HI,
I think mark is correct with his seperation.
I think the basic idea in all derivative pricing is the risk free interest
rate which is choosen as the martingale due to its convenience in
mathematical ways.
Therefore it doesnt matter which maturity you have if you use derivatives
formulas like prices for caplets you "always" use a riskfree rate in the
formula.
(There are measure changes to get somewhere so please dont take the
"always" too literally :)).
The 3M libor/euribor is only taken as the riskfree rate since it is the
shortest maturity actively handeled and at which no "market misbehaviour"
occure as in the overnight rate.
If you look at the detailed derivation of Blacks formula for caplets for
T1-T2 (for example see Brigo & Mercurio "Interest Rate Models - Theory and
Practice p.200-202) you see that it starts out with E[F_2(T1)-X)]+ (the
plus indicating only positive values) under the measure Q^2 (so not in the
real world).
Basically put very not mathematically:
In derivatives pricings you use an coordinate system where the
x-Axis is not the regular x-Axis, but the riskfree rate of return.
Therefore you always use that one, and the maturity of an
instrument is only introduced by other variables.
Best Regards
Matthias
Von:
markleeds@verizon.net
An:
bogaso.christofer@gmail.com
Kopie:
r-sig-finance@stat.math.ethz.ch
Datum:
01.06.2009 20:29
Betreff:
Re: [R-SIG-Finance] [R-sig-finance] Newbie question on risk free
Interest Rate
Gesendet von:
r-sig-finance-bounces@stat.math.ethz.ch
Hi: When you say the 'black" formula, I think you might be talking about
the martingale valuation approach as opposed to the replication approach
? if that's the case, then what changes in the expectation ( in the
martingale approach ) is the distribution of how the price evolves under
the risk neutral probability distribution. i.e: the distribution that
makes the price a martingale .
this doesn't have anything to do with the factor that you discount with to
get the present value. as long as the r that you discount back with is
the same r that you take the expectation of P with respect to, then r can
be anything you want. they use r but it's just a parameter . in fact, for
me , using r confuses the derivation. i just think of r as the parameter
in the transformed diffusion that makes S a martingale . in the continous
hedging derivation, the r
that they used to hedge ends up being the same as the "risk free r" in the
martingale derivation because in the
hedging derivation they hedge the stock process to create a risk free
portfolio and end up with the differential
equation whose solution is BS.
someone on this list that deals with options more than me , please feel
free to correct me on this or say more because it's been a long time
since i looked at this material and my memory/thinking could be wrong. I
just replied because this is the second time I saw this question and I
didn't see a reply.
On Jun 1, 2009, Bogaso wrote:
any view pls?
Bogaso wrote:
>
> Hi, I have come across one more question. I understood that for BS
options
> pricing, I should take short rate i.e. overnight rate because BS derive
> option price through some replicating portfolio which is changed
> instantaneously. However if I price an instrument using Black formula,
> wherein only the distribution of underlying at maturity period is
> considered
> i.e. in this case there is no replicating portfolio story, shouldn't I
> consider risk free rate for longer horizon i.e. a rate whose maturity
> period
> exactly matches with the life of the instrument?
>
> I mean to say, under Black's framework, one only needs to calculate
> expected
> value of the instrument like E[max(0, S[T] - K)] at maturity and then to
> calculate the present value of that. In this case there is nothing abt
> replicating portfolio. Therefore I feel that to calculate PV I should
> consider LIBOR with maturity [o, T].
>
> What you feel on that? If I am correct i.e. if I price same option using
> BS
> and Black, there must be some fundamental difference in theoretical
option
> price.
>
> -----Original Message-----
> From: glenn [mailto:g1enn.roberts@btinternet.com]
> Sent: 29 December 2008 17:33
> To: bogaso.christofer
> Subject: Re: [R-SIG-Finance] Newbie question on risk free Interest Rate
>
> Further to Mahesh's answer Christofer, think of it like this;
>
> The rate in the BS calculation represents a rate that any portfolio
> consisting of an option and the delta equivalent of the underlying (in
> your
> example a swap maybe) MUST earn. Think about how long the portfolio will
> remain delta neutral (risk free) for before a re-balence is needed.
That's
> the rate you want i.e the short rate.
>
> Glenn
>
>
> On 28/12/2008 21:53, "bogaso.christofer"
> wrote:
>
>> Hi,
>>
>>
>>
>> I would like to ask one newbie question on risk free interest rate.
This
> is
>> the essential part to price any financial derivatives, like options,
>> Interest Rate only [IO] strip etc. My question is standing at time "t"
> which
>> risk free interest rate I should consider? 3 month, 6 month, 10 year
> t-bill
>> or t-bond ? for example suppose, I need to price a call option using BS
>> formula, whose remaining life time is 2 years and another option whose
> life
>> time is 5 months. Which interest rate I need to take to value those 2
>> options? After some goggling it is suggested to take 3 month t-bill as
> risk
>> free rate. What is the logic behind that?
>>
>>
>>
>> Again suppose, an Investor is to purchase an IO strip for 7 years, on a
>> 10
>> years mortgage. In this case, I saw one book [by Cuthbertson],
suggested
> to
>> take annual yield on 10-year t-bond to calculate NPV of all future
> Interest
>> payment against mortgage. However again it did not say why to take
>> 10-year
>> bond not, 3-month t-bill.
>>
>>
>>
>> Can anyone here please clarify me on above doubts? Your help will be
> highly
>> appreciated.
>>
>>
>>
>> Thanks and regards,
>>
>>
>> [[alternative HTML version deleted]]
>>
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