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  1. 1. Proprietary&Confidential ForUsebyPermissionOnly Guided Tour Inside the Random Walk with Declensions of Black, Scholes & Merton Frederic Siboulet fsiboulet(at)nyu(dot)edu 28March2016 1
  2. 2. Proprietary&Confidential ForUsebyPermissionOnly Objective The objective of this presentation is to compare and contrast some of the theories and techniques that may be used to demonstrate either the Black-Scholes-Merton partial differential equation (BSME), or the Black-Scholes- Merton option pricing formulae (BSMF) or both. Considering the extensive work of many scholars, there are at least 12 methods that can lead to the BSME or the BSMF. We limit our presentation to 5 methods, and refer the reader to specialized text books on the matter if required (see for example Paul Wilmotts Frequently Asked Questions in Quantitative Finance or CQF classes). This work borrows extensively from many research papers, finance classes and other conversations. In particular, it is largely inspired by Riaz Ahmad, Peter Carr, Raphael Douady, Randeep Gug, Seb Leo, Fabio Mercurio, Zari Rachev, Paul Wilmott and many more, all of whom I am grateful to. 28March2016 2
  3. 3. Proprietary&Confidential ForUsebyPermissionOnly TOC 1. Partial Differential Equation 2. Binary Tree and Cox Ross Rubinstein 3.1 Martingales and Probabilities 3.2 Martingales and Change of Numeraire 4. Kolmogorov and Fokker Planck Equations 28March2016 3
  4. 4. Proprietary&Confidential ForUsebyPermissionOnly General Introduction BSM: there are at least 12 methods (see Paul Wilmotts FAQQF) expressed either in Equation (Partial Differential Equation, infinitesimal) or Formulae (Option Pricing, integral) BSM Equation is more general than BSM Formulae For derivatives pricing on traded (Equities, Ccies, Futures) or non-traded underlying assets (Interest Rate, Credit Spread) Generic or specific formulations (i.e. may or may not be easily generalized) In continuous time or discrete time With the Gaussian distribution assumed - or not With or without incremental or friction costs, such as bidask spread, dividend, carry, transaction cost, liquidity premium With constant, deterministic or stochastic parameters (e.g. volatility) Some solutions focus solving for the drift , others for the diffusion . 28March2016 4
  5. 5. Proprietary&Confidential ForUsebyPermissionOnly 1/ Partial Differential Equation: Classic BSM Equation and Formulae 28March2016 5
  6. 6. Proprietary&Confidential ForUsebyPermissionOnly Our Assumptions (see Black and Scholes, or Markowitz) Short selling, Fractional trading always possible No market friction (such as BA spread, transaction fee, dividend, tax, illiquidity) Continuous trading and dynamic hedging assumed At this stage Constant rate r for the RFA At=ert Constant stock return Constant volatility The only derivative payoff is at the horizon T (V(T), Euro- style) with no path dependency Underlying stock asset (S) follows a GBM and assumed to be traded. In that complete market, any such contingent claim Vt (i.e. derivative) can be replicated @t with a replicating portfolio of 1.Stock (St) & 2.Risk Free Asset (At). 28March2016 6 1/ Partial Differential Equation
  7. 7. Proprietary&Confidential ForUsebyPermissionOnly 28March2016 7 1/ Partial Differential Equation The risky term (not to be confused with credit risk) of the PDE is the Brownian Motion (BM), aka Weiner Process (WP). A WP is a Random Process (RP), aka a time dependent Random Variable (RV) iff it has the four following properties: [ An alternative definition, the Levy Characterization, states that a RP is a WP iff it is an almost surely continuous martingale, with zero origin and quadratic variation [W]t =t ] is continuous 2* 2* (1) X(0) 0 =1 Xis continuous in time (2) / w/proba one , , 0, (3) , 0, X's Variance is the time interval , (4) , independent ' 0, t X t t T t s X N t s t s X t s X t t t Markovian of ' property X t
  8. 8. Proprietary&Confidential ForUsebyPermissionOnly 1/ Partial Differential Equation 1. Assume a Stock S following a GBM S is the time dependent RV and the underlying of the derivative, where X is the WP: 2. Construct the portfolio , long 1xOption and short x stocks tbd: 3. Move forward in time, from t to t+t, assuming constant: 4. It Lemma (or Taylor expansion order o(dt), with S2~t~dt): 28March2016 8 , , ,S t V S t S S t , ,t t t tS t V S t S Sdt dXdS S 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 dt 1 1 d 2 t 2 t 1 d 2 2 Tay d lo tX r d o V V dt dt dS dV dS o dS V dS t S dS S V dS S dt S dt V V dt t S V S V d t t dt t S S S dX o
  9. 9. Proprietary&Confidential ForUsebyPermissionOnly 1/ Partial Differential Equation 5. Neglecting the high order terms (HOT), d is s.t. (w/=cte): 6. Now, pick so that is risk-free over dt (dynamic hedging argument). There should be no arbitrage, therefore the return of that portfolio over dt should be that of the RFR: 7. And therefore over dt, (5)=(6) gives: 28March2016 9 d r dt r V S dt 3 2 2 2 2 2 2 dV d dV dS O V d V S S dS S V dt d S tt t risk free, determini 2 2 2 2 2 2 2 2 2 2 2 ti 2 s 2 2 2 dS dS V S V dt dt r V S dt t S V S V dt Sdt dt Sdt r V S dt t S V S V r V S S S dt t S SdX SdX V d S S V S S dS V risky, random termcterm 0 V S dX S
  10. 10. Proprietary&Confidential ForUsebyPermissionOnly 1/ Partial Differential Equation 8. According to the risk-free argument the random term is null, eliminating the random term dX; therefore write: 9. Plug the new-found value of into the deterministic term: 10. Which gives the Black Scholes Merton Equation: 28March2016 10 2 2 2 2 risk discount free termdiffusion drift BSME 2 V S V V rS rV t S S : 0 0 SdX S dX V V S S 2 2 2 2 0 2 0 V S V V r V S S S t S S V V S S
  11. 11. Proprietary&Confidential ForUsebyPermissionOnly To go from the BSM Equation to the BSM Formulae for option pricing, solve the BSM PDE: 11. Formulate a function U which is the forward value @T of the present value @t of the derivative V(S,t): 12. Derive V by t and by S and express V as a function of U: 13. Replace V by its expression of U in the LHS of the BSME : 28March2016 11 1/ Partial Differential Equation "numeraire" @T "numeraire" @t simplified notation w/o T , , , ,r T t r T t TU S t e V S t V S t e U S t 22 2 2 , , , , , , r T t r T t r T t r T t r T t U S tV re U S t e t t V S t e U S t U S t U S tV V e e S S S S 2 2 2, , 2 2 2 2 2 BSME 2 Backward Kolmogorov Equation BSME 0 BKE identical to BSME w/o discount2 r T t V S t e U S t r T t r T t rV U S U U re U e rS rV t S S U S U U rS t S S
  12. 12. Proprietary&Confidential ForUsebyPermissionOnly 14. First change of variable, on time t: 15. With replacing t, the BKE becomes the FKE (or Forward Kolmogorov Equation, aka Fokker Planck Equation): 16. Second change of variable, on the stock S: 28March2016 12 1/ Partial Differential Equation chain rule limitsvariable . . . ; 0 parameter t T t tt T T t 2 2 2 2 Notice the sign of BSME 0 FKE(1) the time derivation2 T t U S U U rS S S chain rule first order 2 2 variable 2 2 2 2second order . . 1 . ln . 1 . 1 . 1 . 1 . . S S S S S S S S S S S S
  13. 13. Proprietary&Confidential ForUsebyPermissionOnly 17. With replacing S, the FKE(1) becomes FKE(2): 18. Third change of variables: 28March2016 13 1/ Partial Differential Equation 2 2 2 2 2ln 2 2 2 2 2 2 1 1 U BSME 0 2 Notice the change to theU BSME 0 FKE(2) drift (& diffusion) terms2 2 S T t U UU t S S U S U U rS S S U U r 2 2 2 2 2 2 2 2 . . . . . 2 , . . . .2 . . . . x r x x x r x x x x x x
  14. 14. Proprietary&Confidential ForUsebyPermissionOnly 19. Replace the old with the new variables into FKE(2) to get W: 20. Therefore finally: with: 28March2016 14 2 2 2 2 2 2 2, ,t 2 U , 2 0 2 2 2 FKE(2) W x U S UU x r W W W W r r x x x 2 2 2 2 2 2, , 2 2 ln 2(classic) Black Scholes Merton Equation Heat Equation BKE FKE 0 2 2 rt V S t e W x x S r T t V S V V W W rS rV t S S x 1/ Partial Differential Equation 2 2 , , derivative's value @t correspondence of variables r x r V S t e W x S e t T
  15. 15. Proprietary&Confidential ForUsebyPermissionOnly To value the derivative V(S,t) is equivalent to solve the Heat Equation. Therefore with the fundamental solution, seek the pdf of the Gaussian RP W for the alternate set of variables x and . 21. May solve the heat equation by similarity reduction, i.e. transform the PDE (two variables) by an ODE (single variable) by the appropriate change of variable 22. For that new variable z, assume , and y three parameters tbd and write: 23. Derive the variable z once wrt and once wrt x: 28March2016 15 1/ Partial Differential Equation 3 , , ,y / , ]0, ] W x f z x T x y z 1 1 z x y x y z z x
  16. 16. Proprietary&Confidential ForUsebyPermissionOnly 24. Derive the function W once wrt and twice wrt x: 28March2016 16 1 1 1 1 1 2 2 2 ' , ' ' 1 ' '' '' f zW z f z z x y f z f z W x f z zf z f z x y z f zW z x z xz x y f z z f zW z x x z x z f z x f z