Price of Option in Binomial Tress

[sudeepdoon]

Hi David,

With reference to Chapter 11 of John C Hull, We are calculating the price of the option at the nodes (The last ones) as Stock Price - Strike Price; which actually is the Intrensic Value of the Option.

The Value of the option is Intrensic Value + Time Value.

What assumption of the mode has lead to Time Value being Zero..

Thanks,
Sudeep Manchanda

 

[David Harper CFA FRM]

Hi Sudeep,

The assumption is that, at the final nodes only (and only at the final nodes), the option expires and, if in the money, must be excercised (just as you say: at expiration, value = intrinsic value only because time value = 0). The basic binomial tree, by design, builds out to the where the final nodes match the maturity.

FYI, I just uploaded "improved" binomial learning XLS, they include almost all of Hull's examples. See here (Suzanne is currently prepping the online Zoho equivalents). So, if you want to see the calcs underling those exhibits, you can find there...

David

 

[sudeepdoon]

Hey David,

Thanks for the explanation. I am comfortable to understand that Expiration the value of the option would be the Intrensic Value.
This would for sure explain things for the European Option.

But, if we look at the example for American Option, then there are cases when we can execute an option before expiration, in this case the value of the option in MAX(intrensic value, value from formula)... In this case the option has some time to expitration so it should have a time value..

 

[David Harper CFA FRM]

Hi Sudeep,

Yes, excellent point. This gets tricky. What you say is true, in the sense that, as an option holder, at iterim nodes (ie., before expiration) the American option has a value of:

MAX[intrinsic value = payoff if immediately exercised, intrinsic value + time value = total option value]

This is why (essentially) Hull says (9.5) that "it is never optimal to early exercise an *American* call option on a non-dividend paying stock"
i.e., the total value at any node must exceed the intrinsic value by the time value.
(the dividends are forgone by the option holder, so they effectively create urgency to excersise, so we cannot say this about dividend paying stock)

okay, but the above, is from the perspective of the holder at the node. Please see the third sheet of the binomial workbook
http://www.bionicturtle.com/premium/spreadsheet/4.b.1_binomial_opm/
(Suzanne loaded the Zoho so no s/w is required)
This is Hull's Fig 11-8 and cell G21 performs the present value at the 1 year node, under the "down jump" (stock = $40 in green).
In the binomial valuation, the value of interim nodes is *not* MAX[intrinsic, intrinsic + time]
but rather: MAX[intrinsic, discounted (PV) of two subsequent nodes]

so, in most cases, the discounted PV of the node is greater, but this G21 shows the other outcome, the intrinsic value ($12) is greater than the discounted value ($9.46). Is $9.46 the option's value at this node? No! the option's value must exceed the intrinsic value as there is still time value...so this is an aspect of the binomial valuation...hope this helps, David