Mountain Range Options

Copyright © Arkus Financial Services - 2014 Mountain Range options

MOUNTAIN RANGE OPTIONS Paolo Pirruccio


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



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



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

Copyright © Arkus Financial Services - 2014 Risk-based Governance Solutions

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.


“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



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



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



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



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



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



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


0255075

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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



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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 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


Should you have any questions…

Paolo Pirruccio

Risk Analyst

[email protected]

Mountain Range Options

Economy & Finance

Transcript of Mountain Range Options