Apply the Fundamental Theorem of Asset Pricing to find the time 0 and 1 prices of a put option with strike price $110 maturing after two steps, given the same scenarios ω1, ω2, ω3, ω4, stock prices S(0), S(1), S(2) and money market prices A(0), A(1), A(2) as in Example 4.5.

Question

Apply the Fundamental Theorem of Asset Pricing to find the time 0 and 1 prices of a put option with strike price $110 maturing after two steps, given the same scenarios [latex]ω_1, ω_2, ω_3, ω_4,[/latex] stock prices S(0), S(1), S(2) and money market prices A(0),A(1),A(2) as in Example 4.5.

Accepted Answer

The put option gives the right (but no obligation) to sell one share of stock for the strike price X = 110 dollars at time 2. We consider an extended model with three assets, the stock, the money market, and the option. The unit prices of these assets are [latex]S(n),A(n), P^{E}(n),[/latex] where [latex]P^{E}(n)[/latex] is the market price of the put option at time n = 0, 1, 2. The time 2 price of the put option is

[latex] P^{E} (2)[/latex]=max{X − S(2), 0}.

According to the Fundamental Theorem of Asset Pricing, the discounted stock and option prices [latex] \widetilde{S}(n) = S(n)/A(n) [/latex] and [latex] \widetilde{P}^{E} =P^{E}(n)/A(n)[/latex] should form a martingale under some probability measure [latex]P_∗[/latex], or else an arbitrage opportunity would arise. From Example 4.5 we know that there is only one probability [latex]P_∗[/latex] turning [latex] \widetilde{S}(n)[/latex] into a martingale. Because of this,[latex] \widetilde{P}^{E}(n)[/latex] must be a martingale under the same probability [latex]P_∗[/latex]. It follows that

[latex] P^{E}(1) =\frac{A(1) }{A(2)} E_{∗}(P^{E}(2)|S(1))[/latex] and [latex] P^{E}(0) =\frac{A(0) }{A(1)} E_{∗}(P^{E}(1)).[/latex]

Using the values of [latex]P_∗[/latex] for each scenario found in Example 4.5, we can compute [latex]P^{E}(1)[/latex] and [latex]P^{E}(0)[/latex]. For example

[latex] P^{E}(1, ω_3) = P^{E}(1, ω_4)=\frac{A(1)}{A(2)} \frac{P_{∗}(ω_3)P^{E}(2, ω_3) + P_{∗}(ω_4)P^{E}(2, ω_4)}{P_{∗}(ω_3)+P_{∗}(ω_4)} [/latex]
[latex]=\frac{110}{120} \frac{\frac{1}{25} imes 20+\frac{1}{100} imes 30 }{\frac{1}{25}+\frac{1}{100} } \cong 20.00[/latex]

dollars. In this manner, we obtain

Scenario	[latex]P^{E}(0)[/latex]	[latex]P^{E}(1)[/latex]	[latex]P^{E}(2)[/latex]
[latex]ω_{1}[/latex]	1.96	1	0.00
[latex]ω_2[/latex]	1.96	1	4.00
[latex]ω_3[/latex]	1.96	20.00	20.00
[latex]ω_4[/latex]	1.96	20.00	30.00

Question 4.8: Apply the Fundamental Theorem of Asset Pricing to find the t...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions