• C++ Programming for Financial Engineering
    Highly recommended by thousands of MFE students. Covers essential C++ topics with applications to financial engineering. Learn more Join!
    Python for Finance with Intro to Data Science
    Gain practical understanding of Python to read, understand, and write professional Python code for your first day on the job. Learn more Join!
    An Intuition-Based Options Primer for FE
    Ideal for entry level positions interviews and graduate studies, specializing in options trading arbitrage and options valuation models. Learn more Join!

Derivative payoff question

Joined
12/11/13
Messages
17
Points
11
Hi everyone,

I am trying to the answer the following question:
A derivative has a payoff of S2/S1 at the end of year 2. S1 and S2 are respectively the underlying price at the end of year 1 and 2. What is the price of the derivative?

The problem is that this question is very vague. Do you think I should try to solve it under the Black and Scholes scope? or I should right down something like:
S0(1+r)²=S1(1+r)=S2
Making the assumption that the interest rate is constant.

Thank you for you help. I must confess that I don't really see how to tackle this problem.
 
I think the main problem is not of what sort of assumptions you need to make regarding the prevailing interest rates. Sure, it would make sense to assume the interest rates as per the interest rate curve at the time of the pricing of the derivative. You most certainly can't use Black and Scholes. There is no trading strategy you can come up with that involves this derivative, the underlying asset and the ability to borrow/lend money that nullifies risk which is the basis of the Black and Scholes model.

You would ideally need to price the derivative assuming risk neutrality and price it as a function of the stochastic volatility and the current price of the asset.

Others could perhaps comment further to either corroborate or determine flaws in this method.
 
I would agree with @moretodo . Basically the payoff based on S2/S1 itself already indicates path dependence. Hence we could not directly apply BS formula. The simplest method I could think of is obviously by monte carlo method. I don't know PDE method much enough to comment on it's feasibility.

Hope this helps.
 
This derivative is very similar to a forward contract normalized to the value of 1 an year into the future.
 
Hi,

Thank you for your answers.
Regarding your last answer moretodo, you suggest me to use my second approach with continuous rate, i.e.
S0*exp(2r)=S1*exp(r)=S2

Is it what you meant?
 
I personally dont think this derivative is hard to price assuming infinite liquidity. Your derivative essentially pays the bearer the rate of return on that asset from the period of year 1 to year 2. Its basically a asset linked bond of the future. The only added advantage of investing in this derivative as opposed to the underlying asset itself is that one can invest in finer multiples if S1 and S2 are really big.

At t = 0, I think the price of this derivative should be just the discount factor for 1 year. If lower, you can buy this derivative and short the underlying asset at t = 1 and earn an arbitrage profit at t = 2 when the derivative contract matures. If greater, vice-versa.

A more interesting payoff structure would personally be S2/S0*e^(r*1year).
 
The risk free rate you'll need to price this will be the forward rate from year 1 to year 2
 
Hey!

So at the end, you think the price is given by:
$$\dfrac{S2}{S1} = S0(1+r_{12})$$
where $r_{12}$ is the forward rate between year $1$ and year $2$.
Is it what you meant?
 
Sorry, I'm new here, but what would the answer be? The expectation of S0(1+r12)? I'm not quite sure I follow...
 
Hi everyone,

I am trying to the answer the following question:
A derivative has a payoff of S2/S1 at the end of year 2. S1 and S2 are respectively the underlying price at the end of year 1 and 2. What is the price of the derivative?

The problem is that this question is very vague. Do you think I should try to solve it under the Black and Scholes scope? or I should right down something like:
S0(1+r)²=S1(1+r)=S2
Making the assumption that the interest rate is constant.

Thank you for you help. I must confess that I don't really see how to tackle this problem.

It does not matter if the actual underlying is not a bond, so think as simple as follows: S1 and S2 are prices of a bond at year end 1 and 2 respectively, then your problem becomes pricing a forward bond starting at time 1 and end at time 2. There are a number of models for bond price in the literature.

Don't think too much, have little to do!

And mr moretodo, you think too much!
 
I think this may be more appropriate.

What a piece of work is a man! How noble in reason,
How infinite in faculty! In form and moving
how express and admirable! In action how like an Angel!
In apprehension, how like a God!
 
But yesterday the word of Caesar might
Have stood against the world; now lies he there.
And none so poor to do him reverence.
O masters, if I were disposed to stir
Your hearts and minds to mutiny and rage,
I should do Brutus wrong, and Cassius wrong,
Who, you all know, are honourable men:
I will not do them wrong; I rather choose
To wrong the dead, to wrong myself and you,
Than I will wrong such honourable men.

Antony
 
There is a tide in the affairs of men,
Which taken at the flood leads on to fortune,
Omitted all the voyage of their life
Is bound in shallows and in miseries.

Brutus
 
“The fault, dear Brutus, is not in our stars, but in ourselves, that we are underlings.”
Julius Caeser
 
In the secret parts of Fortune? Oh most true she is a strumpet.
 
Back
Top