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The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Calculate values for u, d, and p when a two-month time step is used. What is the value of a four-month European call option with a strike price of $80 given by a two-step binomial tree. Suppose a trader sells 1,000 options (10 contracts). What position in the stock is necessary to hedge the traders position at the time of the trade

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Tundexi

Purchasing 500.00 shares of the non dividend paying stock is necessary to hedge the traders position at the time of the trade

What is Stock hedging?

Stock hedging means the act of buying investment which are designed to reduce the loss risk from another investment.

T = 4 months = 4/12 = 1/3 year  

n = 2

t = T/n = 2/2 = 2 months = 2/12 = 1/6 years

u = 1.1303

d = 1/u = 1/1.1303 = 0.8847

p = (ert - d) / (u - d)

p = (e^3% x 1/6 - 0.8847) / (1.1303 - 0.8847)

p = 48.98%

Su = S0 x u

Su = 78 x 1.1303

Su = 88.16

Sd = S0 x d

Sd = 78 x 0.8847

Sd = 69.01

Suu = u x Su

Suu = 1.1303 x 88.16

Suu = 99.65

Sud = Su x d

Sud = u x Sd

Sud = 78

Sdd = d x Sd

Sdd = 61.05

Cuu = max (Suu - K, 0)

Cuu = max (99.65 - 80, 0)

Cuu = 19.65

Cud = max (Sud - K, 0)

Cud = max (78 - 80, 0)

Cud = 0

Cdd = max (Sdd - K, 0)

Cdd = max (671.05 - 80, 0)

Cdd = 0

Cu = [p x Cuu + (1 - p) x Cud]e-rt

Cu = [48.98% x 19.65 + (1 - 48.98%) x 0]e-3% x 1/6

Cu = 9.58

Cd = [p x Cud + (1 - p) x Cdd]e-rt

Cd = 0

Value of a four-month European call option = [p x Cu + (1 - p) x Cd]e-rt = [48.98% x 9.58 + (1 - 48.98%) x 0]e^-3% x 1/6

Value of a four-month European call option = 4.67

Since the trader has sold call options, he need to buy:

= (Cu - Cd) / (Su - Sd) * N

= (9.58 - 0) / (88.16 - 69.01) * 1,000

= 500.00 shares.

In conclusion, purchasing 500.00 shares of the non dividend paying stock is necessary to hedge the traders position at the time of the trade.

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