Consider a security that pays off a dollar amount equal to log ST at expiration time T (with no strike) on a non-paying dividend stock. Use risk-neutral valuation to calculate the price of the security at time t.
The Price of the stock which is a non dividend paying one, at expiration is given as LOG ST.
The price of the stock at a certain time "t" will be assumed to be LOG St
Since we are assuming a risk free neutral rate with no arbitrage as the conditions of our valualtion, let us suppose that the risk free neutral rate is
R (risk free rate)
The time duration between expiration and present time will be (T-t) assumed in years.
As per the risk free / neutral measurement model, the current price of the stock should be equal to the discounted expected expiration / maturity price of the stock discounted at the risk free rate R. In formula terms this will mean the following
LOG St = LOG ST / (1+ R/100)^(T-t).
Given the values of T, t and R we can fine the current value of the stock as its assumed in the question that there is no strike.
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