In: Finance
Based on a spot price of $96 and strike price of $98 as well as the fact that the risk-free interest rate is 6% per annum with continuous compounding, please undertake option valuations and answer related questions according to following instructions:
Binomial trees:
Additionally, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%.
Note: When you use no-arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the different nodes of the binomial tree.
4 month is equivalent to one period . The stock price and the options value at maturity is shown below
81.319 | 13.3186 | 0 | ||
73.3 | 65.934 | 0 | 2.066 | |
66 | 59.4 | 53.46 | 0 | 14.54 |
t=0 | t=1 | t=2 | Value of Call option at t=2 | Value of put option at |
a) Under No arbitrage approach,
From t=1 to t=2 when stock price is $73.26
Let X shares be purchased and one call option be shorted to create the no arbitrage portfolio
So, X*81.3186- 13.3186 = X*65.934 -0
=> X = 0.8657
So, Value of option(C1h) at t=1 when stock price is $73.26 is given by
0.8657*73.26 - C1h = 0.8657*65.934/exp(0.06*4/12)
=> C1h= $7.4724
Similarly From t=1 to t=2 when stock price is $59.4
X*65.934- 0 = X*53.46 -0
=> X = 0
So, Value of option(C1L) at t=1 when stock price is $59.4 is given by
0*59.4 - C1L= 0*53.46/exp(0.06*4/12)
=> C1L= 0
and From t=0 to t=1when stock price is $66
X*73.26- 7.47 = X*59.4 -0
=> X = 0.5391
So, Value of option(C) at t=0 when stock price is $66 is given by
0.5391*66 - C= 0.5391*59.4/exp(0.06*4/12)
=> C= $4.19
b) Under No arbitrage approach,
From t=1 to t=2 when stock price is $73.26
Let X shares be purchased and one put option be purchased to create the no arbitrage portfolio
So, X*81.3186 + 0 = X*65.934 +2.066
=> X = 0.1343
So, Value of option(P1h) at t=1 when stock price is $73.26 is given by
0.1343*73.26 + P1h = 0.1343*81.3186/exp(0.06*4/12)
P1h = 0.866
Similarly From t=1 to t=2 when stock price is $59.4
X*65.934+2.066= X*53.46 +14.54
=> X = 1
So, Value of option(P1L) at t=1 when stock price is $59.4 is given by
1*59.4 + P1L= (1*53.46+14.54)/exp(0.06*4/12)
=> P1L= 7.2535
and From t=0 to t=1when stock price is $66
X*73.26 + 0.866 = X*59.4 + 7.2535
=> X = 0.46086
So, Value of option(P) at t=0 when stock price is $66 is given by
0.46086*66 +P= (0.46086*59.4+7.2535)/exp(0.06*4/12)
=> P= $3.53
c) From put call parity
C+K*exp(-rt) = P +S
LHS = 4.19+ 68*exp(-0.06*8/12) = $69.53
RHS = 3.53+66 = $69.53
As LHS = RHS , the Put call parity holds
d) Under risk neutral valuation ,the risk neutral probability is given by a)
p = (exp(0.06*4/12)- 0.9)/(1.11-0.9) = 0.5724
So, Value of European call option
= (p^2*value of option when stock is $81.3186 + 2*p*(1-p)*value of option when stock is $65.934 + (1-p)^2*value of option when stock is $53.46) / exp(0.06*8/12)
= $4.19
e) Under risk neutral valuation
Value of European put option
= (p^2*value of option when stock is $81.3186 + 2*p*(1-p)*value of option when stock is $65.934 + (1-p)^2*value of option when stock is $53.46) / exp(0.06*8/12)
= (2*0.5724*0.4276*2.066+0.4276^2*14.54)/exp(0.06*8/12)
=$3.53
f) Thus, we get the same value of the Call and the Put option from both no-arbitrage approach and the risk-neutral valuation.