HOW? 

To illustrate this relationship, we will take the case of two investors, each having Rs.5,00,000 to invest. 

Investor 1 buys INFY CE1000, which gives him the right to buy 500 shares of Infosys at the strike price of 1000. He pays only a small premium and retains the cash with him. 

Investor 2 buys 500 shares of Infosys at Rs.1000 each. Now he has exhausted all his funds. So, he purchases a put option of Infosys at a strike rate of Rs.1000 

Put-Call Parity Example:

Two investors, each with Rs. 5,00,000, illustrate the relationship:

Investor 1:

• Buys INFY CE 1000 (call option) for a small premium

• Gets the right to buy 500 Infosys shares at Rs. 1000 strike price

• Keeps the remaining cash

Investor 2:

• Buys 500 Infosys shares at Rs. 1000 each (exhausting funds)

• Buys a put option at Rs. 1000 strike price to cover potential losses

Both investors have similar exposure to Infosys shares, but different strategies. The put-call parity theory states that the costs of these two strategies should be equal, resulting in an inverse relationship between call and put prices.

Which investor had a better strategy?

Comparing Strategies:

Investor 1's strategy:

• Waited and watched the price movement before investing

• Kept their Rs. 5 lakhs safe in a risk-free investment

• Had the option to buy at Rs. 1000 and sell at the market price if it rose above Rs. 1000

• Could let the option lapse if the price fell below Rs. 1000, protecting their investment

This strategy allowed Investor 1 to manage risk effectively, while Investor 2 exhausted their funds buying shares and had to buy a put option to cover potential losses.

Investor 2's Strategy:

Investor 2 aimed to protect their Rs. 5 lakhs invested in Infosys shares from a potential price drop. To achieve this, they:

• Bought a put option with a strike price of Rs. 1000

• Could exercise the option to sell at Rs. 1000 if the price fell below Rs. 1000 at expiry, limiting their loss

• Would let the option lapse if the price rose above Rs. 1000, as the put option would expire worthless

By buying the put option, Investor 2 effectively insured their investment against a potential decline in Infosys' share price.

Call-Put Parity:

Both investors employed risk-free strategies, protecting their assets. The theory states that:

• Call + cash (Investor 1) = Put + underlying stock (Investor 2)

At expiry, both portfolios have the same value. Therefore, their present value should also be equal, making the Call-Put parity value zero. This relationship ensures that the call-and-put options are fairly priced relative to the underlying stock and cash, eliminating arbitrage opportunities.

In the strategies of two investors we mentioned above, one was, buying a call and investing an amount equal to the strike price value in risk-free assets. The other was buying a put and also the stock at spot price. The pay-off in both strategies was identical. Now that we have seen the relationship let us illustrate it as a formula. 

Call-Put Parity Formula:

The strategies of the two investors illustrate the relationship between call and put options, which can be represented by the formula:

CE + PV = S + PE

Where:

• CE = European Call option

• PV = Present Value of cash invested in risk-free assets

• S = Spot price of the share

• PE = European Put option

This formula demonstrates the parity between call and put options, allowing us to derive the value of one variable if the others are known.

Constructing and Reading Formulas:

To understand and work with formulas, remember:

• Interchanging letters on either side of the equal sign changes the mathematical relationship.

• Moving a positive figure to the other side of the equal sign makes it negative.

Example:

• Original formula: CE + PV = S + PE

• To find PE, rearrange the formula: PE = CE + PV - S

By doing so, S changes from a buy spot price to a sell spot price.

This rearranged formula shows that buying a call, investing in risk-free assets, and selling in the spot market is equivalent to buying a put option.

Why do we need such combinations?

We need these combinations, also known as synthetic options, when:

• Options for a specific stock are not available in the market

• We want to create a customized option that suits our investment strategy

• We want to replicate the behavior of an option without actually buying it

By combining calls, puts, stock, and risk-free assets, we can create synthetic options that mimic the behavior of actual options, allowing us to manage risk and invest more flexibly.

Here is an example.

XYZ three three-month call option with a strike price of Rs.400 is sold for Rs 36. The spot price of the stock is Rs.380 right. The risk-free return is 8% per annum. What would be the theoretical value of an XYZ put, having the same maturity and strike price?

To find the theoretical value of the XYZ put option (PE), we can use the modified formula:

PE = CE + PV - S

We are given:

• CE = Rs. 36 (call option price)

• S = Rs. 380 (current market price of the stock)

To calculate PV (present value of Rs. 400 invested at 8% for 3 months), we can use the formula:

PV = Rs. 400 / (1 + (8%/4))

PV ≈ Rs. 392.98

Now, plug in the values:

PE = Rs. 36 + Rs. 392.98 - Rs. 380

PE ≈ Rs. 48.98

So, the theoretical value of the XYZ put option with the same maturity and strike price is approximately Rs. 48.98.

How do we find the present value?

· PV = FV / (1+r) ^ N

= 400 / (1+0.08) ^ 0.25

= 392.38

And then, you've used this PV value to calculate the theoretical value of the put option (PE):

PE = CE + PV - S

= 36 + 392.38 - 380

= 48.38

This means that if you buy a call option (CE) and invest the strike price (400) in a risk-free asset earning 8% interest, and then sell the stock, you've effectively created a put option worth 48.38.

As you mentioned, if the actual put option price is:

• Below 48.38, it's undervalued, and you should buy it.

• Above 48.38, it's overvalued, and you should sell it.