A1
Fixed leg: the cash flow will be 150000 at month 2, 150000 at month 5 and 10150000 at
month 8. The present value of the fixed payments would be:
150,000*e^(-0.045*2/12) + 150,000*e^(-0.047*5/12) + 10,150,000*e^(-0.048*8/12) =
10,126,312.09
Assuming that the 3 month SOFR reference rate in the last payment date can be
calculated from the 2 month continuous compounding risk free rate and past 1 month
risk free rate:
e^(0.045 * 2/12) * e^(0.04 * 1/12) = e^(SOFR * 3/12)
SOFR = 4.333%
The present value of the float payments would be:
10,000,000*(1+4.333%/4)*e^(-0.045*2/12) = 10,108,333.33
PHB Corp will pay fix and receive float, thus the value of the swap to PHB Corp is:
Vfloat – Vfix = 10,108,333.33 - 10,126,312.09 = -17,978.76
A2
a, Present value of dividend: 2*exp(-0.05*3/12) = 1.975
Put call parity for European options with dividend:
c + K*e^(-rt) = p + S0 – d
c = p + S0 – d - K*e^(-rt) = 4.5 + 70 – 1.975 – 72*e^(-0.05*6/12) = 2.3025
The price of the European call option with maturity of 6 months and strike price of 72
should be 2.3025
b, If the European call price is 1.5, there will be arbitrage opportunities as the actual call
price is lower than the theoretical call price. Since the call is underpriced, to make the
arbitrage profit, we should buy the call, short the put and short stock.
Action now:
-
Buy call for 1.5
Short stock for 70
Short put for 4.5
invest the proceed of 73
Action in 6 months:
-
-
Get the investment proceed: 73*e^(0.05*6/12) = 74.848
If spot price is less than 72, the put option will be exercise and we will buy the
stock for 72. If spot price is more than 72, exercise the call option to buy the
stock for 72.
Pay the dividend. The future value of dividend in 6 months is: 2*exp(0.05*3/12) =
2.025
Thus, the profit after 6 months is 74.848 – 72 - 2.025 = 0.823
A3
a, Up movement u = e^(sigma * sqrt(delta t)) = e^(0.24*sqrt(1/12)) = 1.0717
Down movement d = 1/u = 0.933
b, Risk neutral probability:
p = (e^(r*delta t) – d)/(u-d) = 0.5098
c, Assume the current IBM stock price is 227.48
Binominal tree for stock price movement:
T=0
T=1
T=2
261.2888
243.7991
227.48
227.48
212.2533
198.0458
d, No arbitrage approach
Call option payoff after 2 month period: max(ST-K,0)
fuu = max(261.2888 – 230,0) = 31.2888
fud = max(227.48 – 230,0) = 0
fdd = max(198.0458 – 230,0) = 0
T=0
T=1
T=2
D
261.2888
31.2888
B
243.7991
15.89068
A
227.48
8.07041
E
227.48
0
C
212.2533
0
F
198.0458
0
Delta when going from Node D and E back to node B:
∆ = (fu – fd)/(Su – Sd) = (31.2888 – 0)/(261.2888 - 227.48) = 0.92546
Payoff at node B: f = S*∆ - (Su*∆ - fu) * e^(-rT)
= 243.7991*0.92546 – (261.2888*0.92546 - 31.2888) * e^(-0.045*1/12) = 15.8907
Payoff at node C should be 0, as the payoff at node E and F are all zero
Delta when going from Node B and C back to node A:
∆ = (fu – fd)/(Su – Sd) = (15.8907 – 0)/(243.7991 - 212.2533) = 0.5037
Payoff at node A: f = S*∆ - (Su*∆ - fu) * e^(-rT)
= 227.48*0.5037 – (243.7991*0.5037 - 15.8907) * e^(-0.045*1/12) = 8.0704
Thus, the call option value using no arbitrage approach is 8.0704
e, Risk neutral approach
p = 0.5098 (from part b)
1-p = 0.4902
Discounting back payoff from node D and E, the payoff in node B is:
fu = e^(-rt) * (p*fuu + (1-p)*fud) = e^(-0.045*1/12) * (0.5098*31.2888 + 0.4902*0)
= 15.89068
The payoff at node C should be 0, as the payoff at node E and F are all zero
Discounting back payoff from node B and C, the payoff in node A is:
f = e^(-rt) * (p*fu + (1-p)*fd) = e^(-0.045*1/12) * (0.5098*15.89068+ 0.4902*0)
= 8.0704
Thus, the call option value using risk neutral approach is 8.0704
f, Put option payoff at expiration: max(0, K-ST)
Payoff at node D: max(0, 230 – 261.2888) = 0
Payoff at node E: max(0, 230 – 227.48) = 2.52
Payoff at node F: max(0, 230 – 198.0458) = 31.9542
T=0
T=1
T=2
D
261.2888
0.0000
B
243.7991
1.2307
A
227.48
8.8719
E
227.48
2.52
C
212.2533
16.8858
F
198.0458
31.95421
Discounting back payoff from node D and E, the payoff in node B is:
fu = e^(-rt) * (p*fuu + (1-p)*fud) = e^(-0.045*1/12) * (0.5098*0 + 0.4902*2.52) = 1.2307
Discounting back payoff from node E and F, the payoff in node C is:
fd = e^(-rt) * (p*fud + (1-p)*fdd) = e^(-0.045*1/12) * (0.5098*2.52 + 0.4902*31.9542)
= 16.8858
Discounting back payoff from node B and C, the payoff in node A is:
f = e^(-rt) * (p*fu + (1-p)*fd) = e^(-0.045*1/12) * (0.5098*1.2307+ 0.4902*16.8852)
= 8.8719
Thus, the price of two-month European put option is 8.8719
g, For the American put option, the payoff at node D, E, F are still 0, 2.52 and 31.9542
Discounting back payoff from node D and E, the payoff in node B is 1.2307.
T=0
T=1
T=2
D
261.2888
0
B
243.7991
1.2307
A
E
227.48
9.2923
227.48
2.52
C
212.2533
17.7467
F
198.0458
31.9542
If exercising the option at node B, the payoff is max(0, 230 - 243.7991) = 0. Thus there
is no incentive to exercise early at node B.
Discounting back payoff from node E and F, the payoff in node C is 16.8858.
If exercising the option at node B, the payoff is max(0, 230 - 212.2533) = 17.7467. Thus
the option should be exercised early at node C, and the payoff at node C is 17.7467.
Discounting back payoff from node B and C, the payoff in node A is:
f = e^(-rt) * (p*fu + (1-p)*fd) = e^(-0.045*1/12) * (0.5098*1.2307+ 0.4902*17.7467)
= 9.2923
If exercising the option at node A, the payoff is max(0, 230 – 227.48) = 2.52. Thus there
is no incentive to exercise early at node A.
Thus, the American put option price is 9.2923
h, The American call option price should be equal to the European call option price if
there is no dividend during the option life, as there is no incentive to exercise the call
option early. Holding the call option would offer several benefits compared to exercising
the call option and holding the IBM share
-
No income is sacrificed as there is no dividend payment
Delay the payment of strike price until expiration date. That amount could be
invested to get some interest
Having insurance if stock price falling below strike price, as option holder may
choose not to exercise the option and limit the loss to the premium paid.
i, Call price: 8.0704; Put price: 8.8719
Put call parity formula: c + K*e^(-rT) = p + S0
Left hand side: 8.0704 + 230*e^(-0.045*2/12) = 236.3519
Right hand side: 8.8719 + 227.48 = 236.3519
Thus, the put call parity holds
j, The actual put option price with strike price of 230 is (16.95 + 18.35)/2 = 17.65, which
is different from the valuation of 8.8719 from risk neutral pricing. Some potential
reasons for the difference is:
-
-
Model assumption: the risk neutral method relies on some assumption and
estimated volatility (24%) and interest rates (continuous compounding of 4.5%),
which may not appropriate with the market condition
Dividends: there may be some dividend payment of IBM during option life, which
is not covered in the calculation
Transaction cost: the risk neutral pricing does not consider transaction costs,
such as commission, bid-ask spread in the calculation.
k,
Delta of 1 call option is 0.52, thus the delta of the portfolio should be -500 * 0.52 = -260.
For every $1 increase in the IBM share, the portfolio value would decrease by $260
Gamma of 1 call option is 0.0092, and the delta of the portfolio should be -500 * 0.0092
= -4.6. For every $1 increase in the IBM share, the portfolio delta will decrease by 4.6
Rho of 1 call option is 0.2123, and the rho of the portfolio should be -500 * 0.2123
= -106.15. For every 1% increase in risk free rate, the portfolio value will decrease by
106.15
Vega of 1 call option is 0.4146, and the vega of the portfolio should be -500 * 0.4146
= -207.3. For every 1% increase in volatility, the portfolio value will decrease by 4207.3
Theta of 1 call option is -0.1248, and the theta of the portfolio should be -500 * -0.1248
= 62.4. The portfolio value would increase by $62.4 by each day, if everything else
remains the same.
A4
The present value of coupon payment:
I = 12*e^(-0.09*0.5) + 12* e^(-0.09*1.5) = 21.9566
Bond forward price
Fb = (B0 – I)*e^rT = (120 - 21.9566) * e^(0.09*2) = 117.3793
d1 = (ln(Fb/K) + (sigma^2)*T/2) / (sigma*sqrt(T)) = -0.4994
d2 = d1 – sigma*sqrt(T) = -0.4994 – 0.08*sqrt(2) = -0.6126
N(-d1) = 0.6913
N(-d2) = 0.7299
p = e^(-rT) * (K * N(-d2) – Fb * N(-d1)) = 8.4362