1. No category

ACTS 4302
Instructor: Natalia A. Humphreys
SOLUTION TO HOMEWORK 8
Lesson 14: Gap, Exchange, and Other Options.
Problem 1
For a stock you are given:
(i)
(ii)
(iii)
(iv)
(v)
The
The
The
The
The
stock’s current price is 80.
stock’s price follows the Black-Scholes framework.
stock pays continuous dividends proportionate to its price at a rate of 0.04.
stock’s annual volatility is 0.2.
continuously compounded risk-free interest rate is 0.02.
A special option expires in 6 months and will pay 0.3 times the price of the stock at that time, if it is
greater than 90. Otherwise it pays nothing.
Determine the price of the option.
Solution. We are given: S0 = 80, δ = 0.04, r = 0.02, σ = 0.2. We need to calculate the price of the
following special option:
0.3 · S, S > 90
SO =
0,
S ≤ 90
The value of the asset-or-nothing call option S|S > K is S0 e−δT N (d1 ). Calculating, we obtain:
80
+ 0.02 − 0.04 + 0.5 · 0.22 0.5
ln 90
ln (S/K) + (r − δ + 12 σ 2 )t
0.1178
√
√
=−
= −0.8329
=
d1 =
0.1414
σ t
0.2 0.5
N (d1 (90)) = N (−0.83) = 0.2033
SO = e−0.04·0.5 · 0.3 · 80 · 0.2033 = 4.7826
Problem 2
For a stock whose price at time t is S(t), you are given:
(i)
(ii)
(iii)
(iv)
S(0) = 60
The stock’s price follows the Black-Scholes framework.
The stock pays continuous dividends proportionate to its price at a rate of 0.02.
The continuously compounded risk-free interest rate is 0.07.
An option will pay S(1) at the end of one year if S(1) > 80. Otherwise it will pay nothing. The price
of this option is 3.35. Determine the volatility of the stock.
Solution. We are given: S0 = 60, δ = 0.02, r = 0.07. We need to find σ. We are given the value of
the following special option:
S1 , S1 > 80
SO =
0, S1 ≤ 80
The value of the asset-or-nothing call option S|S > K is S0 e−δT N (d1 ). Calculating, we obtain:
SO = 60e−0.02 · N (d1 (80)) = 58.8119N (d1 (80)) = 3.35 ⇒
3.35
N (d1 (80)) =
= 0.05696 ≈ 0.0571 ⇒ d1 (80) = −1.58
58.8119
Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. SOLUTION TO HOMEWORK 8.
By definition of d1 :
ln 60
+ 0.07 − 0.02 + 0.5 · σ 2 1
ln (S/K) + (r − δ + 12 σ 2 )t
80
√
√
d1 =
=
= −1.58 ⇔
σ t
σ 1
−0.2377 + 0.5σ 2
− 1.58 ⇔ 0.5σ 2 + 1.58σ − 0.2377 = 0
σ
D = b2 − 4ac = 1.582 + 4 · 0.5 · 0.2377 = 2.9718 = 1.72392
−1.58 + 1.7239
σ1 =
= 0.1439 σ2 < 0 ⇒ rejected
1
Thus, the volatility is σ = 0.1439 .
Problem 3
For a stock whose price at time t is S(t), you are given:
(i) S(0) = 50
(ii) The stock’s price at the end of 6 months follows a lognormal distribution.
(iii) The stock pays continuous dividends proportional to its price at a rate of 0.03.
(iv) The continuously compounded risk-free interest rate is 0.08.
An asset-or-nothing call option with strike price 40 expiring in 6 months has price 37.89. Calculate the
price of an asset-or-nothing put option with strike price 40.
Solution. The value of the asset-or-nothing call (AONC) option S|S > K is S0 e−δT N (d1 ). The value
of the asset-or-nothing put (AONP) option S|S < K is S0 e−δT N (−d1 ). Therefore,
AON P = S0 e−δT N (−d1 ) = S0 e−δT (1 − N (d1 )) = S0 e−δT − AON C =
= 50e−0.03·0.5 − 37.89 = 49.2556 − 37.89 = 11.3656
Problem 4
You are given the following prices for a stock:
Date
Price
Apr. 1
57
Apr. 2
71
Apr. 3
63
Apr. 4
52
Consider the following portfolio of options purchased on Apr. 1:
1. 3-day Asian arithmetic average price call, strike price 61.
2. 3-day Asian arithmetic average strike put.
3. 3-day up-and-in barrier call with barrier of 65, strike price 55.
4. 3-day up-and-in barrier put with barrier of 65, strike price 55.
5. 3-day gap put option with trigger 65, strike price 55.
6. 3-day gap call option with trigger 45, strike price 57.
Calculate the total payoff on Apr. 4 from this portfolio of options.
Solution. Let us calculate the payoff of each option in the portfolio
1. 3-day Asian arithmetic average price call, strike price 61.
71 + 63 + 52
= 62, 62 − 61 = 1
3
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Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. SOLUTION TO HOMEWORK 8.
2. 3-day Asian arithmetic average strike put.
max(62 − 52, 0) = 10
3. 3-day up-and-in barrier call with barrier of 65, strike price 55. The barrier is triggered, but the
payment is still 0:
max(0, 52 − 55) = 0
4. 3-day up-and-in barrier put with barrier of 65, strike price 55. The barrier is triggered, and the
payment is:
max(0, 55 − 52) = 3
5. 3-day gap put option with trigger 65, strike price 55.
S = 52 < K2 = 65 ⇒ K1 − S = 55 − 52 = 3
6. 3-day gap call option with trigger 45, strike price 57.
S = 52 > K2 = 45 ⇒ S − K1 = 52 − 57 = −5
Thus, the total payoff is:
1 + 10 + 0 + 3 + 3 − 5 = 12
Problem 5
For a 3-month European gap call option on a stock:
(i) The stock’s price is 37.
(ii) The strike price is 45.
(iii) The trigger is 35.
(iv) The stock’s annual volatility is 25%.
(v) The stock pays continuous dividends proportional to its price at a rate of 0.03.
(vi) The continuously compounded risk-free interest rate is 0.07.
Determine the Black-Scholes option premium.
(A) −3.49
(B) 0.15
(C) 0.85
(D) 3.18
(E) 4.01
Key: A
Solution. In this problem K1 = 45 and K2 = 35.
C = Se−δt N (d1 (K2 )) − K1 e−rt N (d2 (K2 )), where
+ 0.07 − 0.03 + 0.5 · 0.252 · 0.25
ln 37
ln (S/K2 ) + (r − δ + 0.5σ 2 )t
35
√
=
=
d1 (K2 ) =
0.25 · 0.5
σ t
0.0734
=
= 0.5871
0.125
N (d1 (K2 )) = N (0.5871) = 0.7214
√
d2 (K2 ) = d1 − σ t = 0.5871 − 0.125 = 0.4621
N (d2 (K2 )) = N (0.4621) = 0.6780
C = 37 · e−0.03·0.25 · 0.7214 − 45e−0.07·0.25 · 0.6780 = −3.4868 ≈ -3.49
The premium is negative because of the possibility of the buyer paying the seller if the stock price is
between 35 and 45.
Problem 6
Assume the Black-Scholes framework.
For a 100 1-year U.S. dollar denominated European gap call options on Canadian dollars:
(i) The spot exchange rate is 0.95 US$/C$.
(ii) The strike price is 0.9.
(iii) The trigger is 1.0.
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Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. SOLUTION TO HOMEWORK 8.
(iv) The continuously compounded risk-free rate for U.S. dollars is 4%.
(v) The continuously compounded risk-free rate for Canadian dollars is 6%.
(vi) The relative volatility of Canadian dollars to U.S. dollars is 9%.
Determine the total value of such gap options.
(A) −1.18
(B) 1.02
(C) 2.95
(D) 4.88
(E) 5.35
Key: C
Solution. We are given: x0 = 0.95, K1 = 0.9, K2 = 1, rd = rU SD = r = 0.04, rf = rCD = 0.06 =
δ, σ = 0.09.
C = x0 e−rf t N (d1 (K2 )) − K1 e−rt N (d2 (K2 )), where
ln 0.95
+ 0.04 − 0.06 + 0.5 · 0.092 · 1
ln (x0 /K2 ) + (r − rf + 0.5σ 2 )t
1
√
d1 (K2 ) =
=
=
0.09 · 1
σ t
0.0672
=−
= −0.7471
0.09
N (d1 (K2 )) = N (−0.7471) = 0.2275
√
d2 (K2 ) = d1 − σ t = −0.7471 − 0.09 = −0.8371
N (d2 (K2 )) = N (−0.8371) = 0.2013
C = 0.95 · e−0.06·1 · 0.2275 − 0.9e−0.04·1 · 0.2013 = 0.2027 − 0.1734 = 0.0295 ⇒
100C = 2.95 Problem 7
For a 6-month European gap put option on a stock:
(i) The stock’s price is 70.
(ii) The trigger is 65.
(iii) The stock’s annual volatility is 30%.
(iv) The stock pays continuous dividends proportional to its price at a rate of 0.02.
(v) The continuously compounded risk-free interest rate is 0.05.
(vi) Options are priced using the Black-Scholes formula.
Calculate the strike price to make the option price 0.
(A) 56.47
(B) 57.68
(C) 62.17
(D) 79.88
(E) 90.97
Key: A
Solution. We are given: T = 0.5, S0 = 70, K2 = 65, σ = 0.3, δ = 0.02, r = 0.05, r − δ = 0.03.
P = K1 e−rT N (−d2 (K2 )) − Se−δT N (−d1 (K2 )), where
2 · 0.5
ln 70
+
0.03
+
0.5
·
0.3
0.1116
ln (S/K2 ) + (r − δ + 0.5σ 2 )t
√
√
= 65
=
= 0.5261
d1 (K2 ) =
0.2121
σ t
0.3 · 0.5
N (−d1 (K2 )) = N (−0.5261) = 0.2994
√
d2 (K2 ) = d1 − σ t = 0.5261 − 0.2121 = 0.314
N (−d2 (K2 )) = N (−0.314) = 0.3768
70 · e−0.02·0.5 · 0.2994
K1 =
= 56.4673 ≈ 56.47 e−0.05·0.5 · 0.3768
Problem 8
For a 1-year European exchange option, you are given:
(I)
(i) Stock I has price S0 = 55.
(II)
(ii) Stock II has price S0 = 65.
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Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. SOLUTION TO HOMEWORK 8.
(I)
(I)
(iii) The option allows acquiring Stock II by paying 1.5S1 , where S1
expiry.
(iv) Stock I pays dividends at a continuous annual rate of 0.04.
(v) Stock II pays dividends at a continuous annual rate of 0.07.
(vi) Stock I has annual volatility 0.25.
(vii) Stock II has annual volatility 0.45.
(viii) The continuous annual risk-free rate rate is 3%.
(ix) The correlation between the 2 stocks is 0.7.
Determine the Black-Scholes option premium.
(A) 2.62
(B) 3.59
(C) 5.22
(D) 9.93
is the price of Stock S (I) at
(E) 21.28
Key: A
Solution. The risk-free rate is irrelevant.
C = Se−δS T N (d1 ) − Qe−δQ T N (d2 ), where
√
ln (S/Q) + (δQ − δS + 0.5σ 2 )T
√
, d2 = d1 − σ T
d1 =
σ T
In our problem: S = S (II) , Q = 1.5 · S (I)
The relative volatility is
1
1
1
2
σ = σI2 + σII
− 2 · ρ · σI · σII 2 = 0.452 + 0.252 − 2 · 0.7 · 0.45 · 0.25 2 = (0.1075) 2 = 0.3279
The Black-Scholes exchange option premium is:
C = S (II) e−δII T N (d1 ) − 1.5 · S (I) e−δI T N (d2 ), where
ln S (II) /1.5 · S (I) + (δI − δII + 0.5σ 2 )T
ln 65 + (0.04 − 0.07 + 0.5 · 0.1075) · 1
√
d1 =
= 1.5·55
=
0.3278 · 1
σ T
0.2147
=−
= −0.6547, N (d1 ) = N (−0.6547) = 0.2563
0.3278 √
d2 = d1 − σ T = −0.9826, N (d2 ) = N (−0.9826) = 0.1629
C = 65 · e−0.07·1 · 0.2563 − 82.5e−0.04·1 · 0.1629 = 15.5349 − 12.9128 = 2.6221 ≈ 2.62
⇒ Answer A. Problem 9
For a stock you are given:
(i) The stock’s price is 98.
(ii) The stock pays continuous dividends proportionate to its price at a rate of 0.04.
(iii) The continuously compounded risk-free interest rate is 0.04.
An option allows you to choose, at the end of 9 months, between 105-strike European put and call
options expiring at the end of 12 months from now.
You are given the following prices for European call options with strike price 105:
Expiry
Option price
3 months
6.30
6 months
10.57
9 months
11.53
12 months
12.85
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Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. SOLUTION TO HOMEWORK 8.
Determine the value of the chooser option.
Solution.
V = C(S, K, T ) + e−δ(T −t) P S, Ke−(r−δ)(T −t) , t
C(S, K, T = 1) = 12.85
C = Se−δt N (d1 ) − Ke−rt N (d2 )
Since
δ = r, r − δ = 0 ⇒ P S, Ke−(r−δ)(T −t) , t = P (S, K, t)
By Put Call parity,
P = C − Se−δT + Ke−rT
P (S, K, t = 0.75) = 11.53 + e−0.04·0.75 (105 − 98) = 18.3231
V = 12.85 + e−0.04·0.25 · 18.3231 = 30.9908 ≈ 30.99
Problem 10
For a nondividend paying stock, you are given:
(i) The stock’s price is 70.
(ii) The stock’s volatility is 30%.
(iii) The continuously compounded risk-free interest rate is 0.05.
An option allows you to choose, at the end of 3 months, between at-the-money (i.e. S=K) European
call and put options expiring at the end of 8 months from now.
Using the Black-Scholes framework, determine the value of this chooser option.
Solution. We are given: S0 = 70, σ = 0.3, t = 0.25, T =
8
12 ,
T −t=
5
12 ,
r = 0.05, δ = 0.
V = C(S, K, T ) + e−δ(T −t) P S, Ke−(r−δ)(T −t) , t
C(S, K, T ) = Se−δT N (d1 ) − K1 e−rT N (d2 ), where
70
+ 0.05 + 0.5 · 0.32 ·
ln 70
ln (S/K) + (r − δ + 0.5σ 2 )T
√
q
d1 =
=
8
σ T
0.3 · 12
8
12
=
0.0633
= 0.2586
0.2449
N (d1 ) = N (0.26) = 0.6026
√
d2 = d1 − σ t = 0.2586 − 0.2449 = 0.0136
N (d2 ) = N (0.01) = 0.5040
=
8
C = 70 · 0.6026 − 70e−0.05· 12 · 0.5040 = 42.182 − 34.1234 = 8.0586
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Copyright ©Natalia A. Humphreys, 2014
ACTS 4302. AU 2014. SOLUTION TO HOMEWORK 8.
5
K1 = Ke−(r−δ)(T −t) = 70e−0.05· 12 = 68.5568
P = K1 e−rt N (−d2 ) − Se−δt N (−d1 ), where
70
ln 68.5568
+ 0.05 + 0.5 · 0.32 · 0.25
√
= 0.2972 ≈ 0.3
d1 =
0.3 · 0.25
N (−d1 ) = N (−0.3) = 0.3821
√
d2 = d1 − σ t = 0.2972 − 0.15 = 0.1472 ≈ 0.15
N (−d2 ) = N (−0.15) = 0.4404
P = 68.5568e−0.05·0.25 · 0.4404 − 70 · 0.3821 = 29.8174 − 26.747 = 3.0703
V = 8.0586 + 3.0703 = 11.129 ≈ 11.13 Page 7 of 7