# Lower Bound Option - American Puts

Can someone help me with this concept? The lower bound of a call option (american/european) takes into account the PV of a risk-free asset as additional “premium” on the lowest price of the call option. Effectively this is the time-value of the option

Why does this not apply to the American puts? On page 107, example-2 part B, it says a 75 Put: MAX(0,75-72) = 3, why is the european option worth less than it’s intristic value of 3? Why does the American put have a lower bound equal to it’s intristic value?

The idea is that European Calls mean you can only exercise on the day of expiration while American Calls you can excise prior to the excise date. However, in most cases most people, even thou they have an American Call, tend to exercise on the date of expiration.

Therefore, for American/European Calls you have to discount the strike price since you missing out on the opportunity to invest that option on Bonds i.e Risk Free Return.

On the other hand, American Put most people want to exercise the strike price as soon as they can such that they can put their money in a Risk Free Security so you DONT discount the strike value at the RFR.

THis is how I remember it.

You just made it click for me.

Essentially it’s the asset that is being discounted against the risk-free rate. I.e, if you could sell 1k worth of XYZ stock today, you could reinvest the sale at the risk-free rate. Therefore, by buying a put that triggers a sale in the future, you’re not receiving a return on the cash from the sale.

thanks a bunch!

I have doubt

Is an euro put with longer duration vs euro put of horter duration ?

I MEAN longer duration can be less ,equal or more than shorter duration ?

I have doubt

Is an euro put with longer duration vs euro put of horter duration ?

I MEAN longer duration can be less ,equal or more than shorter duration ?

Theoretical lower bound for an in the money European call:

Let’s say I am holding a 1 year call option at 50 on a stock that is trading at 55. The risk free rate is 5%

The lower bound is essentially, the amount of money I can pocket right now for locking in AT LEAST a zero or greater payoff at maturity. Here’s how:

I short the stock at 55, and then using the proceeds, I buy a bond that pays off 50 at maturity for 50 / (1.05) = \$47.62

I now have 55 – 47.62 = \$7.38 in my pocket, and no matter what happens my position cannot lose money at maturity. Therefore the lower bound on this option is \$7.38.

Here are some potential outcomes:

Stock at maturity is trading at \$45

• Call option value = 0
• Purchase Stock = - 45
• Proceeds from bond = 50

Position value = \$5 (0-45+50)

Stock at maturity is trading at \$50

• Call option value = 0
• Purchase Stock = - 50
• Proceeds from bond = 50

Position value = \$0 (0-50+50)

Stock at maturity is trading at \$55

• Call option value = 5
• Purchase Stock = - 55
• Proceeds from bond = 50

Position value = \$0 (5-55+50)

Notice that the outcomes pay off either zero or greater.

For an American call option, we say the lower bound = the European lower bound, because no rational investor will exercise the above position early for a \$5 payoff, when he could do the above and pocket \$7.38 at a minimum.

Theoretical lower bound for in the money European put:

Let’s say I am holding a 1 year put option at 50 on a stock that is trading at 45. The risk free rate is 5%

The lower bound is essentially, the amount of money I can pocket right now for locking in AT LEAST a zero or greater payoff at maturity. Here’s how:

I borrow 50 / (1.05) =\$47.62 and then using the proceeds, I buy the stock at 45

I now have 47.62 – 45= 2.62 in my pocket, and no matter what happens my position cannot lose money at maturity. Therefore the lower bound on this option is \$2.62

Here are some potential outcomes:

Stock at maturity is trading at \$45

• Put option value = 5
• Repay Bond = - 50
• Sell Stock = 45

Position value = \$0 (5-50+45)

Stock at maturity is trading at \$50

• Put option value = 0
• Repay Bond = - 50
• Sell Stock = 50

Position value = \$0 (0-50+50)

Stock at maturity is trading at \$55

• Put option value = 0
• Repay Bond = - 50
• Sell Stock = 55

Position value = \$5 (0-50+55)

Notice that all outcomes either pay off zero or greater

For an American put option, we say the lower bound = the intrinsic value, because in the above example, a rational investor would rather exercise early and collect \$5 right away.

Hope this is correct and all makes sense.

I think about it this way, owning an in the money American put is like being able to sell the underlying for more than the market price, so you can immiediatly make X - S profit if you want. So this is the minimum price any one would sell it to you.

A European Put is the same except that it is the ability is to sell the underlying at time t in the future. Since you are selling it at time t in the future, you have to discount that cash flow by t. But you could buy the underlying immediatly to lock in that profit, so S does not need to be discounted.

For calls you would always want the value of the exercise price you are paying to be minimized, so you would want it to be discounted as much as possible.

Hi

If expected cash flows of underlying asset are expected to increase will call value go up or down?

If the expected cash flows of the underlying increase before the excersise date then the value will go down because the time value of the option will go down.

Afucfa - the expected cash flows do not have anything to do with time value.

but the relationship you suggest is correct.

If expected cash flows increase, say by an increase in dividends, the market will price this new higher cash flow into the call option by decreasing its value.

This is because we know that when a dividend goes ex, the stock falls by the amount of the dividend in theory. This has downward pressure on the stock, which reduces the price of call options which, of course become less valuable when the stock price falls.

I’m not sure that’s right. According to wikipedia…

“More specifically, TV reflects the probability that the option will gain in IV — become (more) profitable to exercise before it expires.[5] An important factor is the option’s volatility. Volatile prices of the underlying instrument can stimulate option demand, enhancing the value. Numerically, this value depends on the time until the expiration date and the volatility of the underlying instrument’s price. TV cannot be negative (because the option value is never lower than IV ), and converges to zero at expiration. Prior to expiration, the change in TV with time is non-linear, being a function of the option price.[6]

So if a divided is lowers the probablity of an increase in the spot price, than it is affecting the Time Value of the option.

There is probably some specific relationship you can derive from black - scholes, but I don’t think it matters much either way for the test. Just know the nature of the relationship.