Valuation of Equity Derivatives

Transcription

1 Valuation of Equity Derivatives Dr Mark Beinker TRONGnet school Bielefeld, 1./. June 011 1

2 What s a derivative? More complex financial products are derived from simpler products What s not a derivative? tocks, interest rates, FX rates, oil prices, Derivatives are pay off claims somehow based on prices of simpler products or other derivatives Derivatives may be traded via an exchange or directly between two counterparties (OTC: over-the-counter) OTC-Derivatives are based on freely defined agreements between counterparties and may be arbitrarily complex

3 Example I: Equity Forward Buying (or selling) stocks at some future date Pay off T at maturity (expiry) T Pay off Long position: you bought the stock (Your counterparty is short) Physical ettlment: get tock, pay Cash ettlment: get T Forward Price or trike Price T tock Price T at maturity y( (expiry) T 3

4 Example II: Plain Vanilla Option Most simple and liquidly traded options (Plain Vanilla) Call Option (Plain Vanilla) Put Option Pay off Pay off Pay off: T Pay off: max, 0 max, 0 T 0 T T T T 4

5 Example III: Bonus Certificate Getting more than you might expect Full protection against minor losses Pay off Zero strike call Pay off H T = + Everywhere at or above stock price, but still could be sold at current stock price level or below Pa ay off Down and out put H T If Protection level H is hit once before expiry, protection gets lost. Put becomes worthless, if barrier level H is hit once before expiry. H T 5

6 Even more complex structures There are no limits to complexity Baskets as underlying imple basket products: Pay off depends on total value of basket of stocks Correlation basket products: Pay off depends on performance of single stocks within the baskets, e.g. the stock that performs worst or best, etc. imulation of trading strategies t Quantos Pay off in a currency different from the stock currency Combination with other risk factors (hybrid derivatives) E.g. Convertible Bonds (bonds that could be converted into stocks) 6

7 tocks now your underlying A tock or hare is the certification of the ownership of a part of a company, the community of shareholders h is the owner of the whole company Issued stock is equivalent to tier 1 equity capital of the corporation The stock may pay a dividend The company can be listed at one or more stock exchanges 7

8 tock process The Geometric Brownian motion of stocks Drift Volatility tandard normal distributed random number d dt dw dw dt d ln dt dw Divends are neglected! 8

9 Time value of money Time is money. But how much money is it? Money today is worth more than the same amount in some distant future Risk of default Missing earned (risk free) interest Cashflow Y at future time T is worth now Y/(1+rT) Discounting Interest earned over time period T: XrT + =X =Y t 9

10 A note on notation I Compounding of interest rates Usually, interest is paid on a regular basis, e.g. monthly, quarterly or annually If re-invested, the compounding effect is significant Number of years Compoundings per year r r r m m m nm times r m nm 10

11 A note on notation II The Continuous compounding limit Continuous compounding is the limit of compounding in infinitesimal short time periods nm r rt lim 1 e, m m nconst. T n e rt Value of one unit at future time T as of today t=0. Also: discount factor or zero bond r: continuous compounding rate This mathematically convenient notation is used throughout the rest of the talk. It is also most often assumed in papers on finance. 11

12 Example I: valuation of the Forward contract First Try: Forward value based on expectation 1. tep: Calculate expectation of forward pay off E T E e T T 0. tep: Discount expected pay off to today V E Forward T e 0 e r rt Fair price for new contract: e TT 0 To avoid losses, bank would have either 1. sell as many Forwards with same strike and maturity as they buy. or have to rely on the correctness of the above formula on average 1

13 Arbitrage Making money out of nothing Arbitrage is the art of earning money (immediately) without taking risk Example: Buy stock at 10 and sell at 15 Because of bid/ask-spreads, spreads broker buys at 10 and sells at in this example is the broker fee ince money earned by arbitrage is easy money, market participants p will take immediate advantage of arbitrage opportunities If the markets are efficient, there are no opportunities for arbitrage If you can replicate a pay off with a strategy built on liquidly traded instruments, this must be the fair value of the pay off > there is no free lunch! 13

14 Example I: valuation of the Forward contract I Replicating the Forward agreement Assumption: 0 e rt 1. tep: Borrow at interest rate r for term T the money amount rt B e. tep: Buy the stock and put the rest of the money aside: rt A e 3. tep: At time T, loan has compounded to : e rt B 4. tep: Exchange stock with strike and pay back loan 0 Amount A has been earned arbitrage free! To avoid arbitrage, the fair strike must be e rt 0 14

15 Example I: valuation of the Forward contract II Lessons learned V Forward 0 e rt The real world expectation of at future time t doesn t matter at all! Hedged counterparties face no market risk Credit risk remains Value of the Forward is equal to the financing cost No fee for bearing market risk Required assumptions: No arbitrage Possible to get loan at risk-free interest rate 15

16 Adding optionality For options, the distribution function matters Plain Vanilla option: cut off distribution function at strike Pay off European Call option pay off: max, T 0 T T Q: Is there any arbitrage free replication strategy to finance these pay offs? T 16

17 Ito s lemma The stochastic process of a function of a stochastic process Process of underlying: Fair value V of option is function of : Ito s lemma: dv V V 1 V t d dt dw V V () V dt dw Caused by stochastic term ~ dt 17

18 Replication portfolio for general claims Replicate option pay off by holding portfolio of cash account and stock Ansatz: V B x with db rbdt Changes in option fair value V dv V V t 1 V dt V dw rbdt x dt x dw V Choose x and insert for B V x V t 1 V rb rv r V With this choice of x, the stochastic term vanishes 18

19 A closer look at the Black-choles PDE The arbitrage free PDE of general claims rv V t r V 1 V Final equation does not depend on µ Replication portfolio is self-financing Additional terms for deterministic dividends have been neglected With time dependent r=r(t) or σ=σ(t), the PDE is still valid 19

20 olving the Black-choles PDE Numerical methods for pricing derivatives Trees Analytic olutions V C rt N( d ) e N ( d ) ( 1 d emi-analytic olutions D( t p ) CM V floor (0) 1 f ( ) P( ) P( x) f ( x) dx L 0? Whichh Method? Finite Difference Monte Carlo Finite Elements 0

21 The famous formula of Black and choles Analytic olution of Black-choles PDE for Call options olve the Black-choles PDE for Plain Vanilla Call options rv V t V r pecific products can be defined as set of end and boundary conditions of the Black-choles PDE 1 V V ( T, ) V ( t,0) 0 limv ( t, ) ( T ) olution: the famous result of Black and choles V d Call 1, N( d ) e 1 rt N( d l( ln( / ) rt 1 T T ) 1

22 Binomial tree single step imulating the stochastic process on a tree like grid Probability for up movement p u tock price at origin tock value after up/down movement 1pp dt d

23 Binomial tree parameters Four parameters and two equations Parameters:, : u d p: dt : Value of stock after up/down move Probability of up move Length of time step (determines the accuracy of the calculation) Equations: E var ( dt ) p (1 p ) u e rdt dt ( dt) p(1 p) e e 1 u d d rdt Derived from lognormal stock price process 3

24 Binomial tree choices Popular choices to determine parameters Cox-Ross-Rubinstein choice: u u and d / u u p e dt rdt ue 1 u 1 Jarrow-Rudd-choice: p 0. 5 u d r 1 dt e dt r 1 dt e dt 4

25 Binomial trees application Applied decision i trees V uu V u Calculate option price by backward induction V pv ( 1 p ) p V u V d pv uu (1 p) V ud (1 p ) pv (1 p ) V ud V dd V d V ud V dd Determined by option pay off function 5

26 Trinomial trees Two-step binomial tree or explicit finite difference scheme ey features of trinomial trees 6 free parameters (three different states, two probabilities, one time step) Otherwise, approach is similar to binomial i trees More flexibility than binomial tree Faster convergence (two time steps in one) For certain geometries, trinomial trees are identical to explicit finite difference method 6

27 Finite differences olving the PDE on a rectangular grid Application of finite difference method 1. Choose one of the well established finite it difference schemes for PDEs of the parabolic type (diffusion equation), e.g. explicit or implicit Euler scheme or Crank Nicholson scheme. Discretise PDE on a rectangular lattice according to finite difference scheme 3. Apply pay off function as final boundary condition 4. Depending on option type, apply Dirichlet or (generalised) von Neumann (e.g. second order derivative is zero) conditions on upper and lower boundaries 5. Roll back throw lattice to get solution 7

28 Comparison of numerical methods Instead of trees, use finite difference method! Binomial tree Option price Number of time steps Trinomial tree Finite differences Analytical 8

29 Monte Carlo simulation The method of last means If everything else false, use Monte Carlo simualtion Easy to implement, but bad performance Typical applications: Underlying is basket of underlying (e.g. many dimension) Multi-factor problems Path dependent problems Implementation methods imulate stochastic differential using small time steps Better: Integrate of longer time period and draw random numbers directly from log-normal distribution function Calculate pay off based on simulated stock prices and discount to today 9

30 Assumptions Which of these assumptions hold in reality? There are no transaction costs No, there are bid-ask preads Continuous trading is possible Markets have infinite liquidity Dividends id d are deterministic i ti Markets are arbitrage free Volatility has only termstructure tocks follow log-normal Brownian motion There is no counterparty risk No, due to technical limitations No, problem for small caps No, company performance dependent d Almost, because of transaction costs No, volatility depends on strike and term No, only approximately, problems of fat tails No, as the last crisis has shown Everybody can finance at risk-free rate No, financing depends varies broadly 30

31 Volatility smile Volitility depends on strike ( moneyness ) and expiry mile vanishes for long expiries tility Volat At-the-money: pot equals trike Using Black-choles: putting the wrong number (i.e. volatility) into the wrong formula to get the right price. 31

32 Local volatility surface Transform σ(,t) into σ(,t) When moves, local volatility moves in wrong direction tility Volat tructure is much more complex Local Volatility: Allows for fit to the whole volatility surface, but behaves badly. till, it is widely used. 3

33 Examples of other methods of modelling volatility I More advanced volatility models Displaced diffusion Assume +d instead of to follow the lognormal process Jumps d ( d) r( d) dt ( d) dw Add additional stochastic Poisson process to spot process d ( r ) dt dw dj Poisson process Rate of poisson process or jump intensity Relative jump size 33

34 Other methods of modelling volatility II More advanced volatility models tochastic Volatility Model volatility as second stochastic factors, e.g. Heston model Instantaneous variance Wiener processes with correlation ρ d rdt dw d ( ) dt dw Mean reversion rate Long term average variance Volatility of volatility Local-toch-Vol Combination of local volatility and stochastic volatility 34

35 Dividends I Dividend payments: interest equivalent for equities Dividends compensate the shareholder for providing money (equity) ome companies don t pay dividends, most pay annually, some even more often In general, dividend is paid a few days after the (annual) shareholder meeting Dividend id d payment amount is loosely l related to the company s P&L Regardless of the above, most models assume deterministic dividends i.e. Dividend amount or rate and payment date are known and fixed 35

36 Dividends II Three methods of modeling dividends Continuous dividend yield q: continuous payment of dividend payment proportional to current stock price Unrealistic, but mathematically easy to handle Discrete proportional dividends: dividend is paid at dividend payment date, amount of dividend is proportional to stock price Tries to model dividend s loose dependency on P&L (assuming P&L and stock price to be strongly correlated) Discrete fixed dividends: fixed dividend amount is paid at dividend payment date Causes headaches for the quant (i.e. the person in charge of modelling the fair value of the derivative) 36

37 Dividends III Impact of dividends on stock process I Continuous dividend yield: d Discrete dividends qdt dw 1. Method: ubtract dividend value from and model without dividends proportional dividends fixed dividends * 1 n D i i1 * n i1 e rt i Di 37

38 Dividends III Impact of dividends on stock process II Discrete dividends. Method: Modelling (deterministic) jumps in the stochastic process with jump conditions defined as Proportional dividend Fixed dividend t t 1 D i t i ti Di i i Methods 1 and assume different stochastic processes, i.e. the volatilities are different! 38

39 Example Ia: Forward contract with dividends Replicating the Forward agreement 1. tep: Borrow the money to buy the stock at price 0. tep: plit loan into two parts a) For time period t D at rate r D the amount b) For time period T at rate r the rest 0 e r D t D e D r D t 3. tep: At dividend payment date t D, receive dividend D and pay back first loan which is now worth D D D 4. tep: At expiry, the second loan amounts to rt r e e D t D D 0 In order to make the Forward contract be arbitrage free, the fair strike must rt r D e e D t D 0 be 39

40 Example Ia: Forward Contract with dividends Formulas for Forward agreements with dividends Continuous dividend yield Fair strike: Fair value: e rq T 0 qtt rtt e 0 e Proportional discrete dividends Fair strike: rt e 1 i Fair value: VForward 0 1 i V Forward n 1 n 1 Di Di e rt 40

41 Example Ia: Forward Contract with dividends Formulas for Forward agreements with dividends Fixed discrete dividends Fair strike: Fair value: e rt 0 n i1 e rt i D i V Forward 0 n i1 e D e rt i rt i Dividends reduce the forward value by reducing the financing cost of the replication strategy! 41

42 Impact of dividends on derivative prices Currently, no well-established established model for stochastic dividends Dividends can be hedged, but not (easily) modelled d stochastically Impact on option price is significant Trick: use dividends to hide option costs Pay off H T Everywhere at or above stock price, but still could be sold at current stock price level since between today and T lies a stream of dividend payments! 4

43 Hedging derivative in practice Hedging makes the difference Hedging: trading the replication portfolio Reduce transaction cost by Frequent, but discrete hedging Hedging derivative portfolio as a whole Improve Hedging performance Hedging g based on Greeks 43

44 Greeks Partial derivatives named by Greek letters V Delta V Most important Greeks V Gamma V V Theta Bucketed tt V ensitivities V Rho rr Vega Volga Vanna Type of hedging strategy often named after Greeks hedged, e.g. Delta-Hedging 44

45 Example IIIb: down-and-out Put Present Value of Down-and-out out Put option Pay off Fair value increases Down-and-out Put H T Fair value falls to zero Put Option 45

46 Example IIIb: down-and-out Put Delta of Down-and-out out Put option Delta Down-and-out Put Delta Put Option 46

47 Example IIIb: down-and-out Put Gamma of Down-and-out out Put option Gamma Down-and-out Put Gamma Put Option 47

48 Example IIIb: down-and-out Put Vega of Down-and-out out Put option Vega Down-and-out Put Vega Put Option 48

49 Counterparty default risk Calculation of the Credit Value Adjustment (CVA) Hazard rate: Probability of default in time intervall dt Positve part of derivative pay off V D V T 1 R he ht e rt E V ( t ) dt 0 Defaultable Fair Value Recovery Rate urvival probability until time t Counterparty risk can be traded by means of Credit Default waps paying periodically a premium for insurance against default of specific counterparty 49

50 Forward fair value with CVA Counterparty risk adds option feature Forward with CVA: V D Forward e rt (1 R) 1 e ht V Call e rt (1 R) ht rt 1 e N( d ) e N( d ) 1 Call option with CVA: pecial case R=0: V D Call V 1 R) V 1 e Call ( Call ht V D Call e ht rt N( d ) e N ( d ( 1 ) 50

51 CVA for the bank s derivatives portfolio Effect of Netting Agreements and Collateral Management Tr rades Before Netting After Netting Tr rades Netting of Exposures Netting ets Relevant for CVA hort Long Collateral 51

52 ummary What you might have learnt What s a derivative Typical equity derivatives and how they work General idea behind the method of arbitrage-free pricing The assumptions of the theory and their validity in reality What s missing in the theoretical framework Impact of counterparty default risk 5

53 Literature Financial Calculus : An Introduction to Derivative Pricing, Martin Baxter & Andrew Rennie, Cambridge University i Press, 1996 Options, Futures & Other Derivatives, John C. Hull, 7 th edition, Prentice Hall India, 008 Derivatives and Internal Models, Hans-Peter Deutsch, 4 th edition, Palgrave McMillan, 009 Derivate und interne Modelle: Modernes Risikomanagement, Hans-Peter Deutsch, 4. Auflage, chäffer-poeschel,

54 Your Contact Dr. Mark W. Beinker Partner d-fine Frankfurt München London Hong ong Zürich Zentrale d-fine GmbH Opernplatz Frankfurt am Main Deutschland T F: d-fine All rights reserved 54

55 Glossar Frequently used expressions r σ T μ V(t) Underlying asset/stock price (default) risk free interest t rate Volatility trike price Expiry Date Drift of underlying Value of derivative at time t tock price process Risk neutral stock price process Pay off function of call/put option d d dt dw rdt dw ll/ t ti max T, 0 T Discount factor = value of zero bond Black-choles equation hort position rv V t e r Position sold V r T t 1 V dw tochastic differential ~ dt Long position Position bought 55