FINS2624_Problem Set 11 Written Solutions
Transcript of FINS2624_Problem Set 11 Written Solutions
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FINS2624 Portfolio
Management
Tutorial 11 Week 12
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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: