Real Options for Engineering Systems

Session 9: Introduction to the modeling of randomly changing values

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 1

Today’s plan Background: In order to use financial options valuation techniques you need to model the random behaviour of the “underlying” value processes, such as stock price oil/gold/copper/zinc price value of cash flows of a passive project

Today you will get a brief intro to the modelling of random processes This will introduce you to some of the technical lingo of the options world (e.g. Brownian motion) I will also show how risk-neutral valuation can be done (in the European options setting) with Monte Carlo simulation

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 2

Mathematical models of uncertain dynamics: Stochastic processes Deterministic dynamic process xt: Number (or vector) that changes with time t

Stochastic dynamic process Xt: Random variable whose distribution changes with time t

Time domains: discrete t=0,1,2,3,… Continuous t ∈[0,T] or t ∈[0, ∞)

Discretization of continuous time domain Observe the process at times t = 0 , ∆ t , 2 ∆ t , 3 ∆ t ,...

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 3

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Example: Random walk Start with X0=0, fix period length ∆t and move according to

X t + ∆t = X t + ε t ∆t where the εt’s are independent standard normal variables (mean 0, std 1)

Let’s look at a simulation (Random walk.xls) Properties Start at 0 Increments over m periods Xt+m∆∆t-Xt are normal with mean 0 and variance m∆ ∆t Increments over non-overlapping time intervals are uncorrelated Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 4

Brownian motion BM is the limiting process of a random walk as ∆t goes to zero Properties are the same as for random walks Start at 0 Increments zT-zt are normal with mean 0 and variance (T-t) Increments of non-overlapping time periods are uncorrelated

BM is also called a Wiener process To simulate it, discretise time and simulate a random walk (for small ∆t)

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 5

A short-hand for the increment BM increment at time t: z t + ∆t − z t = ε ( t ) ∆ t

Short-hand for BM increment for (infinitesimally) small ∆t: dz ( t ) = ε ( t ) dt Here, dt denotes a (infinitesimally) small time increment ∆t Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 6

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Modelling dynamics through increments: ODE Ordinary differential equations (ODE) Describe a process via its increment dx(t) during small time increment dt

dx(t ) ≈ x(t + dt ) − x(t ) Example: each member of a population produces a offspring over a unit time interval

dx ( t ) = ax ( t ) dt

Solving ODE means finding the form of the process x(t) (a function of time t) Can sometimes find closed-form solution O/w discretize the ODE and simulate the process (“numerical integration of ODE) Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 7

Modelling dynamics through increments: SDE Stochastic differential equations (SDE) Describe a process via its increment dx(t) during small time increment dt, including a random component in the description Brownian E.g. a generalized Wiener process motion increment

dx(t ) = adt + bdz(t )

Solve SDE to find the solution x(t) (a random process) Can sometimes find closed form solution O/w discretise the SDE and simulate the process (Monte Carlo) Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 8

Interpreting a continuous random process

Example: a so-called “Ito process” is of the form

dx(t ) = a ( x(t ), t ) dt + b( x(t ), t )dz (t )

dz(t) is Brownian motion increment

εt are independent standard normals (mean = 0, variance = 1)

Discrete form of an Ito process x (t + ∆ t ) − x (t ) = a ( x (t ), t ) ∆t + b ( x (t ), t )ε t ∆ t ≈ dx ( t )

≈ dt

≈ dz ( t )

This is a random walk (discretization of BM)

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 9

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Mean-reversion property Some quantities, e.g. oil prices or interest rates, are believed to have a “natural home” about which they vary randomly Oil: marginal cost of production (competition argument)

Example of a mean reversion model:

dx ( t ) = ( µ − x ( t )) dt + σ dz If µ>x(t) then expected change in x is µ -x(t)>0 and vice versa Strong tendancy to revert to mean if far off.

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 10

Modelling stock price behaviour Assumption: Log-returns on stock prices S behave like a generalized random walk ln(

S t + ∆t ) = ν∆ t + σ ε t St

∆t

uncertaint y driver is a random walk

The drift ν corresponds to the expected increase per unit time The volatility σ changes the size of the jumps of the random walk

Assumption goes back to Louis Bachelier, “The theory of speculation”, PhD thesis 1900 Notice: assumes that historic price path gives no information about future returns What about technical trading? Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 11

Continuous time description Notice that ln(

S t + ∆t ) = ln( S t + ∆t ) − ln( S t ) = ν∆ t + σε t ∆t St increment of ln( S t )

Continuous time version (i.e. period length

0)

d ln S ( t ) = ν dt + σ dz ( t )

Hence ln(S(t)) is a generalized Wiener process

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 12

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Geometric Brownian Motion Description of ln(S(t)): d ln S (t ) = ν dt + σ dz Can we find a SDE description of S(t) (change of variables)? Ito’s Lemma is a “change of variables” formula for stochastic differential equations Given a SDE for x(t) and suppose y(t)=F(x(t),t), what is the SDE for y(t)? See e.g. Luenberger p. 312,313

If Ito’s Lemma is applied, the above formula for ln(S(t)) results in the following formula for S(t): dS ( t ) = (ν + σ S (t )

2

2

) dt + σ dz

µ

dS(t)/S(t) is called the instantaneous return of the stock µ is the mean return per time period, while ν is the mean logreturn per time period Such a process is called a geometric Brownian motion Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 13

Simulating geometric Brownian motion Fix a period length and step the process forward period by period Form 1: d ln S ( t ) = ν dt + σ dz Discretize: ln(S(t + ∆t )) − ln(S (t )) = ν∆t + σεt ∆t

Form 2:

2 dS ( t ) = (ν + σ ) dt + σ dz 2 S (t )

µ

Discretize: S (t + ∆t ) − S (t ) = µ ∆t + σε t ∆t S (t )

The two continuous processes are the same (only different descriptions), but the discretizations are different Does not play a role in practice since differences in discretizations are small and tend to cancel out in the long run Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 14

Summary A random walk is defined by the increment X

t+∆t

− X

t

= εt

∆t

Random walk is a useful building block for modelling random value processes (e.g. stock prices) An often used model is the Ito process which is of the discretized form x (t + ∆ t ) − x (t ) = a ( x (t ), t ) ∆t + b ( x (t ), t )ε t ∆ t ≈ dx ( t )

≈ dt

≈ dz ( t )

Special case: geometric Brownian motion S (t + ∆t ) − S (t ) = µS (t ) ∆t + σS (t )ε t ∆t a ( S ( t ),t )

Richard de Neufville Stefan Scholtes

b ( S (t ), t )

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 15

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Risk-neutral valuation of options revisited How can we price a European call option on an underlying stock that follows a geometric Brownian motion? First trial: Price of option = E[max{0,ST-K}]e-δδT Can be found by simulating the stock price ST (see Option Pricing I.xls) What discount rate δ should be used? This approach is essentially equivalent to matching expected returns in the gambling setting of Session 4 Under what conditions is this equivalent to Black-Scholes? This is equivalent to changing the probabilities in a lattice to the riskneutral ones

Mathematical result: Black-Scholes formula is equivalent to pricing the stock at its discounted expected return, provided mean µ of stock returns replaced by the risk-free interest rate r discounting is done at risk-free rate r

This is called risk-neutral pricing see Luenberger p.357 for a mathematical explanation and p.344 for the relation to risk-neutral pricing in a lattice Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 16

Simulation Continuous risk-neutral process dS ( t ) = rdt + σ dz S (t )

dt

Stock price at time t (discretization): S (t ) − S (0) = S (0)(1+ rt + σε t )

ε is a standard normal variable

See spreadsheet Option Pricing II.xls

Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

Slide 17

Conclusions MCS is useful for options analysis Have seen that stock price movements, modelled as geometric Brownian motion, can be easily simulated Same is true for other processes (e.g. mean reverting processes) Have seen the analogue of risk-neutral valuation in a lattice for geometric Brownian motion

MCS has limitations in valuing flexibility (as in American options or with most real options) If used in conjunction with decision rules it will give conservative value estimates as the decisions are not necessarily optimal Alternative is the lattice model (so-called “dynamic programming”) Richard de Neufville Stefan Scholtes

Civil & Environmental Engineering, MIT Judge Institute of Management, CU

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