Finance-AW-Q156

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Black Scholes formula
 

ND_1>=ND_2. Explain if there is arbitrage in this equation.
Written part
Assume for a non dividend paying stock So= 80, vol= 30% p.a, r= 3% pa.cc, time= 9 months.
{40 if St <=70
{100 – St, If 70<St<110 {0 Otherwise In R (program) Value x, early exercise -> E style option not allowed. 250 step C-R-R binomial tree
* Value= 25.85 -> How?
 

Can x be created by combination of bond (face value 40) short call at strike 70 and long call at strike 110? Explain!
 

 St<=7070<St<=110St>100
Bond404040
SC70070(k)-St70- St
LC11000St – 110
 40110-St0

 
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If you price separately EC70= 14.78 EC110= 1.52
39.11-14.78+1.52= 25.85 -> price of x
 
Show x can be created by only 2 EP options separately by using binomial tree
 
Value the 2 option with 250 steps. Show that the portfolio of these 2 options has the same value as X
 

Excel
 

3 months “down and in” call option N-D paying stock, Stock price= 100, Strike price =100, rf= 8%, vol=30% pa, barrier 95, time intervals= 200, 500 simulations.
Calculate the 95% confidence interval of the option price.
 

12 months Bermudian style put option N-D paying stock
Stock price =100, Strike price= 100, rf=3% pa.cc, vol= 30% pa. Early exercise on first day of 3rd, 6th and 9th month.
 

252 day in one year. Price the option. 100 step binomial tree.
2% dividend yield pa.cc on the stock?
 

Theory question – S follows a process dS= mSdt + oSdz where m and o are constant. What is the probability followed by Y=〖Se〗^((r-t)). If S follows a process dS= k(b-S)dt + oSdz where k,b,o are constant. What’s the process followed by Y =S^2?
 

12 months, EC option N-D paying stock , rf= 3% pa.cc, vol= 30% pa
 

In R:
Plot the gamma of the call option by assuming that the strike price =100 and it varies from $0 to $200 with a step size of 0.01. Label the x-axis, y-axis as Gamma and the title as Gamma options.
 
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