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Consider the task of pricing at time 0 a European put option i. The Black-Scholes-Merton pricing formula is. The other two variables are. It would be nice if we could simply carry out the additions, multiplications, divisions, etc. The situation is a little more difficult than that, however. It is true we can calculate the numerator of the expression for d 1 , using scalar operations where appropriate, and probabilistic operations to add the last two terms together.
Evaluating the price probabilistically could be a major challenge. A different way of presenting the same problem gives the answer without difficulty. The present value of the strike price is just Xe —rT , an expression that involves only one random variable, r , and can be readily computed. To illustrate, Figure 1 shows the distributions of the present values of X and S T.
The present value of the strike price X is narrow because there is not much uncertainty in the risk-free rate r. In contrast, the present value of the stock price S T is much broader because of its volatility. Distributions of the present values of the final stock value 1 and the strike price 2. The difference between the present values can be positive or negative. The put option has a 0 value if the stock price is higher than the strike price. The distribution of the value of the put, given that value is greater than 0, is shown in Figure 2 3.
However, the probability of the put option having a non-zero value is only 0. Therefore, the value of the put is 0. Distributions of the present values of the final stock value 1 , the strike price 2 and the benefit from cashing in the put option, if it was positive 3.
Home Growing investment example Option valuation example Bayesian analysis example Underlying theory. Distributions of the present values of the final stock value 1 and the strike price 2 The difference between the present values can be positive or negative.