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MOUNTAIN RANGE OPTIONS Paolo Pirruccio
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Mountain Range options
► 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).
Introduction
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Mountain Range options
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.
Usage
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Mountain Range options
Himalayan - based on the performance of the best asset in the portfolio.
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.
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.
Atlas - in which the best and worst-performing securities are removed from the basket prior to execution of the option.
Everest - a long-term option in which the option holder gets a payoff based on the worst-performing securities in the basket.
3.300 m -
4.167 m -
8.000 m -
8.091 m -
8.848 m -
Definition and types
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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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“Altiplano con Memoria” Options
𝑷𝒂𝒚𝒐𝒇𝒇 𝒊 = 𝜼 𝑵 ∗ 𝑪 ∗ 𝒊 𝒊𝒇 𝒊 = 𝟏
𝑷𝒂𝒚𝒐𝒇𝒇(𝒊) = 𝜼 (𝑵 ∗ 𝑪 ∗ 𝒊 − 𝑪𝒏)
𝒊−𝟏
𝒏=𝟏
𝒊𝒇 𝒊 = 𝟐, 𝟑, . . , 𝒏
𝜼 = 𝟏 𝒊𝒇 𝑴𝒊𝒏 𝟏≤𝒋≤𝒏,𝒕𝟏≤𝒕≤𝒕𝟐
𝑺𝒋𝒕
𝑺𝒋𝟎 ≥ 𝑳
𝟎 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
Payoff structure
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“Altiplano con Memoria” Options
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
𝑷𝒂𝒚𝒐𝒇𝒇(𝒊) = 𝜼 (𝑵 ∗ 𝑪 ∗ 𝒊 − 𝑪𝒏)
𝒊−𝟏
𝒏=𝟏
𝒊𝒇 𝒊 = 𝟐, 𝟑, . . , 𝒏
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
Pricing – The algorithm
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“Altiplano con Memoria” Options
Consider an integral on the unit interval [0,1]:
𝑰 = 𝒈 𝒙 𝒅𝒙𝟏
𝟎
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:
𝑰𝒎 =𝟏
𝒎 𝒈(𝑼𝒊)
𝒎
𝒊=𝟏
The strong law of large numbers implies that, with probability 1, 𝒍𝒊𝒎𝒎 → + ∞
𝑰𝒎 = 𝑰
Pricing: the idea behind Monte Carlo Integration
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“Altiplano con Memoria” Options
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
Pricing – Generating normal random variates through the Inverse Transform Method
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“Altiplano con Memoria” Options
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 = 𝑴𝑻𝐌
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 + 𝑴𝑻𝐙 , 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 =𝟏 𝝆𝝆 𝟏
It can be verified that the Cholesky Matrix is 𝐌 =𝟏 𝟎
𝝆 (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
Pricing – Correlated Random Numbers: Cholesky Factorization
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“Altiplano con Memoria” Options
The pricing structure is primarily dependent on the correlation between the constituent stocks.
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.
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
Pricing – the correlation estimation problem and the impossibility to use a closed form
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“Altiplano con Memoria” Options
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:
𝒅𝒍𝒐𝒈𝑺𝒕 = (𝝁 −𝟏
𝟐𝝈𝟐)𝒅𝒕 + 𝝈𝒅𝑾𝒕
The last equation is particularly useful, as it can be integrated exactly and discretized, yielding to:
𝑺𝒕 = 𝑺𝟎𝒆(𝝂𝜹𝒕+𝝈 𝜹𝒕𝜺)
Pricing – Path Generation & the Geometric Brownian Motion
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0255075
100125150175200225250275300325350375400425450475500525550575600625650675700725750775800825850
T1 T2 T3 T4 T5 T6
Leve
l of
the
Un
der
lyin
g
Barrier Observation Dates
Case A: All coupons paid
S1
S2
S3
S4
B1
B2
B3
B4
“Altiplano con Memoria” Options
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0255075
100125150175200225250275300325350375400425450475500525550575600625650675
T1 T2 T3 T4 T5 T6
Leve
l of
the
Un
der
lyin
g
Barrier Observation Dates
Case B – Some coupons paid (C1,C2 & C3) and some not (C4,C5 & C6)
S1
S2
S4
B1
B2
B3
B4
S3
“Altiplano con Memoria” Options
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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)
𝝏𝑺𝟎= 𝐥𝐢𝐦
𝜹𝑺𝟎→𝟎
𝒇 𝑺𝟎 + 𝜹𝑺𝟎 − 𝒇 𝑺𝟎𝜹𝑺𝟎
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 =𝒇 𝑺𝟎 + 𝜹𝑺𝟎, 𝝎 − 𝒇 𝑺𝟎 − 𝜹𝑺𝟎, 𝝎
𝟐𝜹𝑺𝟎
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Should you have any questions…
Paolo Pirruccio
Risk Analyst