Question

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

CONSIDER THE FOLLOWING STOCK

A stock is currently priced at $50. (S)

In six months, it will be either $55 (Su ) or $45 (Sd)

The risk-free rate is 10% (r)

Price of a call option = $3.60 (f) (i.e. expected value of payoff) -fput = $1.16

Delta = 0.50 and V1 (v hat) = $22.50 in arb model

p = 0.7564 in the risk neutral case.   (probability of UP S.O.N., also shown as PR(increase)

What are the values of a six-month call and put option with K=$50??? (strike price)

Answer 1

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)	0.1
9		Current Price, S0	50
10		Exercise Price, X	50
11
12		Period	1
13		Step Period (h)	=6/12	=6/12
14
15		Possible Stock Prices
16
17				B
18				55
19		A
20		=D9
21
22				45
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