In: Finance
(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?? |
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 |
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85 | |||||||||||
19 | A | ||||||||||||
20 | =D9 | ||||||||||||
21 |
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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 |
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=E18 | =MAX(E32-D10,0) | =MAX(E32-D10,0) | |||||||||
33 | A | ||||||||||||
34 | =C20 | ||||||||||||
35 |
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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 |