Question

he current price of a non-dividend paying stock is 40 and the continuously compounded risk-free interest rate is 4%. The following table gives call and put option premiums for three-month European-style options of various exercise prices.

Exercise price

Call Premium

Put premium

35

5.75

0.40

40

2.29

1.90

45

0.50

5.05

A trader interested in speculating on volatility is considering two investment strategies. The first is a long 40-strike straddle. The second is a long strangle consisting of a long put option at strike 35, and a long call option at strike 45.

Determine the range of stock prices (ST) in 3 months for which the strangle outperforms the straddle.

Select one:

The strangle never outperforms the straddle.

35.90 < ST < 44.10

The strangle always outperforms the straddle.

36.71 < ST < 43.29

34.10 < ST < 45.90

Answer 1

The strangle always outperforms the straddle.

The straddle involves buying a $40 strike call option and buying a $40 strike put option

This costs 2.29 + 1.90= 4.19 and grows to 4.19*exp(0.04/4) = 4.232 in three months.

The strangle consists of buying a $35 strike put option and a $45 strike call option

This costs 0.40 + 0.50 = 0.90 and grows to 0.90exp(0.04/4) = 0.9090 at three months.

Let S be the stock price in three months time

For S < 40, the straddle has a profit of 40 – S – 4.232 = 35.768 – S.
For S > 40, the straddle has a profit of S – 40 – 4.232 = S – 44.232
For S < 35, the strangle has a profit of 35 – S – 0.9090 = 34.091 – S.
For 35 < S < 45, the strangle has a profit of –0.9090.
For S > 45, the strangle has a profit of S – 45 – 0.9090= S – 45.909.
For S < 35 the strangle underperforms the straddle

For 35 < S < 40, the strangle outperforms the straddle if –0.9090> 35.768 – S or S > 36.677.

As a check, for 40 < S < 45, the strangle outperforms the straddle if –0.9090> S – 44.232 or
S < 43.323.
For S > 45, the strangle outperforms the straddle if S – 45.909> S – 44.232 , which isn't possible.

Hence, The strangle always outperforms the straddle.