Description
Verification of the Adjustment-Method for Pricing
European Down-and-Out Continuous-, and European
Down-and-Out Discrete-Barrier Options via Monte Carlo
Simulation
Instructions
1. If you worked with someone on a specific problem, make sure you
tell me about it. If you do not do this and I notice a verbatim
reproduction of material, I consider it as plagiarism. There can
be serious consequences to this.
2. If you picked-up the solution from some source, make sure you
mention it. If you do not do this and I notice a verbatim reproduction of material, I consider it as plagiarism. There can be
serious consequences to this.
3. Each submitted document should be personalized. I expect to
see (1) C++ code and header files (if any), (2) A two page writeup with information on your implementation that accompanies
each submission by Midnight, 14 December, 2018 on Compass.
There are two parts to this programming assignment that involves the simulation of price-paths using the material of Lesson 9. The first part is about
pricing an European Down-and-Out Continuous Barrier Option, and the second part is about pricing an European Down-and-Out Discrete Barrier Option.
You are in effect going to verify the adjustment-terms covered in section 1 of
Lesson 7, and section 2 of Lesson 7. But this time, you are going to deal with
the simulated price-paths assuming it follows the standard geometric Brownian
motion model for asset-price. Let us review the details.
Review of the adjustment-term for the Continuous Barrier Option
According to Baldi, Caramellino and Iovino [?], suppose the initial asset-price
is S0, the final asset-price is ST , the volatility is σ, and the barrier price is B,
then the probability that an asset-price path that starts at S0, ends up at ST
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in time T, breaches the barrier price of B is given by
pc =
(
1 if S0 ≤ B or ST ≤ B
e
−
2 log S0
B
log
ST
B
σ2T otherwise.
(1)
So, the probability that the asset price path that starts at S0 and ends up in
ST in time T does not breach the barrier B is 1 − pc.
Baron-Adesi, Fusari and Theal [?] suggest that we just price the option
without taking the Barrier into consideration, and then discount the price by
multiplying it by (1 − pc). You will verify this assertion in the first-half of this
programming assignment, and simultaneously, you will implicitly verify equation
1 shown above.
Review of the adjustment-term for the Discrete
Barrier Option
A Brownian bridge is a continuous-time stochastic process B(t), where
B(t) = X(t) − (t × Z),
where Z is a unit-normal variate.
The Brownian bridge is very useful for the following task – suppose we have
generated a number of points X(0), X(1), X(2), . . . X(n) of a Wiener process
path by computer simulation. We now wish to interpolate between the samples.
The solution is to use a Brownian bridge that is required to go through the
sample points X(0), X(1), X(2), . . . X(n).
In general when X(t1) = a and X(t2) = b, the distribution of X(ti) for
ti ∈ (t1, t2) is normally distributed with mean µi
µi = a +
ti − t1
t2 − t1
× (b − a),
and variance
σ
2
i =
(ti − t1)(t2 − ti)
t2 − t1
.
We have the initial price S0, and we obtain the final price ST at expiration
T through a simulated price-path, we can check if ST > B, the barrier-price.
Following this, we can effectively compute the probability that the price path
never touched any discrete barrier located at t1, t2, . . . , tm(= T), as
pd =
mY−1
i=1
1 − N
B − µi
σi
(2)
Following what we did for the continuous barrier option, we price the discrete
barrier option by ignoring the barrier entirely. Following this, we multiply the
price by pd shown in equation 1.
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You will verify this claim in the second-part of the programming assignment. As before, verifying the adjustment-term is an indirect verification of the
Brownian Bridge concepts introduced above.
What I want from you: First-Part
To get full points for the assignment, I want you to meet the following requirements1
:
1. Your program should take the following command-line input (in the following order; this will make the grading-part easy):
(a) Duration T,
(b) Risk-free interest rate r,
(c) Volatility σ,
(d) Initial stock price S0,
(e) Strike Price K,
(f) n, the number of trials/repetitions of the Monte Carlo simulations,
(g) m, the number of sample-points in each sample price-path, and
(h) Barrier Price B (assume the user will input a value of B such that
B < S0),
2. Your code will simulate a price-path of m-many equally spaced points.
That is, your code will generate values for S0, S1, S2, . . . , Sm(= ST ). You
should exploit the “get-four-paths-for-the-price-of-one-path” method. If
any Si ≤ B, the price path gets knocked-out, and gives you a zero-payoff.
Otherwise, the payoff is max(0, ST − K) for a call, and max(0, K − ST )
for a put. Present the average payoff over the n-many trials as the price
of the option, which is obtained via explicit-simulation.
3. Using the same code, and just the values of S0 and ST , compute pc as
according to equation 1, and use a payoff max(0, ST −K)(1−pc) for paths
did not get knocked-out; all other paths have a payoff of zero. Present the
average of these “adjusted” prices over the n-many trials as (an alternate)
price of the option, which is obtained using just the initial- and final-price
along with the adjustment-term.
4. Use the closed-form formulae in section 1.1 of lesson 7 to compute the
theoretical price of the down-and-out option, and present it as the output
too.
For the first-part, I am looking for an output that looks something like what
is shown in figure 1. When all these three prices are approximately similar, we
have effectively verified the formula shown in equation 1.
1
If you do not follow my instruction, you will loose points unnecessarily.
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Figure 1: A sample output for the first part of the assignment.
What I want from you: Second-Part
To get full points for the assignment, I want you to meet the following requirements2
:
1. Your program should take the following command-line input (in the following order; this will make the grading-part easy):
(a) Duration T,
(b) Risk-free interest rate r,
(c) Volatility σ,
(d) Initial stock price S0,
(e) Strike Price K,
(f) n, the number of trials/repetitions of the Monte Carlo simulations.
(g) m, the number of (equally-spaced) discrete barriers from 0 to T (note,
the m-th barrier is at time T),
(h) Barrier Price B (assume the user will input a value of B such that
B < S0),
2. Your code will simulate a price-path of m-many equally spaced points,
which corresponds to the price of the stock at the location of each discrete
2
If you do not follow my instruction, you will loose points unnecessarily.
4
barrier. Here too, you should exploit the “get-four-paths-for-the-price-ofone-path” method. If any Si ≤ B, the price path gets knocked-out, and
gives you a zero-payoff. Otherwise, the payoff is max(0, ST −K) for a call,
and max(0, K − ST ) for a put. Present the average payoff as the price
of the option over the n-many trials. This is the price obtained by the
explicit simulation of the price-paths.
3. Using the same code, and just the values of S0 and ST , compute pc as
according to equation 1, and use a payoff max(0, ST − K)pd, where pd
is given by the expression in equation 2, for paths did not get knockedout; all other paths have a payoff of zero. Present the average of these
“adjusted” prices as (an alternate) price of the option over the n-many
simulations. This is the price of the option obtained by the brownian bride
correction term.
For the second-part, I am looking for an output that looks something like
what is shown in figure 2. When all these three prices are approximately similar,
Figure 2: A sample output for the first part of the assignment.
we have effectively verified the formula shown in equation 2.
Everything that I asked for has to be implemented – no partial
credit for partial implementations. I want these two programs uploaded
on Compass by Midnight, December 13, 2019.
1. If your code works correctly on my data-sets (and this will require you to
follow the input-format instructions rigorously), you get 100 points.
2. I will permit one back-and-forth iteration, if your submitted code does not
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work3
. That is, if I notice something wrong with your code, I will e-mail
you right away. You have 24 hours to fix it and send it back to me.
(a) If your revised code works correctly you will get 80 points.
(b) Anything else, will get you zero points.
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It is not hard to verify things – get approximately the same prices for all variations of the
problem!
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