Question

Explain the concept of Put-Call Parity using both the formula and by plotting the payoffs as a function of the stock price.

b. Option traders often refer to “straddles” and “butterflies”. Straddle: buy call with exercise price of $100 and simultaneously by put with exercise price of $100. Butterfly: Simultaneously buy one call with exercise price of $100, sell two calls with exercise price of $110, and buy one call with exercise price of $120.

i. Draw position diagrams for the straddle and the butterfly, showing the payoffs from the investor’s net position.

ii. Each strategy is a bet on variability. Explain Briefly the nature of this bet.

c. A stock’s current price is $160, and there are two possible prices that may occur next period: $150 or $175. The interest rate on risk-free investments is 6% per period. Assume that a (European) call option exists on this stock having an exercise price of $155.

i. How could you form a portfolio based on the stock and the call so as to achieve a risk-free hedge?

ii. Compute the price of the call. iii. What would change if the exercise price was $180?

Answer 1

Put-call parity is a principle that defines the relationship between the price of European put options and European call options, all with the same underlying asset, strike price, and expiration date.

Formula:

C + PV(x) = P + S

where:

C = price of the European call option

PV(x) = the present value of the strike price (x), discounted from the value on the expiration date at the risk-free rate

P = price of the European put

S = spot price or the current market value of the underlying asset

b

i.

Straddle payoff diagram

Butterfly payoff diagram

ii.

Straddle is a stratergy based on the volitility of the market. If an investor expects a sharp move in the market but is not sure about the direction, he/she can enter into a straddle to get the maximum payoff for the movement.

Butterfly is a stratergy that bets on low volitility. It pays off when the market volitility is low and the market is stable i.e. low upward or downward movement.

c

Current stock price = 160 = S

Upward movement = 175

Percentage change = 175/160 -1 = 9.4 %

Downward movement = 150

Percentage change = 150/160 = 9.4% = σ

Hence, volitility = 9.4%

Risk free rate = 6% = r

Strike price = 155 = K

Number of periods = 1 = T

We can calculate the price of the call using 2 step binomial

U = e ^ σ * ( T ^1/2 )

U = e ^ .094 * 1 = 1.098

D = 1 / U = 1 / 1.098 = 0.91

Probability of Up = P(U) = ( e ^ r * T ) - D / ( U - D )

= e ^ 0.06 - 0.91 / ( 1.098 - 0.91 ) = 0.80

Probability of Down = 1 - P(U)

= 0.20

Since it is a call option, payoff is expected when price moves up

Payoff = ( 175 - 155 ) * 0.80 = 16

Present value = 16 * ( e ^ - r * T ) = 16 * ( e ^ -0.06 ) = 15.04

In order to form a risk free hedge, we simply have to sell the stock short and buy the call option with the same strike price on which the stock is sold i.e. S = K. That way , in case of upward movement, the loss on short sell stock would be compensated by the call and in case of downward movement, the loss on call would be nullified by the profit from the short sell

The price of the call on exercise price of 155 is 15.04 ( as calculated above )

If the exercise price was 180 , the payoff on the call option would be 0 and hence the price of the call option would also be 0.