Option Anatomy. Copenhagen University March Jesper Andreasen Danske Markets, Copenhagen

Transcription

1 Option Anatomy Copenhagen University March 2013 Jesper Andreasen Danske Markets, Copenhagen

2 Outline Acknowledgements. Kwant life. Introduction. Volatility interpolation. Wots me Δelδa? Conclusion. 2

3 References Andreasen, J and B Huge (2011a): Volatility Interpolation. Risk March, Andreasen, J and B Huge (2011b): Random Grids. Risk July, Andreasen, J and B Huge (2013a): Expanding Forward Volatility. Risk January, Andreasen, J and B Huge (2013b): Wots me Δelδa? Unfinished wp, Danske Markets. Balland, P (2006): Forward Smile. Presentation, ICBI Global Derivatives. Carr, P. (2008): Local Variance Gamma. WP, Bloomberg. 3

4 Dupire, B (1994): Pricing with a Smile. Risk January, Dupire, B (2006): Skew Modeling. Presentation, ICBI Global Derivatives. Hagan, P, D Kumar, A Lesniewski, D Woodward (2002). Managing Smile Risk. Wilmott Magazine September, Nabben, R. (1999): On Decay Rates of Tridiagonal and Band Matrices. SIAM J. Matrix Anal. Appl. 20,

5 Thanks to... Copenhagen University and the scientific committee (Rolf Poulsen, Michael Sørensen, Mogens Steffensen) for awarding me the title of Adjungeret Professor in Mathematical Finance at the Department of Mathematics. The many great colleagues that I have had the pleasure and privilege to work with over the past 16 years. Not least co-authors Leif Andersen and Brian Huge. The great teachers and mentors: Maria Arys, Jørgen Aase Nielsen, and William Keirstead. 5

6 Kwant Life In the late 90s and early 00s the main focus of quant work was the pricing of ever more complicated products and pay-offs. The general approach was to take the vanillas as direct inputs and focus the modelling of the exotics. The financial crisis (luckily) changed that. Liquidity dried up and the model focus over the past 5 years has been much more fundamental. Particularly, a lot of quant energy has gone into answering the questions: - What is the forward? Libor basis... - How to discount? X-ccy basis, collateral, xva,... 6

7 In fact, quant technology is more in demand and in wide spread use today, than it ever was. Quant clients have been extended to include the whole universe of derivatives trading and more. In this talk, we are going to consider the next step up in complexity from forwards and discounting: - How to price and hedge an option? 7

8 Introduction Andreasen and Huge won the Risk Magazine Quant of the Year 2012 award for the two papers Volatility Interpolation and Random Grids....The pair turned conventional approaches on their head, to the bafflement of their colleagues in the quant community, with some highly original thinking on the subject of implied volatility calibration... I could not have said it better myself. The general idea is to work directly with discrete finite difference based models rather than with the continuous time stochastic differential equations. 8

9 The focus is on stability, computer implementation, and consistency with market quotes, rather than perceived accuracy relative to some theoretical stochastic differential equations....which are irrelevant because the discrete model is calibrated to market quotes. The idea can be applied to other types of approximations than the finite difference method. Specifically, we have over the past 3-4 years worked a lot on expansions used on their own or in combination with finite difference methods. Today, we will consider two very fundamental problems: - How to interpolate discrete option quotes without introducing arbitrage? 9

10 - What Delta should I use to hedge an option? For the first problem we will use finite difference machinery and for the second, short maturity expansions. Note: This is a very compressed presentation where we omit a (large) number of proofs, technicalities, extensions, applications, references, and deeper thoughts. 10

11 Volatility Interpolation Someone said that the Black-Scholes formula is the wrong formula c( t, k) s ( ) k ( ), ln s/ k 1v t vt 2 (1)...but if you stuff in a wrong number, for the (implied) volatility v, then you might get the right option price. Suppose you have a discrete set of implied volatilities v ( t, k ), i 1,, n (2) i i... corresponding to the discrete set of option prices { (, )} 1,, c t k. i i i n 11

12 It seems like a trivial exercise to interpolate and extrapolate these implied volatilities into a continuous surface of option prices in expiry and strike { c ( t, k )}... t 0, k 0... except that it very much isn t! The trouble is that it exceptionally easy to break the arbitrage conditions c t k t (, ) 2c( t, k) 2 0, 0 k (3) For a given interpolation scheme for the implied volatilities one can compute the resulting surface of forward volatilities using Dupire s forward volatility formula: 12

13 ds t dw c 1 E t s t k t 2 ( ) 0 [ ( ) 2 ( ) ] 2c k2 (4) The point here is that if the arbitrage conditions (3) are broken then the forward volatility surface produced by (4) will have spikes and imaginary values. 13

14 Spiky Forward Volatility Surface % 80.00% 60.00% 40.00% 20.00% 0.00% % % % % % SX5E example out to 5y. Complex numbers represented as negative. Everybody knows about it nobody talks about it. 14

15 Arbitrage in the Interpolated Option Prices Spikes in the forward volatility surface is a problem for a number of reasons: - Forward volatility models for exotics will not work. - Forward volatility information is distorted and useless => forward volatility is not used in the same way as forward rates. - Problems in risk management of options and particularly spreads. - The risk of arbitrages in prices shown to clients => wider bid/offers. The problem has been unresolved for nearly 20 years. Personally, I have thinking about an efficient solution for more than 15 years. 15

16 Surprisingly, the solution comes from discrete math and matrix algebra and not from a Stiefel Manifolds, Malliavin calculus, Varadhan s lemma, or general relativity. Maybe that is why it has been overlooked... 16

17 The FD Machine: Constructing Arbitrage Free Option Prices Fully implicit finite difference discretisation of the Dupire equation (4) leads to 1 2 f ( k) 1 [ f ( k k2 k) 2 f ( k) f ( k k)] [1 t ( t, k) 2 ] c( t, k) c( t, k) h kk h 1 h kk (5) This can be represented in matrix form as c t A c t (6) ( 1 h 1 ) ( h ) Equation (6) is a recipe for generating option prices from forward volatilities. 17

18 Here A is the tridiagonal matrix 1 0 a 1 2a a a 1 2a a A [1 t ], a 1 t ( t k ) 2 2 k a 1 2a a n 1 n 1 n , 2 kk i 2 h i The matrix A is diagonally dominant with positive diagonal and nonnegative off-diagonal elements. It is a well-known mathematical result that the inverse has only nonnegative entries 18

19 A 1 0 (8) Using the positivity of the inverse it is straightforward to show that option prices generated by (5) satisfy the no-arbitrage properties kk c( t, k ) 0 (9) h 1 c ( t, k) h 1 ( t) 0 (10)... this leads to our volatility interpolation scheme. 19

20 Volatility Interpolation Scheme Generate option prices by the Dupire FD t t 1 t k c t k c t k t t t c k s k (11) 2 [1 ( ) (, ) 2 ] (, ) (, ),, (0, ) ( (0) ) h h kk h h h 1 It is important to note that for all t h t t h 1 we roll to t from t in one step. h This (of course) includes the next expiry t. h 1 Option prices generated by the finite difference machine are arbitrage consistent, i.e. 20

21 c 2c c tk 2 t (, ) k k (12) 21

22 Forward Volatility Using Volatility Interpolator % 80.00% 60.00% 40.00% 20.00% 0.00% % % % % % Calibrates to approx 100 options in about 0.01 seconds. Yes, not perfectly smooth. But the important point is that there are no spikes or poles. 22

23 Nuclear Submarines In the 50s, USA and CCCP competed in being the first to put a nuclear reactor inside a submarine. One of the main difficulties in doing so is to dimension the cooling system accurately: not too big and not too small. For the related calculations on the heat equation (the Black-Scholes PDE), the Americans used the Crank-Nicolson method....whereas the Russians used the fully implicit method. Like we do. The CN method is superior in terms speed of convergence but its solution exhibits small oscillations. 23

24 The fully implicit method shows slower convergence but it has no oscillations. If you have a good computer you will use Crank-Nicolson, and that is what we have been taught to do in skool. But if your computer is a bunch of class enemies sitting in Siberia,...or if you need to calibrate the parameters of your finite difference grid to observed option prices,... then the stability matters more than (theoretical) speed of convergence and you will use the fully implicit method. 24

25 Wots me Δelδa? It s the most pressing question for any option trader. The quant s standard answer is: That depends on the model -- and I have ten of them, you pick one... Hagan et al (2002) argues that the delta can be fine tuned in stochastic local volatility models by changing the correlation versus the local volatility component when the volatility smile is kept the same. Dupire (2006), however, counters that and argues that close to ATM, the minimum variance delta is virtually invariant to the choice of correlation versus local volatility when the smile is kept the same. 25

26 The minimum variance delta is the position in the underlying stock that (locally) hedges as much variance of the option, i.e. including volatility risk, as possible: c s c v cov[ dv, ds] var[ ds] (13) Here we provide a general proof of Dupire s statement in the context of short maturity expansions. We then show that a model free MV gamma also can be produced, and that the ATM theta as well as behaviour of away from ATM can be linked to the value of options on volatility. 26

27 Short Maturity Expansion Consider the following (very) general class of stochastic volatility model ds ( s, z) dw dz ( s, z) dz dw dz ( s, z) dt (14) Clearly the family of models (14) is rich enough to include several models that fit the same smile. As an example, one can think of the smile generated by a Heston model but fitted by a pure local volatility model. Let () ct be the time t price of a European option on s. Suppose we write the option price as 27

28 c( t) g( t, s( t), v( t )) (15)... where g () is Bachelier s option price formula and v is the implied normal volatility. I.e. g( t, s, v) ( s k) ( x ) v ( x ), x s k v, T t (16) Ito expansion of the option price c g( t, s, v ) yields the following arbitrage condition on the implied volatility 0 ( dx2 dt) 2 Et[ dv] v (17) As 0 we get the condition that x needs to be of unit diffusion, i.e. 28

29 dx 2 2 x s x s x z x z x s k dt 2 1, ( ) 0 (18) This is the short maturity arbitrage condition on the implied volatility v ( s k)/ x. Equation (18) is the so-called Eikonal equation. It is a non-linear first order partial differential equation on the diffusion rather than the linear second order partial differential equations on the drift that we are used to in finance. The tactic from here is normally to guess on a solution x x( s, z ) of the Eikonal equation and then find the implied volatility as v ( s k)/ x. 29

30 The short maturity expansions are (generally) not sufficiently accurate for calibration of dynamic models. But they are generally robust and can be used for market making of an option book and/or for gaining intuition about the qualitative effect of various model parameters. Well-known examples include the so-called SABR model. Rather than deriving new pricing equations, the aim here is come up with general asymptotic results for the Greeks, at and around ATM that are valid for all models hitting the initial smile. 30

31 Minimum Variance Delta The MV delta is the position in the underlying stock that minimises the noise of the portfolio of option and stock: minvar t[ dc ds ] (19) The idea first appeared in papers by Follmer, Schweizer, and Sondermann (80-90s) under the name of locally risk minimizing strategies. The (formal) solution is 31

32 c c g g [ v v ] (20) s z s v s z naive sticky vega delta correction minvardeltaof for vol corr strike delta theimplied vol onlydepend on smile So for a given smile, the minimum variance delta of the option price is given from the minimum variance delta of the implied volatility. In this talk, we will identify the ATM MV delta of the implied volatility from the volatility smile.... and consider what can be said for higher order derivatives. 32

33 Short Maturity Minimum Variance Delta Using the relation x ( s k)/ v we can rewrite the Eikonal equation as an equation (directly) in the implied volatility 0 [ 2( ) 2 2(1 2) ]( ) 2 [2 2( )]( ) [ z ] Dv v k s Dv k s v v (21)...where we have defined the MV operator as D [ ], s z (22) Equation (21) and differentials of this equation is what we will use for derivations of all our results. 33

34 Differentiating (21) wrt k and evaluating at k result for k s: v v v v Dv k s z s yields lead to the following (23) For at-the-money the minimum variance delta of the implied volatility is equal to the slope of the smile.... for any model without jumps. This is more or less the statement of Dupire (2006). 34

35 Relation to Options on Volatility Without loss of generality we can rewrite the model as ds zdw dz dz O( dt) (24)...where Z is a new Brownian motion and ( sz, ) [( ) ( ) 2( ) ( )] s 2 2 1/2 z s z dw dz s z dt ( s, z) dt (25) In that case we have at k s: 35

36 v w 1 w k 2 v 2 v v k 2 v v 2 v k 2 v (26)...where w is the implied volatility of an ATM option on volatility.... in the short maturity sense and if no jumps. Further, one can define the implied correlation between the underlying and its volatility as 2vv w k, k s (27) where w is the implied volatility of an ATM option on the volatility of the stock. 36

37 ATM MV Gamma Applying the operator ( D) to (21) and evaluating at k s yields, k k followed by hardcore (Huge) manipulations now lead to D2v [ ] 2v v, k s kk s z (28) The ATM MV gamma is determined by the ATM slope and curvature of the smile. 37

38 ATM Theta If we assume ATM theta is: is a martingale and use the backward PDE we have that the / c v2( D2c) v2( D ) g, z (29) t v Again, Huge manipulations lead to the result that the ATM theta can be written as v c v g g v v v 2 v g 2v 2 1 2( 2 ) 2[ 3 k 1w t ss sv ] k kk 3 v (30) An ATM theta estimate can be produced from ATM smile slope and curvature combined with information from options on volatility. 38

39 Away from ATM So far we have only considered the ATM case. Using the notation v( k) v( s, z; k ), we can re-arrange equation (21) as Dv( k) vk () 1 { v( s) v( k)[1 (1 ) vz( k) ( k s) ] 1/2 } v() s s k vk () 4 0 (31) As the term under the square-root is positive for all stochastic volatility models and zero for pure local volatility models we can conclude that... For a stochastic volatility model relative to a pure local volatility model, the MV Delta is uniformly higher (lower) for k s (k s). 39

40 An example of MV Delta as function of strike in LV and SLV models: SABR LV v Sabr parameters: ( s, z) 0.1z, 0.5, ( s, z) 3z. s z 1, 1/12. The exact picture depends on the model specification through the term 2(1 2) v ( k ) 2 which depends on the finer structure of (God s) model. z 40

41 Conclusion The Russian finite difference machine is an effective way for interpolation of discrete option price quotes. It allows for the use of forward volatility as a tool of spotting value and mispricing in specific option quotes and volatility smiles. The can also be used for constructing very stable dynamic models for exotic options. See Andreasen and Huge (2011b). The approach can be further refined using expansion as in Andreasen and Huge (2013a) to give parametric control over the extrapolation of the wings and smile dynamics. 41

42 We have produced model free short maturity limits of - ATM minimum variance delta and gamma. If we add in the implied volatility of an ATM option on volatility we can also produce - ATM Theta. MV delta is always higher (lower) for low (high) strikes in stochastic volatility models than in local volatility models. Future research: - Identify trading strategies that lock in the ATM MV Delta. - Quantify the effect of jumps. 42