WEMBA 2000Real Options60

WEMBA 2000 Real Options 60

Call Option Delta

Callvalue

S: Price of Underlying Asset

K

Time to expirationdecreases

Call Price Curve:The Call Price as a function of the underlying asset price

As time to expiration decreases,the call price curve tends towardsthe payoff function (the payoff fromthe call at expiration).

Callvalue

S: Price of Underlying Asset

Call pricecurve

Call Delta

The call "delta" is the slope of thecall price curve:

Delta () = c S


Replicating the Rigby Option Payoff by delta-hedging

2841.77

18.77

62.32

28

12.58

t = 1 yearrf = 6.3% = 40%u = e t = 1.4918d = 1/u = 0.6703

92.96138.68

8.435.65

62.32

28

12.5818.77

41.77

14.84 0.8

27.20 0.91

5.320.58

48.65 0.98

10.830.76

0.930.24

83.39 1.00

136.42

00 0

44.78

3.6

0 1.80 0.36

22.45 0.95

Stock Price Tree

Call Option Tree

Price

Delta

Date Price Delta

T=0 28 0.8

T=1 41.77 0.91

T=2 62.32 0.98

T=3 92.96 1.00

T=4 138.68 1.00

Sell delta * 1.2 barrels = $26.88

Reserve $14.8(Call option value)

Invest remainder: $12.08 at 6.3%

Re-hedge: sell further(0.91 - 0.8) * 1.2 barrels0.11*1.2*41.77 = $5.51

$12.84 at year end +

= $18.35 Invest at 6.3%

$19.51 at year end


+= $24.75 Invest at 6.3%

$26.30 at year end

Re-hedge: sell further(1 - 0.98) * 1.2 barrels0.02*1.2*92.96 = $2.23

+= $28.53 Invest at 6.3%

$30.33 at year end

Buy back 1.2 barrels1.2*138.68 = - $166.42+

= - $136.09


Exercise Option: (138.68 - 25)*1.2 = $136.41

Date Price Delta

T=0 28 0.8

T=1 41.77 0.91

T=2 28 0.76

T=3 18.77 0.36

T=4 28 1.00

Sell delta * 1.2 barrels = $26.88

Reserve $14.8(Call option value)

Invest remainder: $12.08 at 6.3%


$12.84 at year end +

= $18.35 Invest at 6.3%

$19.51 at year end

Re-hedge: buy back(0.76 - 0.91) * 1.2 barrels - 0.15*1.2*28 = - $5.04

+= $14.47 Invest at 6.3%

$15.38 at year end

Re-hedge: buy back(0.36 - 0.76) * 1.2 barrels-0.4*1.2*18.77 = -$9.01

+= $6.37 Invest at 6.3%

$6.77 at year end

Buy back remaining barrels- 0.36*1.2*28 = -$12.096+ = -$5.326


Exercise Option: (28 - 25)*1.2 = $3.6


Sources of Error

Inevitable concerns when using the replicating portfolio (apply to financial options as well)

How often to update the delta-hedge? Tradeoff between transactions costs and imperfect hedging

Borrowing/lending rate: we cannot guarantee that our borrow/lend costs will remain unchanged

Implied volatility: if we use the "wrong" implied vol, the delta will be wrong (e.g. set = 0.85 in first leg of Rigby hedge and re-evaluate the delta-hedging strategy)

Short selling: we need to short-sell oil to hedge our exposure to a drop in oil prices.

Further concerns in the case of real options: inputs in the Black-Scholes model

Strike price: are we sure about the exact extraction costs throughout the project? (consider the example in the homework)

Time to expiration: are we sure we have the right time frame?(suppose we are uncertain that the entire extraction can be completed in one year)

Is the traded underlying asset a good proxy for the "real" underlying asset?Tracking risk...


Case Study: Natural Resources Abandonment Option

Abbeytown Copper Refinery has two-year lease over a copper deposit. We have the following information:

Deposit contains eight million pounds of copper

Mining involves a one-year development phase, at a cost of $1.25 million immediately

Extraction costs (outsourced) at $0.85 / pound at beginning of extraction phase (one yearafter development phase is initiated)

Sale of copper would be at spot price of copper as of beginning of extraction phase

Current spot price of copper is $0.95 / pound

Copper prices are normally distributed with mean 7% and standard deviation 20% (p.a.)

Abbeytown's required rate of return for this project is 10%, and the riskless rate is 5%


Case Study: Natural Resources Abandonment Option (2)

Traditional NPV analysis:

Expected NPV = -1.25 + 8(E[S1] - 0.85) 1.1

where E[S1] = Expected price of Copper in one year's time

Current price of Copper, S0 = 0.95Expected rate of return on copper, r = 7%Expected price of copper in one year, S1 = 0.95e0.07 = 1.0189

Hence E[NPV] = -1.25 + 8(1.0189 - 0.85)/1.1 = - 0.022 Reject Option Analysis

S = 0.95 * 8 = 7.6 Call value = 1.3

K = 0.85 * 8 = 6.8 Option cost = 1.25

r = 5% O-A PV = 0.05 Accept

T = 1 year

= 20%

Case Study: Natural Resources Abandonment Option (3)

Why does the option to abandon have value?

Can choose to abandon the project if the price of copper is low after one year.

Probability that we will abandon = 1 - Prob(exercise)

= 1 - N(d2)

= 1 - 0.76 = 0.24

The usual caveats….

How certain are we about the inputs? -- size of deposit ("real" option caveat)-- development costs and timing ("real" option caveat)-- extraction costs and timing ("real" option caveat)-- volatility of copper price over the next year (applies in any option pricing context)-- riskless borrow/lend rate (applies in any option pricing context)


Case Study: Prudence Properties

Prudence Properties holds an option to sell a piece of commercial property that it currently owns, in six weeks time, for a price of $50 million. We have the following information:

Ownership of the property is in the form of a REIT (Real Estate Investment Trust) hence we canobserve the price of the property in the financial markets.

Current price in the market is $50 million, and historic price volatility is 40% per year.

The riskfree rate of interest is 10% per annum.

Scenario 1: The option is European (can only be exercised at the end of the six week period).

Current Stock Price 50.00Exercise price 50.00Interest rate 10.00%Time to expiration 0.417Volatility 40.00%Dividend yield 0.00%

Model Inputs

Put Value (Black-Scholes) 4.08


Case Study: Prudence Properties (2)

Now we evaluate the option using a Binomial Tree model, as follows:

S = 50K = 50rf = 10%T = 5/12 = 40%

Take this on trust.The mathematical

justification is ugly!t = length (in yrs) of one "time-step" = 1/12u = proportional change in S on an up-move = et = 1.224d = proportional change in S on a down-move = 1/u = 0.8909p = (ert - d)/(u - d) = 0.5076

50.00 4.32

56.12 2.11

62.99 0.63

70.70 0

79.35 0

62.99 056.12

1.3050.00 3.67 44.55

6.17 39.69 9.90

50.00 2.66

35.3613.81 31.50

18.08

44.55 6.66 39.69

9.86

89.07 0

70.70 056.12 0

44.55 5.45

35.3614.64

28.0721.93

Scenario 1 cont.: The put option is European (can only be exercised at the end of the six week period).

Why is this resultdifferent from the

Black-Scholes formula?


Scenario 2: The put option is American (can be exercised at any time during the six week period).


50.00 4.32

56.12 2.11

62.99 0.63 56.12

1.3050.00 3.67 44.55

6.17 39.69 9.90

50.00 2.66

35.3613.81 31.50

18.08

44.55 6.66 39.69

9.86

44.55 5.45

35.3614.64

28.0721.93

50.00 4.49

56.12 2.16

62.99 0.63 56.12

1.3050.00 3.77 44.55

6.38 39.6910.31

50.00 2.66

35.3614.64 31.50

18.50

44.55 6.96 39.69

10.36

44.55 5.45

35.3614.64

28.0721.93

Compare the original tree…

… with the value of immediateexercise at each node

Binomial method for Amer putwith 100 steps: put price = 4.30



How do we treat the lease payment prior to option expiration date:(a) add it up-front to the value of the option?(b) add it to the present value of the property?(c) subtract it from the strike price of the option?


(a)


Add PV(2.6) upfrontTotal option value:

4.08 + 2.06 =$6.14


(b)


Add PV(2.06) to value of property

Total option value: $3.36


(c)


Subtract PV(2.06)from strike

Total option value: $3.20

Scenario 3: The put option is European, and ownership of the property yields rental income of $2.06 MM in 3.5 months time.

PV of rental income = 2.06e0.2917*0.1 = 2.00


Scenario 4: The put option is American, and ownership of the property yields rental income of $2.06 MM in 3.5 months time. The market price of the property today is $52.


Step 1: Construct a tree for S*, the property value without the rental income: S* = 50(this is the same as the original tree in Scenario 1)

Step 2: Adjust the tree by the present value of the dividend at each node

28.07

50.0056.12

62.9956.12 50.0044.55 39.69

50.00

35.3631.50

44.55 39.69

44.55

35.36

28.07

52.0058.14

65.0258.17 52.0346.60

39.69

50.00

37.41

31.50

46.57 41.72

44.55

35.36


Scenario 4 cont.: The put option is American, and ownership of the property yields rental income of $2.06 MM in 3.5 months time. The market price of the property today is $52.


28.0721.93

52.00 4.43

58.14 2.15

65.02 0.63 58.17

1.3052.03 3.76 46.60

6.3739.6910.31

50.00 2.66

37.4114.22

31.5018.50

46.57 6.85 41.72

10.1544.55 5.4535.3614.64

Price the American put by checking whether early exercise is optimal at each point…

… and compare with the European put with the rental income (Scenario 2)

52.00 3.56

58.36 1.62

65.51 0.43 58.36

0.8852.00 2.87 46.33

4.96 41.28 8.31

52.00 1.79

36.7812.40 32.77

16.82

46.33 5.62 41.28

8.55

46.33 3.6736.7813.2229.1920.81



Case Study: Prudence Properties -- Summary

European-style American-style

Rental Income Rental Income$0 $2.06$0 $2.06 $0 $2.06$0 $2.06

4.08 3.36 n/a n/a

4.32 3.56 4.49 4.43

Black-Scholes

Binomial tree 5 Steps


Estimating Historical Volatility


Estimating Historical Volatility

Define:n+1 = # of observationsSi = asset price at end of ith interval (i = 0,1,2,…, n)T = length of time interval (years)

Now let ui = ln ( Si / Si-1 )Then ui is the continuously compounded return on the asset in the ith interval

The standard deviation s of the returns vi is given by:

s = [1/(n-1) n ( ui - û )2 ] where û is the mean of the ui ‘s.

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WEMBA 2000Real Options60