**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.