Understanding Carry and Dividends in Put-Call Parity

# 57227


# Under Put-Call Parity, why do we add the cost of carry to Call prices but subtract them from the Stock price and Put prices?












In Natenberg (1994) Chapter 11 he outlines the Put-Call parity relationships.

```
Stock Price = Call Price - Put Price + Exercise Price - Carrying Costs + Dividends

Call Price = Stock Price - Exercise Price + Put Price + Carrying Costs - Dividends

Put Price = Call Price + Exercise Price - Stock Price - Carrying Costs + Dividends
```

Why do we add the cost of carry to the call price but then subtract it from the stock price and also from the put price? Shouldn't we be doing the opposite?

E.g. Add the cost of carry to the stock price but subtract from both the call and put prices?

I say this because in Chapter 3. pg 42, Natenberg says this:

> A trader who buys stock will have to pay out carrying costs, but he will receive the dividends. If we again assume that a stock trade will break even, the expected return at the end of the holding period must be identical to the carrying costs less the dividend. In an arbitrage-free market, where no profit can be made by either buying or selling a contract, all credits and debits, including the expected return, must exactly cancel out. If we assume an arbitrage-free market, we must necessarily assume that forward price, the average price of the contract at the end of the holding period is the current price, plus an expected return which will exactly offset all other credits and debits. If the holding costs on a \$100 stock over some period are \$4, the forward price must be \$104. If the stock also pays a \$1 dividend, the forward price must be \$103. In both cases the credits and debits will exactly cancel out.

And also in Chapter 3 he also says the following and gives us an example of why cost of carry should be subtracted from a supposed "bet" at a roulette wheel (an analogy to buying a Call Option or Put Option and why cost of carry should be subtracted from it, I suppose):

> Where did the player get the 95¢ he used to place his bet at the roulette wheel? In the immediate sense he may have taken it out of his pocket. But a closer examination may reveal that he withdrew the money from his savings account prior to visiting the casino. Since he won’t receive his winnings for two months, he will have to take into consideration the two months interest he would have earned had he left the 95¢ in his savings account. If interest rates are 12% annually (1% per month), the interest loss is 2% X 95¢, or about 2¢. If the player purchases the bet for its expected return of 95¢, he will still be a 2¢ loser because of the cost of carrying a 95¢ debit for two months. The casino, on the other hand, will take the 95¢, put it in an interest-bearing account, and at the end of two months collect 2¢ in interest. Under these new conditions the theoretical value of the bet is the expected return of 95¢ less the 2¢ carrying cost on the bet, or about 93¢. If a player pays 93¢ for the roulette bet today and collects his winnings in two months, neither he nor the casino can expect to make any profit in the long run. The two most common considerations in a financial investment are the expected return and carrying costs.

If that is the case in the passages above, under Put-Call Parity shouldn't we add the carrying costs to only the stock price? And then subtract the carrying costs only from the call and put?

Since in the the risk-neutral world and in efficient markets, the expected return under "Q" of every equity is the risk-free rate. The stock (forward price) should have the carrying costs added to it, not subtracted. As Natenberg says, we should assume the forward price of the stock at the end of the holding period is the current stock price, plus the expected return (carrying cost from placing the price you pay in an interest bearing account) and subtracted by any dividends.

Because why would you want to buy a call or put (take a bet) at a negative expected return, the carrying costs should be deducted from the prices you pay for an Option so you can break-even, not added to it.

I am aware of the relationship between interest rates, dividends, and calls/puts. Rates go up, calls increase and puts decrease, etc etc. But even then, subtracting cost of carry from the stock price and put price but adding it to the call price still doesn't make sense and contradicts what Natenberg wrote.

Thank you in advance.

## Answer by kurtosis (score 0, accepted)

https://quant.stackexchange.com/a/57230

You seem to ask the following many times: "Under Put-Call Parity shouldn't we add the carrying costs to only the stock price? And then subtract the carrying costs only from the call and put?"

Those three equalities are not definitions. In the first equation, you do not add carry to the stock price unless you move it from RHS to LHS by adding it to both sides.

Apart from that semantic confusion, it helps to remember that the idea is to compensate for effects which make one position (stock minus a bond with face equal to the exercise price) equal to the other (call minus put).

The holder of a stock has to pay carry costs; option holders avoid that cost. That makes the stock-minus-bond relatively less attractive than the call-minus-put. To eliminate the relative difference, we either add the carry to the stock-minus-bond or subtract it from the call-minus-put.

The opposite goes for dividends: the holder of a stock receives dividends while options holders avoid that benefit. Dividends makes the stock relatively more attractive than the call-minus-put. To eliminate the relative difference, we either subtract the dividends from the stock or add them to the call-minus-put.