三只看跌期权价格套利

解析需会员查看。

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英文解析

If an arbitrage exists, it will be due to the fact that the convexity of put option values with respect to the strike price is violated.

In the plane $(K,y)$ , the line passing through the points $(K = 40,P(40) = 10)$ and $(K = 70,P(70) = 30)$ is given by

$$y = \frac{70 - K}{30}\cdot 10 + \frac{K - 40}{30}\cdot 30$$

The point on this line corresponding to strike 50 is obtained by substituting $K = 50$

in (3.57), and has $y$ coordinate equal to

$$\frac{2}{3} \cdot 10 + \frac{1}{3} \cdot 30 = \frac{50}{3}.$$

Since $P(K)$ is a strictly convex function of $K$ , a noarbitrage value of the put option with strike 50 should be below the line passing through the price points of the options with strikes 40 and 70. However, $P(50) = 20 > \frac{50}{3}$ . Thus, the put option with strike 50 is overpriced, and an arbitrage exists.

Using a "buy low, sell high" strategy, we can take advantage of this arbitrage opportunity as follows: buy 2 put options with strike 40, buy 1 put option with strike 70, and sell 3 put options with strike 50. There is a $10 positive cash flow when setting up this portfolio, since$

$$3 \cdot \$ 20 - 2 \cdot \$ 10 - \$ 30 = \$ 10$$

The value $V(T)$ of the portfolio at the maturity $T$ of the options is

$$\begin{array}{c}{V(T) = 2\max (40 - S(T),0)}\\ {+\max (70 - S(T),0)}\\ {-3\max (50 - S(T),0)} \end{array}$$

Note that $V(T)$ is nonnegative for any value $S(T)$ of the underlying asset at $T$ . If $70 \leq S(T)$ , then all options expire out of the money and

$$V(T) = 0.$$

If $50 \leq S(T) < 70$ , then

$$V(T) = 70 - S(T) \geq 0.$$

If $40 \leq S(T) < 50$ , then

$$\begin{array}{c}{V(T) = (70 - S(T)) - 3(50 - S(T))}\\ {}\\ {= 2S(T) - 80}\\ {}\\ {\geq 0} \end{array}$$

If $S(T) < 40$ , then

$$V(T) = 2S(T) - 80 + 2(40 - S(T)) = 0.$$

In other words, we set up a portfolio with positive cash flow at inception which does not lose money regardless of the value of the underlying asset at time $T$ . The risk- free profit is equal to the future value at time $T$ of the $10$ cash flow from setting up the portfolio.