Assume the stock return for the next month is a random variable that follows a Normal distribution with the mean 1.5% and the standard deviation 5%, which is the true probability. The interest rate is 0.25% for the next month. Suppose you are the owner of a big investment bank, and one of your VIP clients wants to buy a customized derivative from you. The payoff of such derivative is $4mil if the stock return for the next month is larger than 0.25%, and $1mil if the stock return for the next month is smaller than 0.25%. Assume the probability that the stock return turns out exactly 0.25% is zero. Compute the price of this customized derivative. Your client will pay you extra fees in addition to the price of this derivative. This computation is extremely important for your business. If your valuation is too high, the client will buy the same derivative from other banks, and you lose the fees. If your valuation is too low, you are selling a product with an expected loss.

Check the explanation

Explanation:

The price of the original asset is the same amount as the expected future price which are being discounted at the risk-free rate.

Price of Customized Derivative= Probability of return>0.2%*Pay off+ Probability of Return<0.2%*Payoff/(1+r)^T

= 0.5*$4000000+0.5*$1000000/(1+0.002)^1

=2000000+500000/1.002

=2000000+499001.99

$2499001.99

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