FINS2624_Problem Set 11 Written Solutions

7/24/2019 FINS2624_Problem Set 11 Written Solutions

1/12

FINS2624 Portfolio

Management

Tutorial 11 Week 12


2/12

Problem Set 11

Q1a. Consider the data in the table:

What is the price of a European call option with the properties asdescribed in the table?

A.

tTdd

tT

tTrXS

d

dNXedNSc

t

tTr

tt

12

2

1

21

2ln


3/12

Problem Set 11

Continued A.

6042

7048010083320132

7048053830

8332096700

538305135096700

96700

51350

512

350030

100

132ln

510302

1

2

2

1

.c

.e.c

..NdN

..NdN

....d

.

..

..

.

d

t

..

t


4/12

Problem Set 11

Q1b. What is the price of the corresponding European put option?

A.

20.6

60.42100132 5.103.0

t

t

t

tTr

tt

p

ep

cXeSp


5/12

Problem Set 11

Q1c. How would you expect the put and call prices to change if the risk freeinterest rate increased to 4%?

A.

For call option:

=+ ve

Call option gives the right to buy stocks at fixed price at time t. Thats a negative CF; webuy the stocks. We would like to discount it as largely as possible. So higher interestrate makes the call options more valuable.

For Put option:

=-ve

Put option gives the right to sell stocks at fixed price at time t. Thats a positive CF; wesell the stocks. We would like to discount it as less heavily as possible. So higher

interest rate makes the put options less valuable.

Greek letter Variable Call option Put option

Delta, St Positive Negative

Theta, t Negative (Typically) negative

Vega, Positive PositiveRho, r Positive Negative

None X Negative Positive


6/12

Problem Set 11

Q1d. Show that, in general, the price of a European put option is:

pt= Xer(T t)N(d2) SN(d1)

A.

12

12

21

21

11

dNSdNXep

dNSdNXep

dNXeXeSdNSp

dNXedNSXeSp

cXeSp

t

tTr

t

t

tTr

t

tTrtTr

ttt

tTr

t

tTr

tt

t

tTr

tt


7/12

Problem Set 11

Q1e. You hold 100 put options. What position would you have to take inthe underlying stock in order to be delta neutral?

A.

To make the portfolio delta neutral, we need to buy 16.68 stocks ( Deltahedging). The movement in put values cancelled out by movement instocks.

68.16)1668.0100(

1668.018332.0

11

Portfolio

put

put

dS

d

dNdS

dp


8/12

Problem Set 11

Q1f. You hold 100 put options. What position would you have to takein the call option in order to be delta neutral?

A.

To make the portfolio delta neutral need to buy 20.02 call options.

0220

83320)16680(1000

833201

.N

.N.

.dNdS

dc

Call

CallPortfolio

Call


9/12

Problem Set 11

Q1g. The market price of the call option is $42.60. What is theimplied volatility?

A. The market price of the option turns out to be the one that we alreadyhave calculated. Therefore, the implied volatility is the volatility that

we used in the calculation of call option (35%). The B-S formula pricedthis option accurately.


10/12

Problem Set 11

Q1h. Suppose you would estimate the historical volatility of S to30%.What implications would that have for the volatility and/or theBlack-Scholes model?

A. Historic volatility is 30%. This does not have to be a contradictionwith Black-Scholes.

In B-S, we assume the volatility is constant for the rest of the life of the

option. If the historic volatility is 30%, it does not necessarily mean that

the volatility for the remaining life of the option would not be 35%. The BS

volatility is forward looking.


11/12

Problem Set 11

Q1i. If we accept the assumptions underlying the Black-Scholesmodel, what should the implied volatility of a European calloption written on S with a time to expiration of 1.5 years and astrike price of $150?

A. We would expect it to be the same. Changes in the strike price is not

expected to change the volatility of the underlying stock.


12/12

Problem Set 11

Q1j. Suppose you find that the implied volatility for the option isactually 40%. What implications would that have for the volatilityand/or the Black-Scholes model?

A. Something is wrong.

Stock return may not be normally distributed.

transaction cost

something is not clicking

More research needed; how could we make the model more realisticwelcome

you guys to explore more.

Volatility smile:

FINS2624_Problem Set 11 Written Solutions

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