Sleepless in L.A. Case Analysis

Sleepless in L.A. Case AnalysisMarch 5

2014

1. How can Black Scholes model is used in pricing the options? And what we understand by call and put options in the context of Merton model?

The BS Model is one of the most important concepts in modern financial theory. Black & Scholes(BS) model is not the most accurate one in pricing the options, but is the closest and widely used model with few assumptions and corrections. It is regarded as one of the best ways of determining fair prices of options. Black & Scholes model prices the options by the BS formula which gives the theoretical estimation of the European options(Which assumes that they should be held till expiration. Ex., PE,CE).

By the above model the option pricing can be done with few assumptions.

No dividends paid until expiry of the option

Only European options are exercised

Markets are not predictable and efficient

Risk free rate are known and constant

Returns are lognormally distributed

Commissions are not charged

In the formula of Theoretical call premium(C), first term SN(d1) gives the expected profit by acquiring a stock. Second term Ke(-rt)N(d2) gives the PV of paying the exercise price on expiration.

Call and put options in the context of Merton model:

Merton model is used to evaluate the credit risk by determining the companys ability to meet its financial obligations by servicing its debt.

Only the residual cash flows are paid to the equity holders after clearing all its debt obligations.

Merton model determines the value of the equity of the levered firm with call option on firms asset as follows.

V= Value of firm

D=Value of debt

Claim of equity = max (V-D,0)

Value received by debt holders at any time t = min (D,V)

Value of firm = max (V-D, 0) + min (D, V)

Similar option based analogy can be used to price the value of a firms debt. Debt is a put option on the companys assets with the same strike price.

The premium on this option is the present value of the yield spread over and above the risk free rate.

Value of a firms risky debt is the difference between the risk free debt and the put option on firms assets with strike price equal to same face value of debt.

2. Referring the case exhibit 3, can you use the Black Scholes model to verify the value of these publicly traded options? Does the model give the same value as the market value? Why or why not?Below are the values for call option calculated using the Black Scholes modes. The model does not give the same values as the market values. The values in some cases are close to the market values.

There are few assumptions of the Black-Scholes model.

The stock pays no dividends during the option's life

European exercise terms are used

Markets are efficient

No commissions are charged

Interest rates remain constant and known

Returns are log normally distributed

These assumptions do not hold true in the market scenario and are prone to many fluctuations and changes. Example the assumption that markets are efficient is one of the theoretical concepts which do not hold any significance in the market.

Call Option Prices(Using Black Scholes Model)

Expiration Date04-Jun04-Sep04-Dec

Strike Price

205.836.036.23

22.53.333.563.78

250.831.091.34

27.5-1.66-1.38-1.11

Call Option Prices(Market Prices)

Expiration Date04-Jun04-Sep04-Dec

Strike Price

205.906.206.30

22.53.404.005.00

251.252.152.5

27.5.25.901.40

Put Option Prices(Using Black Scholes Model)

Expiration Date04-Jun04-Sep04-Dec

Strike Price

20-5.83-6.03-6.23

22.5-3.33-3.56-3.78

25-0.83-1.09-1.34

27.51.661.381.11

Put Option Prices(Market Prices)

Expiration Date04-Jun04-Sep04-Dec

Strike Price

20.05.20.35

22.5.15.60.65

25.551.251.75

27.51.952.453

Working Sheet

3. Can option pricing help justify why Microcomps market capitalization is not zero?

From our view we agree that option pricing does help justify why Microcomps market capitalization is no zero. Microcomps outstanding debts were $150.0 million while, the market value of the firm (assets) were at $115.5 million. In this case the market capitalization should have gone to a zero value. However, it stood at $26.5 million. This may be, when shareholders borrowed money by issuing bonds, they actually sold the firm or the part of the firm for cash but, also obtained an option to repurchase the firm at the time of maturity of the debt. The bond buyers have a Call option on these bonds, which is a Right to sell the bond but, not an obligation to sell at the strike price. Hence, this facility comes with a premium and thus gives some value to bond. This in turn might have turned up the market capitalization of Microcomp to $26.5 million.Alternatively, investors future expectations that company will perform well after a certain period of time can give some value to the market capitalization. This factor might have considered while pricing the option in the bond and hence gave a value of $26.5 million on market capitalization.The pricing of a call option bond can explained with the help of Black-Scholes model.

Value of call option = SN(d1)Ke^r(T-t)N(d2)

Value of put option = Ke^r(T-t)N(-d2)SN(-d1)

Where, d1 = lnSK+r+122(T-t)T-t

d2 = d1 - T-t

S Value of underlying share

K Exercise price of the option

r Risk free rate

T-t Remaining time to maturity

2 Variance of share price

N(.) - Cumulative distribution function of the standard normal distributionUsing this formula, a future gain in price can be predicted and according to that a probability of good performance is expected from the firm and hence a market capitalization value of $26.5 million.**Market capitalization is the product of market value per share and the outstanding shares.4. Can options pricing be used to value Microcomps risky bonds and How?

The Company has assets that are financed with bonds and equity. The bonds mature in 2 years at which time a principal payment of $150 mn is required. If the assets are more than $150 mn in 2 years, the equity holders choose to repay the bondholders. However in this case the market value of assets is less than repayment value of the bonds (reflecting financial distress), it has to choose bankruptcy and the bondholder end up in owning the company.

The value of equity in 2 years is therefore a max of (AT-K, 0), where AT is the value of the companys assets and K is the principal payment of the bond at that time. This shows that the equity holders have a 2-year European call option on the assets of the company with strike price of K. The bondholders get min(AT, K) in 2 years. This is the same as K max (K-AT, 0) which shows that today the bonds are worth the present value of K minus the vale of a 2-year European put option on the assets witha strike price of K.

If C and P are the value of call and put options on ht e Companys assets at time T, then

Value of equity = C & Value of debt = PV(K) P

If A0 is the value of assets today, then it will be equal the total value of the instruments used to finance the assets. Summarising the above statement the equation would be,

A0= C + [PV(K) P]

=> C + PV(K) = A0 + P

The above equation would result in put-call parity and thus option pricing can be used for pricing of the bonds.

Submitted by,

Anadi Kaistha

Nabin Basha

Naveen Kumar

Sanoop S

Sreenandan Nambiar P

Sheet1

3300

0.146666666712

=150*.080.08

0.14666666670.080.04932603620.0042424242

0.03067396380.1106739638

5.61248608023.5

0.01173333335.9160797831

0.6

0.08655

111.913450.08655

5.9160797831

11.8321595662

46.8321595662

Sheet2

1

019.96461216641

Price of the stock 25.79

0.8328255708Call Option PricesPut Option prices

Expiration Date4-Jun4-Sep4-Dec4-Jun4-Sep4-Dec

Strike Price

205.82538783366.02984342726.2278752256-5.83-6.03-6.23

22.53.32984669823.55982386823.7826096289-3.33-3.56-3.78

250.83430177741.0898042991.3373440321-0.83-1.09-1.34

0.832825570827.5-1.6612431434-1.3802152701-1.10792156471.661.381.11

T bill Rates4.04%4.04%4.04%

Days to expiry1610920016109200

Historical Volatility24.14%

Dail Volatility1.52%

Calculation of d1

Call Option PricesPut Option prices

Expiration Date4-Jun4-Sep4-Dec4-Jun4-Sep4-Dec

Strike Price

209.52379251636.89761351168.3562738016-9.5237925163-6.8976135116-8.3562738016

22.54.27271248026.15573550057.8085889233-4.2727124802-6.1557355005-7.8085889233

252.54058081645.49210300577.3186681051-2.5405808164-5.4921030057-7.3186681051

27.50.97367713064.8917743916.8754808165-0.9736771306-4.891774391-6.8754808165

T bill Rates4.04%4.04%4.04%

Days to expiry1610920016109200

Historical Volatility24.14%

Dail Volatility1.52%

Calculation of d2

Call Option PricesPut Option prices

Expiration Date4-Jun4-Sep4-Dec4-Jun4-Sep4-Dec

Strike Price

209.46296543386.73885016528.1412175889-9.4629654338-6.7388501652-8.1412175889

22.54.21188539775.99697215417.5935327107-4.2118853977-5.9969721541-7.5935327107

252.47975373395.33333965937.1036118925-2.4797537339-5.3333396593-7.1036118925

27.50.9128500484.73301104466.6604246038-0.912850048-4.7330110446-6.6604246038

Sheet3