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Risk Neutral Method

 

  • The assumption is that investors are indifferent to risk.
    • Step 1: Determine the probabilities associated with the different pay-offs
    • Step 1: Determine expected cash flow under the assumption that investors are indifferent to risk
    • Step 2: Discount the expected cash flow at the risk-free rate to arrive at the present value
  • In our example, since the risk-free rate for six months is 2%, and investors are indifferent to risk, it follows that:
    • Expected return= [probability of rise * 33.33] + [(1- probability of rise) * (-25)] = 2.0 percent
    • Therefore the probability of rise, p, = 0.463 or 46.3%
  • Expected future value of the call option after six months is given by
    • [Probability of rise * 28.33] + [(1- probability of rise) * 0]
    • = (0.463 * 28.33) + (0.537 * 0)
    • = Rs. 13.16
  • The value today therefore is PV(13.16) = Rs. 12.86

Example

Currently, shares of ABC Corp. trade at USD 100. The monthly risk neutral probability of the price increasing by USD 10 is 30%, and the probability of the price decreasing by USD 10 is 70%. What are the mean and standard deviation of the price after 2 months if price changes on consecutive months are independent? (FRM 2010 Sample Paper)
 

Solution

Develop a 2 step tree.
Mean = 9% (120) + 42% (100) + 49% (80) = 92
Variance = 9% (120 – 92)2 + 42% (100 – 92)2 + 49% (80 – 92)2 = 168
Thus, standard deviation = 12.96
 

 





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