<p>A put option yields the holder the right to sell the stock at the strike price <span class="math-container">$K$</span>, only at expiration date since it is a European one.</p>
<p>At expiration date, after discounting for the time value of money, they would receive <span class="math-container">$K e^{-rT}$</span> worth of today’s dollars, since they sold the stock for <span class="math-container">$K$</span>.</p>
<p>Note: Here the time value of money represents the amount of money needed today required to have a given amount of money in a future period after making a risk-free investment (bank deposit).</p>
<p>If they bought the stock now, they needed to pay <span class="math-container">$S_0$</span> for it. Therefore, the action of exercising the option at expiry would net them a today’s profit of <span class="math-container">$K e^{-rT} - S_0$</span>.</p>
<p>So to answer your questions:</p>
<p>a. The <span class="math-container">$38$</span> dollars in the example come from this: <span class="math-container">$37$</span> dollars to buy the stock today at market price <span class="math-container">$+$</span> <span class="math-container">$1$</span> dollar to buy the put option today.</p>
<p>b. In order to execute the arbitrage trade, he needs to buy the stock today.</p>
<p>If he just buys the put but not the stock, there would be no guaranteed profit, he could lose money if the following happens:</p>
<p>If the stock ended above <span class="math-container">$K = 40$</span> at expiration date, the put wouldn’t be worth exercising, it would expire worthless, making the investor lose the put’s initial value.</p>
<p>By the way, if you plan on exercising any put option, you need to buy the stock at some point in time, as exercising it makes you sell the stock. If you exercise it without owning the stock, it’ll leave you with a short position.</p>
<p>A short position is when you borrow the stock and sell it waiting to buy it at a later point in time. This is a strategy done by speculators who expect the stock to have a massive crash. The investor makes a profit when the stock goes down and loses money when the stock goes up. If unhedged, this is an unlimited risk strategy as there is no limit to how much the stock can go up, but limited profit because the stock can only go down to <span class="math-container">$0$</span>.</p>

CC BY-SA 4.0
2023-03-06T12:16:41.310Z
2023-03-06T12:21:51.177Z
2023-03-06T12:21:51.177Z
Nicolas Torres
<p>I have the 9th edition of hull and reading up on options.
Put options. I am reading Chapter 11 on page 240, section,on Lower
Bound for European Puts on Non-Dividend-Paying Stocks.</p>
<p>I'm confused by the specific numerical example that Hull chooses.
He states that the lower bound is</p>
<p><span class="math-container">$$
K e^{-rT} - S_0
$$</span></p>
<p>which is fine. Then he goes on to show us the example.
Suppose <span class="math-container">$S_0 = 37, K = 40, r = 0.05, T= 0.5$</span>
Thus,
<span class="math-container">$$
K e^{-rT} - S_0 = 2.01.
$$</span></p>
<blockquote>
<p>Now he assumes that the European put price is 1.00 instead of the theoretical minimum of 2.01 USD.</p>
</blockquote>
<p>Here is the double confusing part for me.</p>
<blockquote>
<p>How does he come up with &quot;An arbitrageur can borrow 38.00 for 6 months ....&quot;</p>
</blockquote>
<p>a) how does he come up with 38 USD dollars? Why not another amount and how does he pick 38?
b) Why is he required to buy the stock and the option?
For instance, if the stock price is below 40 USD, the arbitrageur exercises the option to sell the stock for 40 USD, repays the loan and makes a profit.</p>
<p>But Why did he need to buy the stock in order to exercise the put?! He didn't need to buy the stock, he could have just bought the put and exercised it! (The idea of the put is that you don't need to own the underlying.)?</p>
<blockquote>
<p>If the stock price is greater than 40.00, the arbitrageur discards the option, sells the stock and repays the loan for an even greater profit.</p>
</blockquote>

<finance><asset-pricing>
[Hull]: Put Lower bound example confusing
user43671
# [Hull]: Put Lower bound example confusing

(Asked by: username_0 on 2023-03-06)

I have the 9th edition of hull and reading up on options.
Put options. I am reading Chapter 11 on page 240, section,on Lower
Bound for European Puts on Non-Dividend-Paying Stocks.

I'm confused by the specific numerical example that Hull chooses.
He states that the lower bound is

$$
K e^{-rT} - S\_0
$$

which is fine. Then he goes on to show us the example.
Suppose $S\_0 = 37, K = 40, r = 0.05, T= 0.5$
Thus,
$$
K e^{-rT} - S\_0 = 2.01.
$$

> Now he assumes that the European put price is 1.00 instead of the theoretical minimum of 2.01 USD.

Here is the double confusing part for me.

> How does he come up with "An arbitrageur can borrow 38.00 for 6 months ...."

a) how does he come up with 38 USD dollars? Why not another amount and how does he pick 38?
b) Why is he required to buy the stock and the option?
For instance, if the stock price is below 40 USD, the arbitrageur exercises the option to sell the stock for 40 USD, repays the loan and makes a profit.

But Why did he need to buy the stock in order to exercise the put?! He didn't need to buy the stock, he could have just bought the put and exercised it! (The idea of the put is that you don't need to own the underlying.)?

> If the stock price is greater than 40.00, the arbitrageur discards the option, sells the stock and repays the loan for an even greater profit.

**Tags:** finance, asset-pricing

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## 1 answers

### Answer by username_1 on 2023-03-06. Score: 2

A put option yields the holder the right to sell the stock at the strike price $K$, only at expiration date since it is a European one.

At expiration date, after discounting for the time value of money, they would receive $K e^{-rT}$ worth of today’s dollars, since they sold the stock for $K$.

Note: Here the time value of money represents the amount of money needed today required to have a given amount of money in a future period after making a risk-free investment (bank deposit).

If they bought the stock now, they needed to pay $S\_0$ for it. Therefore, the action of exercising the option at expiry would net them a today’s profit of $K e^{-rT} - S\_0$.

So to answer your questions:

a. The $38$ dollars in the example come from this: $37$ dollars to buy the stock today at market price $+$ $1$ dollar to buy the put option today.

b. In order to execute the arbitrage trade, he needs to buy the stock today.

If he just buys the put but not the stock, there would be no guaranteed profit, he could lose money if the following happens:

If the stock ended above $K = 40$ at expiration date, the put wouldn’t be worth exercising, it would expire worthless, making the investor lose the put’s initial value.

By the way, if you plan on exercising any put option, you need to buy the stock at some point in time, as exercising it makes you sell the stock. If you exercise it without owning the stock, it’ll leave you with a short position.

A short position is when you borrow the stock and sell it waiting to buy it at a later point in time. This is a strategy done by speculators who expect the stock to have a massive crash. The investor makes a profit when the stock goes down and loses money when the stock goes up. If unhedged, this is an unlimited risk strategy as there is no limit to how much the stock can go up, but limited profit because the stock can only go down to $0$.


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