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Malmentier SA stock is currently priced at $85, and it does not pay dividends. The instantaneous risk-free rate of return is 5%. The instantaneous standard deviation of Malmentier SA stock is 25%. You want to purchase a put option on this stock with an exercise price of $90 and an expiration date 30 days from now. According to the Black-Scholes OPM, you should hold __________ shares of stock per 100 put options to hedge your risk.

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Answer:

you should hold 76 shares of stock per 100 put options to hedge your risk.

Explanation:

Current stock price, S = $85

Risk-free rate of return, r = 5%

Standard Deviation, v = 25%

Exercise price, X = $90

expiration date, t (in years) = 30 days = 1 month = 1/12 = 0.083333 years

The option price (OP) is given by the formula:

[tex]OP = Xe^{-rt} * N(-d_{2} ) - S*N(-d_1)[/tex]

[tex]d_1 = [ln(S/X) + (r + v^{2} /2)t]/vt^{0.5}\\d_1 = [ln(85/90) + (0.05 + 0.25^{2} /2)*0.08333]/(0.25*0.08333^{0.5})\\d_1 = -0.6982[/tex]

[tex]d_2 = d_1 - (vt^{0.5})\\d_2 = -0.6982 - (0.25*0.08333^{0.5})\\d_2 = -0.7704[/tex]

Using the pro-metric calculator for the cumulative normal distribution:

N(-d1) = N(- (-0.6982)) = N(0.6982) = 0.75747

N(-d2) = N(-(-0.7704)) = N(0.7704) = 0.77947

[tex]OP = Xe^{-rt} * N(-d_{2} ) - S*N(-d_1)[/tex]

[tex]OP =[ 90e^{(-0.05*0.08333)} * 0.77947] - (85*0.75747)\\OP = 5.48[/tex]

Note that N(-d₁) = 0.76

This means that 76/100 (i.e to hedge your risk, you should hold 76 per 100 put options )

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