Page 1. Real Options for Engineering Systems. Today s plan. Mathematical models of uncertain dynamics: Stochastic processes

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1 Real Opions for Engineering Sysems Session 9: Inroducion o he modeling of randomly changing values Sefan Scholes Judge Insiue of Managemen, CU Slide 1 Today s plan Background: In order o use financial opions valuaion echniques you need o model he random behaviour of he underlying value processes, such as sock price oil/gold/copper/zinc price value of cash flows of a passive projec Today you will ge a brief inro o he modelling of random processes This will inroduce you o some of he echnical lingo of he opions world (e.g. Brownian moion) I will also show how risk-neural valuaion can be done (in he European opions seing) wih Mone Carlo simulaion Sefan Scholes Judge Insiue of Managemen, CU Slide 2 Mahemaical models of uncerain dynamics: Sochasic processes Deerminisic dynamic process x : Number (or vecor) ha changes wih ime Sochasic dynamic process X : Random variable whose disribuion changes wih ime Time domains: discree =0,1,2,3, Coninuous [ 0, T] or [0, ) Discreizaion of coninuous ime domain Observe he process a imes = 0,, 2,3,... Sefan Scholes Judge Insiue of Managemen, CU Slide 3 Page 1

2 Example: Random walk Sar wih X 0 =0, fix period lengh and move according o X + = X + ε where he ε s are independen sandard normal variables (mean 0, sd 1) Le s look a a simulaion (Random walk.xls) Properies Sar a 0 Incremens over m periods X +m -X are normal wih mean 0 and variance m Incremens over non-overlapping ime inervals are uncorrelaed Sefan Scholes Judge Insiue of Managemen, CU Slide 4 Brownian moion BM is he limiing process of a random walk as goes o zero Properies are he same as for random walks Sar a 0 Incremens z T -z are normal wih mean 0 and variance (T- Incremens of non-overlapping ime periods are uncorrelaed BM is also called a Wiener process To simulae i, discreise ime and simulae a random walk (for small Sefan Scholes Judge Insiue of Managemen, CU Slide 5 A shor-hand for he incremen BM incremen a ime : z+ z = ε( Shor-hand for BM incremen for (infiniesimally) small : dz ( = ε ( Here, d denoes a (infiniesimally) small ime incremen d Sefan Scholes Judge Insiue of Managemen, CU Slide 6 Page 2

3 Modelling dynamics hrough incremens: ODE Ordinary differenial equaions (ODE) Describe a process via is incremen d during small ime incremen d d + d Example: each member of a populaion produces a offspring over a uni ime inerval dx ( = ax ( d Solving ODE means finding he form of he process (a funcion of ime Can someimes find closed-form soluion O/w discreize he ODE and simulae he process ( numerical inegraion of ODE) Sefan Scholes Judge Insiue of Managemen, CU Slide 7 Modelling dynamics hrough incremens: SDE Sochasic differenial equaions (SDE) Describe a process via is incremen d during small ime incremen d, including a random componen in he descripion Brownian E.g. a generalized Wiener process dx ( = ad + bdz( moion incremen Solve SDE o find he soluion (a random process) Can someimes find closed form soluion O/w discreise he SDE and simulae he process (Mone Carlo) Sefan Scholes Judge Insiue of Managemen, CU Slide 8 Inerpreing a coninuous random process Example: a so-called Io process is of he form dx ( = a(, d + b(, dz( dz( is Brownian moion incremen Discree form of an Io process ε are independen sandard normals (mean = 0, variance = 1) + = a(, + b(, ε d ) d dz( This is a random walk (discreizaion of BM) Sefan Scholes Judge Insiue of Managemen, CU Slide 9 Page 3

4 Mean-reversion propery Some quaniies, e.g. oil prices or ineres raes, are believed o have a naural home abou which hey vary randomly Oil: marginal cos of producion (compeiion argumen Example of a mean reversion model: dx ( ) = ( µ x ( )) d + σ dz If µ> hen expeced change in x is µ ->0 and vice versa Srong endancy o rever o mean if far off. Sefan Scholes Judge Insiue of Managemen, CU Slide 10 Modelling sock price behaviour Assumpion: Log-reurns on sock prices S behave like a generalized random walk S + ln( ) = ν + σ ε S uncerain y driver is a random walk The drif ν corresponds o he expeced increase per uni ime The volailiy σ changes he size of he jumps of he random walk Assumpion goes back o Louis Bachelier, The heory of speculaion, PhD hesis 1900 Noice: assumes ha hisoric price pah gives no informaion abou fuure reurns Wha abou echnical rading? Sefan Scholes Judge Insiue of Managemen, CU Slide 11 Coninuous ime descripion Noice ha S + ln( ) = ln( S + ) ln( S ) = ν + σε S incremen of ln( S ) Coninuous ime version (i.e. period lengh 0) d ln S ( ) = ν d + σdz ( ) Hence ln() is a generalized Wiener process Sefan Scholes Judge Insiue of Managemen, CU Slide 12 Page 4

5 Geomeric Brownian Moion Descripion of ln(): d ln S ( = ν d + σdz Can we find a SDE descripion of (change of variables)? Io s Lemma is a change of variables formula for sochasic differenial equaions Given a SDE for and suppose y(=f(,, wha is he SDE for y(? See e.g. Luenberger p. 312,313 If Io s Lemma is applied, he above formula for ln() resuls in he following formula for : ds ( ) 2 = ( ν + σ ) d + σ dz S ( ) 2 µ d/ is called he insananeous reurn of he sock µ is he mean reurn per ime period, while ν is he mean logreurn per ime period Such a process is called a geomeric Brownian moion Sefan Scholes Judge Insiue of Managemen, CU Slide 13 Simulaing geomeric Brownian moion Fix a period lengh and sep he process forward period by period Form 1: d ln S ( = ν d + σdz Discreize: ln( + ) ln( ) = ν + σε Form 2: ds ( ) 2 = ( ν + σ ) d + σdz S ( ) 2 µ + S ( Discreize: = µ + σε The wo coninuous processes are he same (only differen descripions), bu he discreizaions are differen Does no play a role in pracice since differences in discreizaions are small and end o cancel ou in he long run Sefan Scholes Judge Insiue of Managemen, CU Slide 14 Summary A random walk is defined by he incremen X + X = ε Random walk is a useful building block for modelling random value processes (e.g. sock prices) An ofen used model is he Io process which is of he discreized form + = a(, + b(, ε d ) dz( Special case: geomeric Brownian moion d + = µ + σ ε a( S (, Sefan Scholes Judge Insiue of Managemen, CU Slide 15 b( S (, Page 5

6 Risk-neural valuaion of opions revisied How can we price a European call opion on an underlying sock ha follows a geomeric Brownian moion? Firs rial: Price of opion = E[max{0,S T -K}]e -δt Can be found by simulaing he sock price S T (see Opion Pricing I.xls) Wha discoun rae δ should be used? This approach is essenially equivalen o maching expeced reurns in he gambling seing of Session 4 Under wha condiions is his equivalen o Black-Scholes? This is equivalen Mahemaical resul: Black-Scholes formula is equivalen o o changing he pricing he sock a is discouned expeced reurn, provided probabiliies in a laice o he riskneural ones mean µ of sock reurns replaced by he risk-free ineres rae r discouning is done a risk-free rae r This is called risk-neural pricing see Luenberger p.357 for a mahemaical explanaion and p.344 for he relaion o risk-neural pricing in a laice Sefan Scholes Judge Insiue of Managemen, CU Slide 16 Simulaion Coninuous risk-neural process ds ( ) = rd + σ dz d S ( ) Sock price a ime (discreizaion): 0) = 0)(1 + r + σε See spreadshee Opion Pricing II.xls ε is a sandard normal variable Sefan Scholes Judge Insiue of Managemen, CU Slide 17 Conclusions MCS is useful for opions analysis Have seen ha sock price movemens, modelled as geomeric Brownian moion, can be easily simulaed Same is rue for oher processes (e.g. mean revering processes) Have seen he analogue of risk-neural valuaion in a laice for geomeric Brownian moion MCS has limiaions in valuing flexibiliy (as in American opions or wih mos real opions) If used in conjuncion wih decision rules i will give conservaive value esimaes as he decisions are no necessarily opimal Alernaive is he laice model (so-called dynamic programming ) Sefan Scholes Judge Insiue of Managemen, CU Slide 18 Page 6

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random