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(Derivatives & Risk Management - BOPM: Binomial Options Pricing Model) CONSIDER THE FOLLOWING STOCK A stock...

(Derivatives & Risk Management - BOPM: Binomial Options Pricing Model)

CONSIDER THE FOLLOWING STOCK

A stock is currently priced at $80. (S)

In 115 days, it will be either $85 (Su ) or $75 (Sd)
The risk-free rate is 5% (r)

The strike price is $83 (K)

f = $1.24 (i.e. expected value of payoff)

fput = $2.92 (price of a put option)

Delta = 0.20 and V1 (v hat) = $15 in arb model

p = 0.628804 in the risk neutral case.  

(probability of UP S.O.N., also shown as PR(increase)

What are the call and put prices for options with 115-day maturities??

Solutions

Expert Solution

Formula Sheet

A B C D E F G H I J K L M
2 Valuation of European call option:
3
4 Call option gives option buyer the right to buy the Stock at a srike price.
5 Payoff of call option = Max(S-X,0) where S is stock price and X is exercise price.
6
7 Given the following data:
8 Risk free rate (rf) 0.05
9 Current Price, S0 80
10 Exercise Price, X 83
11
12 Period 1
13 Step Period (h) =115/365 =115/365
14
15 Possible Stock Prices
16
17 B
18
85
19 A
20 =D9
21
22 75
23
24 Calculation of upside and downside change ratio
25 u =upside factor =E18/C20 =E18/C20
26 d=downside factor =E22/C20 =E22/C20
27
28 Call option gives option buyer the right to buy the Stock at a srike price.
29 Payoff of call option = Max(S-X,0) where S is stock price and X is exercise price.
30
31 B Payoff of call option.
32
=E18 =MAX(E32-D10,0) =MAX(E32-D10,0)
33 A
34 =C20
35
36 =E22 =MAX(E36-D10,0) =MAX(E36-D10,0)
37
38
39
40 Probability of rise is given by following equations:
41
42 Probability of rise, p = (Exp(rf*h)-d)/(u-d)
43
44
45 Probability of rise, p =(EXP(D8*D13)-D26)/(D25-D26)
46 1-p =1-D45
47
48
49 Value of call option is the present value of expected payoff of call option in future.
50
51 Value of call option =(p*Payoff in case of rise+(1-p)*Payoff in case of fall in Price)*EXP(-r*h)
52 =(D45*F32+(1-D45)*F36)*EXP(-D8*D13) =(D45*F32+(1-D45)*F36)*EXP(-D8*D13)
53
54 Hence value of call option is =D52
55
56 Put call parity for non-dividend paying stock is given by
57 c + PV(X)= p + S0
58 Where c and p is value of call and put option at srike price of X and period T and PV(X) is the present value of strike price,
59 Using the above equation and given the call value, value of put option can be calculated as below
60 p=c + PV(X)- S0
61
62 Using following data,
63 Risk free rate (rf) =D8
64 Current Price, S0 =D9
65 Exercise Price, X =D10
66 Maturity(Years) =D13
67 Value of call option =D54
68
69 Value of put option can be calculated as follows:
70 Value of put option (p) =c + PV(K)- S0
71 =D67+D65/((1+D63)^D66)-D64 =D67+D65/((1+D63)^D66)-D64
72
73 Hence value of put option is =D71
74

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