In financial modeling, there is the concept of the “risk-free world” assumption, which allows for the introduction of probability in the development of events and is also used to determine the fair value of options.

The risk-free world assumption method is based on the idea that when there is an arbitrage opportunity, the market disregards risks and immediately takes advantage of this opportunity.

Interestingly, mathematically, this method of option valuation yields the exact same figures as the portfolio replication method (from the previous note).

The logic is as follows:

  1. If we assume the existence of a risk-free world, then in this world, the expected return on a stock should be equal to the risk-free rate.
  2. Since we are following a simple model, we have only two possible future scenarios: either the stock price will increase by X% or decrease by Y% (specific thresholds are predefined in the simple model, as they were in the replication method).
  3. Therefore, if the probability of an increase is P, the probability of a decrease will be (1-P), and their weighted combined return equals the risk-free rate:

[ \text{Expected return} = (\text{probability of rise} \times \text{upside change}) + ((1 – \text{probability of rise}) \times \text{downside change}) = \text{Risk-free Rate} ]

  1. In the formula, all parameters are known except for P. Therefore, we calculate P, which allows us to determine the value of the Call option.
  2. If the probability is known and the outcomes of the scenarios are known, determining the weighted outcome is simple. Suppose the probability of a stock price increase is 40%, and the profit in case of an increase is $100. This means the value of the call in case of an increase will be $100, and in case of a decrease, it will be $0 (60% * 0 – the option becomes worthless). Therefore, the expected weighted return and the future value of the Call option will be $40:

[ 100\$ \times 40\% + 0\$ \times 60\% = 40\$ ]

  1. Finally, to determine the present value (PV) of the option, we discount this $40 by the risk-free rate, obtaining the current value of the Call (the development time of the scenarios is predefined in the model).

P.S.
Substantively, it is the same procedure as when we adjust actual expected cash flows by guaranteed flows (Certainty Equivalent) and then discount them at the risk-free rate. P is not the real probability of the expected price change; we adjusted it and turned it into a “guaranteed” probability.

Source:

Principles of Corporate Finance – by F. Allen, R. A. Brealey, & S. Myers