Question

1. The current price of one share of a certain stock is 100.

The price of a one-year call option on the stock with a strike price of 105 is 10.

The risk-free interest rate is 5.884% and the stock’s dividend yield is 2.02%

Mr. John would like to create a synthetic long put option on the stock with a strike price of 105 and one year to maturity, using:

• Stock
• Call option
• Cash

Determine the amount of cash and the number of shares of stock Smith will need to create the synthetic put and demonstrate that the portfolio replicates the payoff of a long put option.  Also, determine the price of the synthetic put.

Answer 1

A synthetic put is an options strategy that combines a short stock position with a long call option on that same stock to mimic a long put option. It is also called a synthetic long put. Essentially, an investor who has a short position in a stock purchases an at-the-money call option on that same stock. This action is taken to protect against appreciation in the stock's price. A synthetic put is also known as a married call or protective call.

Unlimited Profit Potential

The formula for calculating profit is given below:

• Maximum Profit = Unlimited
• Profit Achieved When Price of Underlying < Sale Price of Underlying - Premium Paid
• Profit = Sale Price of Underlying - Price of Underlying - Premium Paid

Limited Risk

The formula for calculating maximum loss is given below:

• Max Loss = Premium Paid + Commissions Paid = 10
• Max Loss Occurs When Price of Underlying = Strike Price of Long Call = 105

Breakeven Point(s)

The underlier price at which break-even is achieved for the synthetic long put position can be calculated using the following formula.

• Breakeven Point = Sale Price of Underlying - Premium Paid

The delta of a put is given by

= Spread of option price / Spread of stock price

where q = dividend yield

T = time to maturity

&

Recall that this Δ is the derivative of the value of the put p with respect to the value of the underlying stock S: ∂p/∂S.

So this means that if the underlying goes up by 1, the price of the put change by Δ. Clearly, you see that for puts Δ≤0, which means that if the value of the underlying goes up, the put value goes down.

Let's say you have asset S0=100 and you want to replicate a put which would have a Δ = −10/5 = -2, then you sell 2 shares of the stock.

Say the value next day is S1=80:

Assume you have a portfolio of S and a put p:

• Value at t=0: S0+p0 = 100+p0
• Value at t=1: S1+p1 = S1+(S1−S0)Δ+p0 = 80+(−20)⋅(−2)+p0=40+p0
• The PnL is (40+p0)−(100+p0)=−60

Assume now you don't buy the put but you replicate by investing Δ of the stock:

• Value at t=0: S0+ΔS0 = (1+Δ)S0=3⋅100=300
• Value at t=1: (1+Δ)S1 = 3⋅80=240
• The PnL is −60 as well.

i.e. You replicated the option.