Risk Management & Real Options X. Financial Options Analysis Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course 2004-05

Risk Management & Real Options

X. Financial Options Analysis

Stefan ScholtesJudge Institute of Management

University of Cambridge

MPhil Course 2004-05

2 September 2004 © Scholtes 2004 Page 2

An option is a gamble…

£ v?

£ 5

£ 2

50%

50%

What’s the value of this gamble?What’s the value of this gamble?


An option is a gamble…

£ v?

£ 5

£ 2

50%

50%

v= expected return = 50%£6+50%£3=£4.50

Naïve valuationNaïve valuation


Market valuation: Assumes there is a market for gambles

£ 4

£ 6

£ 3

50%

50%

£ v?

£ 2

£ 550%

50%

Gamble inthe market(stock)

New gamble

Assume payoff determinedAssume payoff determinedby the same coin flipby the same coin flip

IncorporatesIncorporates““marketmarket

risk premium”risk premium”


First approach…

Valuation principle: Since the two gambles are based on the same uncertainty, they should have the same expected returns

9

13

25.1

5.3 v valuea gives gamble new the

togambleincumbent theofn expectatioreturn theApplying

5.32*5.05*5.0 gamble new of payoff Expected

125.14

3*5.06*5.0 gambleincumbent ofreturn Expected

R

Rinc


Change the new gamble

£ 4

£ 6

£ 3

50%

50%

£ w?

£ 2+x

£ 5+x50%

50%

Adding £x to both payoffs of the new gamble Adding £x to both payoffs of the new gamble should change its value to w=£v+xshould change its value to w=£v+x

But for x=1, both gambles are the sameBut for x=1, both gambles are the sameHence 4=v+1, i.e., v=3 is consistent with the existing gamble, not v=3 1/9!Hence 4=v+1, i.e., v=3 is consistent with the existing gamble, not v=3 1/9!

Surely, w=v+xSurely, w=v+x


Second approach:Market based valuation

33

11

v?v?

55

22

22

-1-1

11

1616

ExistentExistentgamble 1gamble 1


New New gamblegamble



33

11

v?v?

55

22

22

-1-1

11

1616




Up/down movements for all gambles areUp/down movements for all gambles aredetermined by the same flip of the coindetermined by the same flip of the coin(underlying fundamental uncertainty)(underlying fundamental uncertainty)



33

11

v?v?

55

22

22

-1-1

11

12

1625

yx

yxEquations for replicating portfolio:Equations for replicating portfolio:

1616







33

11

v?v?

55

22

22

-1-1

11

12

1625

yx


Solution:Solution: 3 ,2 yx1616







33

11

v?v?

55

22

22

-1-1

11

12

1625

yx


Solution:Solution: 3 ,2 yx

Price of replicating portfolio:Price of replicating portfolio: 91*33*2

1616







33

11

v?v?

55

22

22

-1-1

11

12

1625

yx


Solution:Solution: 3 ,2 yx

Price of replicating portfolio:Price of replicating portfolio: 91*33*2

1616





Replicating portfolio has precisely the same payoffs as the new gambleReplicating portfolio has precisely the same payoffs as the new gambleErgo: Price for the new gamble = price of replicating portfolio (v = 9)Ergo: Price for the new gamble = price of replicating portfolio (v = 9)


In-class example

33

11

v?v?

55

22

11

11

00

11

StockStock

CashCash

Call optionCall optionat strikeat strikeprice 4price 4


Why is this conceptually correct?Arbitrage and Equilibrium

Option and replicating portfolio have the SAME future payoffs, no matter how the future evolves

In equilibrium, option has a buyer AND seller• If price of the option < price of the replicating portfolio then no-one

will sell the option<̵ O/w someone buys the option, sells the replicating portfolio and pockets

the difference Risk-less profit (arbitrage)

• If price of the option > price of the replicating portfolio then no-one will buy the option

<̵ O/w someone sells the option, buys the replicating portfolio and pockets the difference Risk-less profit (arbitrage)

The only option price that is consistent with the existing market prices is the price of the replicating portfolio


Why is this conceptually correct?Consistency

A weaker argument then no-arbitrage is that of “valuation consistency”

• This argument does not require the existence of a market but replaces it by assumptions on valuations

Recall the problem: • Two chance nodes with different payoffs but the same underlying

“random experiment” (“same flip of the coin”)• We have already valued one of the chance node

1st key assumption: Linear valuation• “Constant returns to scale”: if all payoffs of a chance node are

multiplied by the same factor then the value of that chance node is multiplied by that factor as well

• “Adding values”: The value for the sum of two chance nodes with the same underlying random experiment is the sum of the value of the chance nodes

2nd key assumption: “Law of one price” • If two chance nodes follow the same underlying random experiment

(“same flip of coin”) and have the same payoffs then their values are the same


Example

33

11

v?v?

55

22

11

11

00

11

ChanceChanceNode 1Node 1



All moves are triggered by the sameAll moves are triggered by the sameflip of the coinflip of the coin


Example

33

11

v?v?

55

22

11

11

00

11




== 22

22 22

++ v?v?

11 00

3x3x



Example

33

11

v?v?

55

22

11

11

00

11




== 22

22 22

++ v?v?

11 00

3x3x


Only “consistent” value for v is Only “consistent” value for v is v=(3-2)/3=1/3v=(3-2)/3=1/3


Option pricing in a binomialmodel: The general case

S

uS

dS

Risky investmentin stock returns u>1, d<1

1

(1+r)

(1+r)

Risk-free Investmentr=one-period risk-free rate

Cu

Cd

Options contractC=?

Call with exercise price KCall with exercise price K

All parameters are known, except for CAll parameters are known, except for CCall option: Call option: Cu=Max{uS-K,0}, Cd=Max{dS-K,0}


Option pricing in a binomialmodel: The general case

Invest £x in stock and £y in bank (negative amounts mean short sales and borrowing, resp.)

Equations for replicating portfolio

Solution

d

u

Cryxd

Cryxu

)1( :match downwards

)1( :match upwards

du

dCuC

rr

uxCy

du

CCx

udu

du

1

1

1


Replicating the option payoffs

Price of the option is the price of the replicating portfolio C=£x+£y

After some simple algebra C=x+y becomes

Notice that 0<q<1, provided u>1+r>d (sensible assumption) Can interpret C as an expected payoff discounted at the risk-

free rate• However, q has nothing to do with the actual probability that the

stock moves upwards!• Can interpret q is the “forward price” for a contract that pays Cu=1

if the stock moves up and Cd=0 o/w

du

drq

r

CqqCyxC du

1

with 1

)1(


Replicating the option payoffs

Price of the option is the price of the replicating portfolio C=£x+£y

After some simple algebra C=x+y becomes

Option pricing principle: Price of the call option is its expected payoff if upwards probability was q, discounted at the risk-free rate

This is called risk-neutral pricing and q is called the risk-neutral upward probability

du

drq

r

CqqCyxC du

1

with 1

)1(


Keeping track of the replicatingportfolio

It is always a good idea to keep track of the replicating portfolio if you value an option

Recall the equations for amount £x in stock and amount £y in risk-less investment (long-term government bond):

Holding £x in stock and £y in risk-less money exactly replicates the option in our model

du

dCuC

ry

du

CCx

ud

du

1

1


Multi-period models

Single-period stock price model• Given a stock price S today, the stock will move over a period t to

uS (upward move) with probability p and to dS (downward move) with probability (1-p)

u and d are numbers with u>1+r>d>0• typically d=1/u

Let us see how this model develops over time…

S

uS

dS

p

1-p t


Example of unfolding of stock price uncertainty(p=50%)

Period Now 1 2 3 4 5 6 7 8

Stock price Probability Stock price Probability Stock price Probability Stock price Probability Stock price Probability Stock price Probability Stock price Probability Stock price Probability Stock price Probability

£98.08 0.4%£92.61 0.8%

£87.45 1.6% £87.42 3.1%£82.58 3.1% £82.55 5.5%

£77.98 6.3% £77.95 9.4% £77.93 10.9%£73.63 12.5% £73.61 15.6% £73.59 16.4%

£69.53 25.0% £69.51 25.0% £69.49 23.4% £69.47 21.9%£65.66 50.0% £65.64 37.5% £65.62 31.3% £65.60 27.3%

£62.00 100% £61.98 50.0% £61.96 37.5% £61.94 31.3% £61.92 27.3%£58.53 50.0% £58.51 37.5% £58.49 31.3% £58.47 27.3%

£55.25 25.0% £55.23 25.0% £55.22 23.4% £55.20 21.9%£52.16 12.5% £52.14 15.6% £52.12 16.4%

£49.24 6.3% £49.22 9.4% £49.21 10.9%£46.48 3.1% £46.46 5.5%

£43.88 1.6% £43.86 3.1% £41.42 0.8%

£39.10 0.4%


What’s the distribution of the value after many periods?

Mathematical result:

Binomial model of moving up by factor u with probability p and down by factor d with probability 1-p is, for many periods, an approximation of log-normal returns, i.e., log(Sn/S0) is approximately normal with mean

22 )]log())[log(1( variance

)]log()1()log([ mean

dupnp

dpupn


Does that make sense?

Suppose returns rt=St/St-1 of a stock over small time periods are independent and have an unknown distribution

Consider t=0,1,…,T (e.g. T=52 weeks). What is the distribution of the long run (say annual) return?

ST/S0=r1*r2*…*rT

ln(ST/S0)=ln(r1)+ln(r2)+…+ln(rT)

Therefore the central limit theorem provides an argument that long-run returns tend to be log-normally distributed, even if short-run returns are not

• A random variable X is called log-normally distributed if log(X) is normally distributed


Histogram of log-normal variable(Simulation of exp(Y), where Y is normal with mean 10%and standard deviation 40%)

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

20.00%

return

Fre

qu

en

cy


Histogram of correspondingnormal variable Y

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

log return

Fre

qu

en

cy


Estimating parameters for the lattice model

Choose base period, e.g. a year, and estimate mean n and variance s2 of the log stock price return over the base period

• e.g. based on historic data Partition base period into n periods of length t=1/n Recall that log-return log(Sn/S0) is approximately normally

distributed with •

Setting n=1/t gives the equations

22 ))log())(log(1(

)log()1()log(

duppt

dpupt

22 )]log())[log(1( variance

)]log()1()log([ mean

dupnp

dpupn


Estimating parameters forthe lattice model

System

consists of two equations in three unknowns p, u, d Can remove the degree of freedom arbitrarily, e.g. by setting

d=1/u Corresponding solution of the system is

With this choice of parameters the binomial lattice is a good approximation of normally distributed log stock price returns with mean n and volatility s

tt edeutp

, ,22

1

22 ))log())(log(1(

)log()1()log(

duppt

dpupt


Alternative parameter choice

System

consists of two equations in three unknowns p, u, d Can remove the degree of freedom arbitrarily, e.g. by setting

p=50% Corresponding solution of the system is

This choice of parameters incorporates the trend in the upwards and downwards moves, as opposed to the earlier choice which incorporates trend in probability p.

tttttttt eeedeeeup )()( , ,2

1

22 ))log())(log(1(

)log()1()log(

duppt

dpupt


Example

Data:• Stock price is currently £62, Estimated standard deviation of

logarithm of return = 20% over a year (T=1)• European call option over 2 months at strike price K=£60• Risk-free rate is 10%, compounded monthly (r=0.1/12, t=1/12)

Conversion of this information to lattice parameters:

Risk-neutral probability: q=((1+r)-d)/(u-d)=0.559 Notice: Risk neutral probability (and therefore the options

price) are independent of the probability p of upward moves • The important parameters, u,d, only depend on the standard

deviation of the log returns (volatility), not on the mean (trend)

944.0 ,059.1 121

121

edeu


In-class example

Given annual volatility of 25%, what are u and d for a lattice with weekly periods?

Given a risk-free interest rate of 5% p.a. what is the weekly risk-free interest rate r? (assume continuous compounding, i.e., capital grows by factor exp(t5%) over a period of length t)

What is the “risk-neutral probability” q?

If the stock price today is £50 and we have a one-week call option with strike price £51, what is the value of the option in a single-period lattice (i.e. only one up or down move)?


2-period stock price

Now Month 1 Month 2

Stock price Stock price Stock price

£69.59£65.68

£62.00 £62.00£58.52

£55.24


Corresponding decision tree

Price movePrice movemonth 1month 1


Exercise?Exercise?

upup

upup

upup

downdown

downdown

downdown

yesyes

yesyes

yesyes

yesyes

nono

nono

nono

nono


Corresponding decision tree



Exercise?Exercise?

upup

upup

upup

downdown

downdown

downdown

yesyes

yesyes

yesyes

yesyes

nono

nono

nono

nono

These two nodesThese two nodesare identical sinceare identical sincemoving up first and moving up first and then down is the samethen down is the sameas moving down firstas moving down firstand then upand then up


Simplified decision tree



Exercise?Exercise?

upup

upup

upup

downdown

downdown

downdown

yesyes

yesyes

yesyes

nono

nono

nono

We will value this decision tree using non-arbitrage valuation of chance nodesWe will value this decision tree using non-arbitrage valuation of chance nodes


Valuation of the final decisionnodes

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24

Euopean call option value process

£9.59

£2.00

£0.00

Max(stock-strike, 0)Max(stock-strike, 0)

Strike price = £60Strike price = £60


Valuation of the final decisionnodes

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24


£9.59 v=?

£2.00

£0.00

Existing gambleExisting gamblein the marketin the market

New gambleNew gamble


Non-arbitrage valuation of month 1 chance nodes

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24


£9.59£6.18

£2.00

£0.00

r

CqqCC du

1

)1( uC

dC


Non-arbitrage valuation of month 1 chance nodes

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24


£9.59£6.18

£2.00

£0.00

r

CqqCC du

1

)1(Value x+y of replicatingValue x+y of replicatingportfolioportfolio

borrowing) investment (negative

account free-risk

in 51.59£1

1

andstock in 69.65£

invest then 1month in £65.68 is price

stock theIf :portfolio gReplicatin

du

dCuC

ry

du

CCx

ud

du


Non-arbitrage valuation of second chance node in month 1

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24


£9.59£6.18

£2.00£1.11

£0.00r

CqqCC du

1

)1( uC

dC

CorrespondingCorrespondingrisky gamblerisky gamblein the marketin the market


Non-arbitrage valuation of today’s chance node

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24


£9.59£6.18

£3.90 £2.00£1.11

£0.00r

CqqCC du

1

)1( uC

dC

Option Option valuevalue

CorrespondingCorrespondingrisky gamblerisky gambleIn the marketIn the market


In-class example





Now value a 3-week call option with the same strike price


No-arbitrage valuation versus discounted expected values

We can, conceptually, also value the decision tree with discounted expected values at the chance nodes

Assuming, in the former example, an annual expectation of log stock returns of 15%, we obtain the upward move probability (see earlier slide)

The expected monthly return on the stock is 1.42% If we value the chance nodes by their expected value,

discounted by the stock return expectation then the obtain the value £4.38

• Why is this the wrong value?

%8.6022

1 tp


The Black-Scholes formula

Black and Scholes have found a formula that allows you to compute the value of a European call option without the use of a lattice

C is the option price, S is today’s stock price, K is the strike price and T is the time to maturity

• Make sure that n, s, r and T refer to the same base unit, i.e. if risk-free interest, means and standard deviation of log-returns are annual then T is measured in years as well

N(x) is the standard normal cumulative distribution function (N(x)=P(X<=x) where X is a standard normal (i.e. mean 0, variance 1)

• Normsdist(x) in Excel

TddT

TrKSd

dNKedSNC rT

12

2

1

21

,)2/()/ln(

with

)()(



First observation: Option value

is independent of mean n of the underlying stock price

Second observation (after some calculus): Option value increases with increasing volatility s

• Do you have an intuitive argument for this observation?

TddT

TrKSd

dNKedSNC rT

12

2

1

21

,)2/()/ln(

with

)()(


In-class example





Now value a 3-week call option with the same strike price

Now calculate the value of the 3-week call option with the B-S formular



The B-S price for the option that we had valued earlier in the two-stage lattice is £ 3.82 (against £3.90 in our model)

If we use more time periods (e.g. a half-monthly or weekly lattice), then the lattice approximation of B-S becomes better and better

Mathematical result: As Dt gets smaller and smaller, the value obtained by a lattice valuation approaches the Black-Scholes value

• See Luenberger, Chapter 13, for more explanations

So why do we do lattices, then? B-S applies only to European option

• European option can only be exercised at maturity American options can be exercised at any time until they

mature• More realistic for real options• American options can be priced by the lattice model!


Decision tree for American option

Move in Move in Month 1Month 1

Exercise?Exercise? Move in Move in Month 2Month 2

Exercise?Exercise?

yesyes

yesyes

nono

nono


Decision tree for Americanoption



Exercise?Exercise?

Can beCan bemergedmergedas beforeas before

yesyes

yesyes

nono

nono


Simplified decision tree



Exercise?Exercise?

We will value this decision tree using non-arbitrage valuation of chance nodesWe will value this decision tree using non-arbitrage valuation of chance nodes

yesyes

yesyes

nono

nono


American call option valuation:Valuing final decision nodes

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24 America call option value process

Option Price Exercise? Option Price Exercise? Option Price

£9.59

£2.00 £0.00

Max(stock-strike, 0)Max(stock-strike, 0)

Strike price = £60Strike price = £60

Final period: exercise only if stock price > strike priceFinal period: exercise only if stock price > strike price


American call option valuation:Simultaneous valuation of month 1 chancenode and decision node

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52



£9.59

£2.00 £0.00

Is the valueIs the valuefrom exercisingfrom exercisingthe option largerthe option largerthan the value than the value from holding it?from holding it?



Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52



£9.59

£2.00 £0.00


68.5£strike-stock :option theexercising from Value

18.6£1

)1( :option theholding from Value

r

CqqCC du



Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52



£9.59 no

£2.00 £0.00


68.5£strike-stock :option theexercising from Value

18.6£1

)1( :option theholding from Value

r

CqqCC du



Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52



£9.59£6.18 no

£2.00 £0.00

Maximum of Maximum of the value of the value of holding the optionholding the optionand the value from and the value from exercising itexercising it


Roll-back to the root of the tree

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52



£9.59£6.18 no

£3.90 no £2.00£1.11 no

£0.00

Max(holding,exercising)Max(holding,exercising)


Aside

One can show mathematically that an American call option on a non-dividend-paying stock is never exercised before maturity

Therefore the additional flexibility of being able to exercise before maturity has no value

American calls on non-dividend-paying stocks have the same value as European calls on the same stock

This is not the case for put options


American Put Option

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24

America put option value process


£0.00£0.44 no

£2.21 no £1.00£4.48 yes

£7.76

Strike price £63Strike price £63

Max(strike-stock,0)Max(strike-stock,0)


American Put Option

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24



£0.00£0.44 no

£2.21 no £1.00£4.48 yes

£7.76



Max(exercise value,holding value)Max(exercise value,holding value)


American Put Option

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24



£0.00£0.44 no

£2.21 no £1.00£4.48 yes

£7.76



Max(exercise value,holding value)Max(exercise value,holding value)stock-strike : valueExercise

1

)1( : valueHolding

r

CqqCC du


American Put Option

Stock price process

Now Month 1 Month 2


£69.59£65.68

£62.00 £62.00£58.52

£55.24



£0.00£0.44 no

£2.21 no £1.00£4.48 yes

£7.76



Max(exercise value,holding value)Max(exercise value,holding value)stock-strike : valueExercise

1

)1( : valueHolding

r

CqqCC du

PrematurePrematureExerciseExercise


Summary of option valuation

Set up the lattice for the underlying stock price Calculate the option price at maturity Value the nodes of the tree successively backwards, using

Value of a node for European option = holding value Value of a node for American option = max(holding value,

exercise value) Value of the origin of the tree is the current value of the option

pricestock -price strike :put a exercising of Value

price strike-pricestock :call a exercising of Value1

)1( :stock theholding of Value

r

CqqCC du


Optimal decision

Notice that for American options you do not only get the value of the option but also an optimal contingency plan when to exercise the option

For real options, the contingency plan is much more important than the precise value, since real options are typically not traded in a market


Group work

Carry out a binomial lattice valuation for an American call option over 5 months with annual volatility 30%, risk free interest 5% p.a., strike price $35 and current stock price $30

Compute the optimal contingency plan for the exercise decision

Compute the initial replicating portfolio and the re-balancing strategy

Compare with the value you obtain from the Black-Scholes formula

Documents

Risk Management & Real Options X. Financial Options Analysis Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course 2004-05