Before we proceed with the analysis of the simple financial market, we consider an exercise that illustrates how puts can be used to reduce the risk of an investment portfolio.

Consider an investor who has 33 euro available. She buys one stock Royal Dutch Shell, traded at 30 euro, and puts the remaining 3 euro on her bank account. The interest rate (on the bank account) is 5%. Our investor has an investment horizon of one year, that is she wants to sell her portfolio after one year. We consider the following two scenarios:

- (u) the stock price after one year is 35 euro
- (d) the stock price has fallen to 22 euro.

(answer IIa)Regardless of the scenario, the investor receives 3x1.05=3.15 euro from her bank account. Selling the stock yields 35 euro for the ‘up’ scenario (u) and 22 euro for the ‘down’ scenario (d). So, the payoff of the portfolio is:

- 38.15 euro for scenario (u);
- 25.15 euro in scenario (d).

(answer IIb) From Exercise 1 we know that the payoff of the put option is max{ 30 - S1, 0}, where S1 denotes the price of the stock after one year. We thus obtain:

- max{ 30- 35,0} = max{-5,0} = 0 euro for scenario (u);
- 30-22,0} = max{8,0} = 8 euro for scenario (d).

(answer IIc)

- 35+0 = 35 euro for scenario (u);
- 22+8 = 30 euro for scenario (d).

- (P1) a stock plus investing money on the bank account;
- (P2) a stock combined with a put option.

(answer IId) Note that portfolio (P1) has more upside potential: if scenario (u) would occur after one year, the payoff is 38.15 euro compared to 35 euro for portfolio (P2). On the other hand, if scenario (d) would occur the payoff is 25.15 euro compared to 30 euro for portfolio (P2). The put option thus offers insurance against a falling stock price: it ensures that the payoff of your portfolio is at least 30 euro.

Which portfolio you prefer depends on the probability of the scenarios and your risk preferences. In this mini course we will not discuss portfolio choice problems and risk preferences. These topics are extensively discussed in our bachelor program.