this as the Sticky Moneyness or Sticky Delta Model. MODERATION (STICKY DELTA) STICKY DELTA : Σ = Σ( K S) = Σ atm bk ( S) Characteristics Atm vol is independent of S. Fixed-strike vol increases with S. > BS. Page 13 of 24

14 How Options Trees Evolve In The Sticky Delta Model. Index Strike Known PART III MODERATION (STICKY DELTA) Atm vol is independent of S. Fixed-strike vol increases with S. Page 14 of 24 > BS.

15 PART IV PART IV FEAR (STICKY IMPLIED TREE) FEAR (STICKY IMPLIED TREE) Page 15 of 24

16 Why The Skew? Fear of Index Declines! PART IV FEAR (STICKY IMPLIED TREE) The skew represent the premium for the fear of a downward market move and an increase in realized and implied volatility. Relation between the current skew and the expected future volatility. Strike Implied Volatility (%) % 99 21% 98 22% 97 23% You can deduce the local volatility at different market levels by treating the implied volatility as an average over local (future at-themoney) volatilities. Index Level Local volatility (%) % 99 22% 98 24% 97 26% Page 16 of 24 These local volatilities are the future at-the-money volatilities feared to occur in a decline. Note that local volatilities increase twice as fast with index changes as implieds increase with strike.

17 Sticky Implied Tree Extracts Local Volatilities There is one market-consistent tree - the implied tree - whose expectations of future volatilities match all current options prices and the skew. In this view, the skew is attributable to an expectation of higher volatility as the market moves (jumps?) down. You can use this tree to price all options consistently off future implied local volatilities. This is similar to pricing all off-the-run bonds off current forwards. PART IV FEAR (STICKY IMPLIED TREE) stock price variable local volatility σ(s,t) in the future time several different constant volatility trees are equivalent to one implied tree When the index moves, to find the new skew, you roll along the local vols. This is similar to rolling along the forward curve to get future yields as time passes. STICKY IMPLIED TREE: Σ( K, S) = Σ atm bk ( + S) Page 17 of 24

18 How Options Trees Evolve In The Sticky Implied Tree Model Strike Index Current Tree PART IV FEAR (STICKY IMPLIED TREE) Page 18 of 24 Fixed-strike volatility decreases as K or S increases. Atm vol falls twice as rapidly as skew. < BS.

19 PART V PART V MODEL SUMMARY MODEL SUMMARY Page 19 of 24

20 The Properties of the Models Stickiness Model Strike PART V MODEL SUMMARY Delta Implied tree Equation for Σ( S, K) Σ atm () t bt ()K ( S 0 ) Σ atm () t bt ()K ( S) Σ atm () t bt ()K ( + S) Fixed-strike Option Volatility independent of index level increases as index level increases decreases as index level increases Behavior of At-the-money Option Volatility decreases as index level increases independent of index level decreases twice as rapidly as index level increases Delta = BS > BS < BS Page 20 of 24

21 PART VI PART VI WHAT REALLY HAPPENS:MODEL REGIMES WHAT REALLY HAPPENS: MODEL REGIMES Page 21 of 24

22 Which Model Reigns in Which Regime? PART VI 60 WHAT REALLY 55 HAPPENS:MODEL REGIMES 50 Three-Month S&P 500 Implied Volatilities 45 40? /1/97 9/11/97 9/23/97 10/6/97 10/17/97 10/29/97 11/10/97 11/20/97 12/3/97 12/15/97 12/25/97 1/7/98 1/20/98 2/2/98 2/12/98 2/24/98 3/6/98 3/18/98 3/30/98 4/9/98 4/21/98 5/1/98 5/13/98 5/25/98 6/4/98 6/16/98 6/26/98 7/8/98 7/20/98 7/30/98 8/11/98 8/21/98 9/2/98 9/14/98 9/25/98 10/7/98 10/19/98 10/29/98 11/10/98 11/20/98 12/2/98 12/14/98 12/24/98 1/6/99 1/18/99 1/28/99 2/9/99 2/19/99 3/4/99 3/16/99 3/26/99 4/7/99 4/19/99 4/29/99 Page 22 of ATM S&P 500 Level sticky strike or sticky implied tree Fear Greed Correction Greed Fear Greed Correction jumpy index: sticky implied tree index trends; should be sticky delta, seems to be sticky strike CORRECTION vols rise to sticky delta level stable index trends; should be sticky delta, seems to be sticky strike jumpy index: sticky implied tree index trends; should be sticky delta, seems to be sticky strike CORRECTION vols rise to sitcky delta level sticky implied tree INDEX Volatility

23 Conclusions Sticky strike (complacency) Sticky delta (moderation) Sticky implied tree (fear) PART VI WHAT REALLY HAPPENS:MODEL REGIMES are intuitively useful ways of thinking about variations in implied volatility that sometimes correspond to modes of market behavior. When times are good, and the index keeps rising, the options market keeps every strike s volatility roughly fixed, and so the pendulum of at-the-money volatility drops. When times get bad, and the index jumps down a few percentage points, the market has to compensate for having let at-the-money volatility drop too far. The pendulum reverses, and moves at-themoney volatility up at twice the rate as the index collapses. On average, over the long haul, the pendulum oscillations between sticky-strike Greed and sticky-implied-tree Fear average out to sticky-delta Moderation. Will these conjectured regimes extend through time and across markets? Is there a model of stochastic volatility that encompasses this? Page 23 of 24

24 Recent Update: July-August 99 PART VI WHAT REALLY HAPPENS:MODEL REGIMES Page 24 of 24

25 Volatility /1/99 6/2/99 skew is about 4 vol pts per 100 S&P pts points 6/3/99 6/4/99 6/7/99 6/8/99 6/9/99 6/10/99 6/11/99 6/14/99 6/15/99 6/16/ Three-Month S&P Implied Volatilities Rising Index, Atm vol falls to 19% Falling index, Atm vol rises to 25% Rising index, Atm vol falls again pt rise in S&P S&P declines 140 pts S&P rises 80 pts vols by strike 25.4 rise about pts vols by strike remain vols by strike 33.26remain roughly unchanged again unchanged Atm vol rises twice 21.3as much, about 6 pts ATM vol again drops ATM vol 33.6drops about pts about 3 pts as index as index 33.73rises rises Date ATM INDEX /17/99 6/18/99 6/21/99 6/22/99 6/23/99 6/24/99 6/25/99 6/28/99 6/29/99 6/30/99 7/1/99 7/2/99 7/5/99 7/6/99 7/7/99 7/8/99 7/9/99 7/12/99 7/13/99 7/14/99 7/15/99 7/16/99 7/19/99 7/20/99 7/21/99 7/22/99 7/23/99 7/26/99 7/27/99 7/28/99 7/29/99 7/30/99 8/2/99 8/3/99 8/4/99 8/5/99 8/6/99 8/9/99 8/10/99 8/11/99 8/12/99 8/13/99 8/16/99 8/17/99 8/18/99 8/19/99 8/20/99 8/23/99 8/24/99 8/25/99 Index