MOUNTAIN RANGE OPTIONS
Paolo Pirruccio
Copyright © Arkus Financial Services - 2014
Mountain Range options
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Mountain Range options
Introduction
► Originally marketed by Société Générale in 1998.
► Traded over-the-counter (OTC), typically by private banks and institutional
investors such as hedge funds.
► These options have combined characteristics of Range (multi – year time ranges)
and Basket options (more than one underlying).
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Mountain Range options
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Mountain Range options
Usage
Being struck on two or more underlying assets, mountain range options are particularly relevant for
hedgers who want to cover several positions with one derivative. Instead of monitoring multiple
options written on individual assets, a basket option can be structured to achieve the same coverage.
The advantage of this feature is that the combined volatility will be lower than the volatility of the
individual assets. A lower volatility will result in a cheaper option price, which can significantly decrease
the costs implied by hedging.
All these features made Mountain Range Options an appealing product which usually offer a minimum
capital guarantee, plus the variable part of returns determined by the stock performances. It can be
useful for investors who want a capital protection.
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Mountain Range options
Definition and types
8.848 m -
8.091 m -
8.000 m -
4.167 m -
3.300 m -
Everest - a long-term option in which the
option holder gets a payoff based on the
worst-performing securities in the
basket.
Annapurna - in which the option holder is
rewarded if all securities in the basket never
fall below a certain price during the relevant
time period.
Himalayan - based on the
performance of the best asset in
the portfolio.
Atlas - in which the best and worstperforming securities are removed from the
basket prior to execution of the option.
Altiplano - in which a vanilla option is combined
with a compensatory coupon payment if the
underlying security never reaches its strike price
during a given period.
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Altiplano Options
Al·ti·pla·no [al-tuh-plah-noh; for 1 also Spanish ahl-tee-plah-naw]
1. A plateau region in South America, situated in the Andes of
Argentina, Bolivia and Peru.
2. Financial instrument in which a vanilla option is combined with
a compensatory coupon payment if the underlying security
never reaches its strike price during a given period.
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Risk-based Governance Solutions
“Altiplano con Memoria” Options
Payoff structure
𝑷𝒂𝒚𝒐𝒇𝒇 𝒊 = 𝜼 𝑵 ∗ 𝑪 ∗ 𝒊
𝒊𝒇 𝒊 = 𝟏
𝒊−𝟏
𝑷𝒂𝒚𝒐𝒇𝒇(𝒊) = 𝜼 (𝑵 ∗ 𝑪 ∗ 𝒊 −
𝑪𝒏)
𝒊𝒇 𝒊 = 𝟐, 𝟑, . . , 𝒏
𝒏=𝟏
𝜼=
𝟏 𝒊𝒇 𝑴𝒊𝒏 𝟏≤𝒋≤𝒏,𝒕𝟏≤𝒕≤𝒕𝟐
𝟎
𝑺𝒕𝒋
𝑺𝟎𝒋
≥𝑳
𝒆𝒍𝒔𝒆
►
The Payoff is thus different from zero only if none of the stocks is below the barrier during the specified time period
►
C is a fixed coupon payment
►
i is the Barrier Observation Date
►
Sj represents the value of the j-th stock
►
𝜼 is a binary variable equal to the condition set for the barrier value
►
L is the predetermined limit
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“Altiplano con Memoria” Options
Pricing – The algorithm
1.
Generate normally distributed random variates through the Inverse Transform Method
2.
Simulate the correlated multi asset path through the Cholesky Decomposition
3.
Check, for each barrier observation date, if each single underlying is above the barrier limit

If one of the underlying assets is below the barrier limit  Payoff(i) = 0

If none of the underlying assets is below the barrier limit:
𝑷𝒂𝒚𝒐𝒇𝒇 𝒊 = 𝜼 𝑵 ∗ 𝑪 ∗ 𝒊
𝒊𝒇 𝒊 = 𝟏
𝒊−𝟏
𝑷𝒂𝒚𝒐𝒇𝒇(𝒊) = 𝜼 (𝑵 ∗ 𝑪 ∗ 𝒊 −
𝑪𝒏)
𝒊𝒇 𝒊 = 𝟐, 𝟑, . . , 𝒏
𝒏=𝟏
4. Store the payoff values into an array and discount each of them back at the appropriate discount rate
5. Sum all the discounted payoffs to get the present value of the option
6. Repeat the first 5 steps 20.000 times, to build a distribution of possible option values
7. Take the average of all simulation outcomes to find the final price
8. Greeks Estimation - Delta
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“Altiplano con Memoria” Options
Pricing: the idea behind Monte Carlo Integration
Consider an integral on the unit interval [0,1]:
𝟏
𝑰=
𝟎
𝒈 𝒙 𝒅𝒙
We may think of this integral as the expected value E[g(U)], where U is a uniform random variable on the interval (0,1) and
estimate the expected value - a number – by a sample mean (which is a random variable).
The only thing we have to do is generating a sequence Ui of independent random samples from the uniform distribution and then
evaluate the sample mean:
𝟏
𝑰𝒎 =
𝒎
𝒎
𝒈(𝑼𝒊 )
𝒊=𝟏
The strong law of large numbers implies that, with probability 1, 𝒍𝒊𝒎 𝑰𝒎 = 𝑰
𝒎→+∞
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“Altiplano con Memoria” Options
Pricing – Generating normal random variates through the Inverse Transform Method
Suppose we are given the CDF F(x) = P(X ≤ x), and that we want to generate random variates according to F. If we are able to
generate random variates according to F, then we could:
1. Draw a random number U ~ U(0,1)
2. Return X = F-1 (U)
It can be shown that the random variate X generated by this method is characterized by the distribution function F.
For example u = 0.975 would return 1.959, because 97.5% of the probability of a normal pdf occurs in the region where X < 1.959
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“Altiplano con Memoria” Options
Pricing – Correlated Random Numbers: Cholesky Factorization
Consider a multivariate normal distribution with expected value μ and covariance matrix Σ (symmetric positive definite).
The Cholesky Matrix M is a lower triangular matrix such that:
𝚺 = 𝑴𝑻 𝐌
Once retrieved this matrix, we may apply the following algorithm to generate correlated random numbers X:

Generate n independent standard normal variates Z1, Z2 ,..., Zn

Return 𝑿 = 𝝁 + 𝑴𝑻 𝐙 , where 𝒁 = Z1, Z2 ,..., Zn
T
is a vector of uncorrelated variables
Suppose we must generate sample paths for two correlated Wiener processes, having covariance matrix 𝚺 =
It can be verified that the Cholesky Matrix is 𝐌 =
𝟏
𝝆
𝟏 𝝆
𝝆 𝟏
𝟎
.
(𝟏 − 𝝆𝟐 )
Hence, to simulate bidimensional correlated Wiener Process, we will create two independent standard normal variates Z1 and Z2
and use:
𝒙𝟏 = 𝒁𝟏 and 𝒙𝟐 = 𝝆𝒁𝟏 + (𝟏 − 𝝆𝟐 )𝒁𝟐
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“Altiplano con Memoria” Options
Pricing – the correlation estimation problem and the impossibility to use a closed form
The pricing structure is primarily dependent on the correlation between the constituent stocks.
Stock
A
B
C
D
E
F
G
A
1
Corr(B,A)
Corr(C,A)
Corr(D,A)
Corr(E,A)
Corr(F,A)
Corr(G,A)
B
Corr(B,A)
1
Corr(C,B)
Corr(D,B)
Corr(E,B)
Corr(F,B)
Corr(G,B)
C
Corr(C,A)
Corr(C,B)
1
Corr(D,C)
Corr(E,C)
Corr(F,C)
Corr(G,C)
D
Corr(D,A)
Corr(D,B)
Corr(D,C)
1
Corr(E,D)
Corr(F,D)
Corr(G,D)
E
Corr(E,A)
Corr(E,B)
Corr(E,C)
Corr(E,D)
1
Corr(F,E)
Corr(G,E)
F
Corr(F,A)
Corr(F,B)
Corr(F,C)
Corr(F,D)
Corr(F,E)
1
Corr(G,F)
G
Corr(G,A)
Corr(G,B)
Corr(G,C)
Corr(G,D)
Corr(G,E)
Corr(G,F)
1
In this example of a 7 asset basket, a small estimation error of 0.5% for 1 set of correlation, would lead to an estimation error of
10.5%, which in turn would make any final option value meaningless.
This is why, together with analytical difficulties in deriving it for higher - dimensions problems, any closed form would be obsolete
when dealing with these options.
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“Altiplano con Memoria” Options
Pricing – Path Generation & the Geometric Brownian Motion
For path-dependent options (like Mountain Range Options), we need the whole path or, at least, a sequence of values
of the underlying at given time events.
The first step in simulating a price path is to choose a stochastic process to model changes in financial asset prices.
Stock prices are often modelled by the GBM:
𝒅𝑺𝒕 = 𝝁𝑺𝒕𝐝𝐭 + 𝝈𝑺𝒕𝒅𝑾𝒕
Using Ito’s Lemma, we may transform the above equation into the following form:
𝟏
𝒅𝒍𝒐𝒈𝑺𝒕 = (𝝁 − 𝝈𝟐)𝒅𝒕 + 𝝈𝒅𝑾𝒕
𝟐
The last equation is particularly useful, as it can be integrated exactly and discretized, yielding to:
𝑺𝒕 = 𝑺𝟎𝒆(𝝂𝜹𝒕+𝝈
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𝜹𝒕𝜺 )
Mountain Range options
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“Altiplano con Memoria” Options
Level of the Underlying
Case A: All coupons paid
850
825
800
775
750
725
700
675
650
625
600
575
550
525
500
475
450
425
400
375
350
325
300
275
250
225
200
175
150
125
100
75
50
25
0
S1
S2
S3
S4
B1
B2
B3
B4
T1
T2
T3
T4
T5
T6
Barrier Observation Dates
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“Altiplano con Memoria” Options
Level of the Underlying
Case B – Some coupons paid (C1,C2 & C3) and some not (C4,C5 & C6)
675
650
625
600
575
550
525
500
475
450
425
400
375
350
325
300
275
250
225
200
175
150
125
100
75
50
25
0
S1
S2
S4
B1
B2
B3
B4
S3
T1
T2
T3
T4
T5
T6
Barrier Observation Dates
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“Altiplano con Memoria” Options
Greeks – Estimation - Delta
The Greeks are the quantities representing the sensitivity of the price of derivatives to a change in underlying
parameters on which the value of an instrument is dependent.
The Delta, in particular, measures the rate of change of option value with respect to changes in the underlying asset
price.
In a Monte Carlo framework, Greeks estimation requires a Finite Difference Approximation approach.
This method is based on the re-calculation of the option value with a slight change of one of the input parameters, so
that the sensitivity of the option value to that parameter can be estimated. The parameter in question is the value of
the underlying.
𝜟=
𝝏𝒇(𝑺𝟎 )
𝒇 𝑺𝟎 + 𝜹𝑺𝟎 − 𝒇 𝑺𝟎
= 𝐥𝐢𝐦
𝜹𝑺𝟎→𝟎
𝝏𝑺𝟎
𝜹𝑺𝟎
This idea is however to naive and it can be shown that taking a central difference may be preferable in order to reduce
the variance of the estimator:
𝜟=
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𝒇 𝑺𝟎 + 𝜹𝑺𝟎 , 𝝎 − 𝒇 𝑺𝟎 − 𝜹𝑺𝟎 , 𝝎
𝟐𝜹𝑺𝟎
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Should you have any
questions…
Paolo Pirruccio
Risk Analyst
[email protected]
Copyright © Arkus Financial Services - 2014
Mountain Range options
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Mountain Range options
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