Assume that you have the rights to a coal mine and the most recent valuation of
ID: 2794166 • Letter: A
Question
Assume that you have the rights to a coal mine and the most recent valuation of the mine was £6.6 million. Because of increasing demand from Asia, the price of similar mines has grown by 15 per cent per annum, with an annual standard deviation of 25 per cent. A buyer has recently approached you and wants an option to buy the mine in the next 12 months for £6.9 million. The risk-free rate of interest is 5 per cent per year, compounded continuously.
How much should you charge for the option?
Assume that you have the rights to a coal mine and the most recent valuation of the mine was £6.6 million. Because of increasing demand from Asia, the price of similar mines has grown by 15 per cent per annum, with an annual standard deviation of 25 per cent. A buyer has recently approached you and wants an option to buy the mine in the next 12 months for £6.9 million. The risk-free rate of interest is 5 per cent per year, compounded continuously.
Explanation / Answer
Buyer wants to buy the right to buy which means it's a call option.
Most recent valuation = Spot rate = S = 6.6 Mn
Option to buy = Strike Price = K = 6.9 Mn
Standard Deviation = s = 25%
Riskfree rate = Rf = 5%
Time period = t = 1 year
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Formulae one should memorise:
Call option value = S * N(d1) - K * e-Rf * t * N(d2)
d1 = [ln(S/K) + (Rf + {s2/2} )* t] / s * underroot(t)
d2 = d1 - s * underroot(t)
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d1 = [ln(S/K) + (Rf + {s2/2} )* t] / s * underroot(t)
= ln (6.6/6.9) + ( 0.05 + 0.252/2) *1 / 0.25 * 1 = 0.1471929497
d2 = d1 - s * underroot(t)
= -0.1028070503
N(d1) = 0.55851 ( calculate using excel function '=normsdist(0.1471929497)
N(d2) = 0.45906 ( calculate using excel function '=normsdist(-0.1028070503)
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Call option value = S * N(d1) - K * e-Rf * t * N(d2)
= 6.6 Mn * 0.55851 - 6.9 Mn * e-0.05 * 0.45906
= 0.673133 Mn = 673133.48 pounds