Question

Q3.

Frank is going to sell his 100,000 Google stock shares in three months. The current price is $5/share. To reduce the risk, he wants to buy a put option. Assume the price of Google share either goes up by 5% or down by 4% each month, though we can’t predict which event will happen exactly. The risk free interest rate is 1% each month. Gina sells Frank a put option. The strike price is 100,000 shares for $500,000.

a) What’s the price of this put option?

b) Describe Gina’s hedging portfolio two months after the option is issued.

Q4.

Carefully state the put-call parity. If put-call parity fails to hold, describe and confirm the arbitrage strategies.

Answer 1

Price   5       3month    1% per month
UP      5%           FSP = Future Spot Price  
Down 4%              
X or strike price   = 5              
Upward FSP   5.25              
Down FSP   4.8              
                  
Situation    FSP   Vc        
I                  5.25   -0.25          
II   4.8   0          
Change       0.45      -0.25          
                  
STRATEGY:                  
WRITE A CALL AND HOLD DELTA SHARES                  
                  
Calculation of Delta Shares                  
Delta shares = Change in VC        =   0.555555556 shares  
                       Change in FSP                  
Future value of portfolio/ maturity value of Portfolio                
Situation   FSP    Share Value   Vc          Portfolio value  
I               5.25      2.916666667   -0.25   2.666666667  
II                4.8      2.666666667   0           2.666666667  
Now we have created a risk less portfolio as we can see that the                  
in both situations the value of portfolio does not change !                  
                  
Present value of portfolio                 
The future value of portfolio is not influenced by FSP and is certain.                  
So to calculate Present Value, we can use Riskfree rate (ie 1%)                  
                  
So value (of portfolio) is discounted using continous discounting model                 
   2.66666667 / 1.01005      = 2.640133327      
                  
                  
Present value of delta shares                 
   Delta shares x Spot price              
   Delta shares       0.555555556      
   Spot Price                          5      
                       =      2.777777778     

Value of call                  
Value of Portfolio = Value of call +    Value of Delta shares              
2.640133 = Vc + 2.777778                  
Vc   0.137644451              
                  
Value of Put                
Put Call Parity equation                  
Vc + PVx = Vp + Vs                  
Present valueof strike price    5 / 1.01005       4.950249988      
                  
So Vp is                  
   0.087894439              

PUT CALL PARITY EQUATION

Vp+Vs = Vc+ PVx (where Vp is value of put, Vs is value of shares, Vc is value of call and PVx is Present value of spot price.

NOTE:

If the prices of the put and call options diverge so that this relationship does not hold good, an arbitrage opportunity exists, meaning that sophisticated traders can THEORETICALLY earn a risk-free profit. Such opportunities are uncommon and short-lived in liquid markets.