MOUNTAIN RANGE OPTIONS

Transcription

1 MOUNTAIN RANGE OPTIONS Paolo Pirruccio Copyright Arkus Financial Services Mountain Range options Page 1

2 Mountain Range options Introduction Originally marketed by Société Générale in 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). Copyright Arkus Financial Services Mountain Range options Page 2

3 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. Copyright Arkus Financial Services Mountain Range options Page 3

4 Mountain Range options Definition and types m - Everest - a long-term option in which the option holder gets a payoff based on the worst-performing securities in the basket m - 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 m - Himalayan - based on the performance of the best asset in the portfolio m - Atlas - in which the best and worstperforming securities are removed from the basket prior to execution of the option m - 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. Copyright Arkus Financial Services Mountain Range options Page 4

5 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. Copyright Arkus Financial Services Risk-based Governance Solutions

6 Altiplano con Memoria Options Payoff structure Payoff i = η N C i if i = 1 i 1 Payoff(i) = η (N C i C n ) if i = 2, 3,.., n n=1 η = 1 if Min 1 j n,t 1 t t 2 0 else S j t S j 0 L 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 Copyright Arkus Financial Services Mountain Range options Page 6

7 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 C n ) 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 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 Copyright Arkus Financial Services Mountain Range options Page 7

8 Altiplano con Memoria Options Pricing: the idea behind Monte Carlo Integration Consider an integral on the unit interval [0,1]: I = 1 g x dx 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 U i of independent random samples from the uniform distribution and then evaluate the sample mean: I m = 1 m m i=1 g(u i) The strong law of large numbers implies that, with probability 1, lim m + I m = I Copyright Arkus Financial Services Mountain Range options Page 8

9 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 = would return 1.959, because 97.5% of the probability of a normal pdf occurs in the region where X < Copyright Arkus Financial Services Mountain Range options Page 9

10 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 = M T M Once retrieved this matrix, we may apply the following algorithm to generate correlated random numbers X: Generate n independent standard normal variates Z 1, Z 2,..., Z n Return X = μ + M T Z, where Z = Z 1, Z 2,..., Zn T is a vector of uncorrelated variables Suppose we must generate sample paths for two correlated Wiener processes, having covariance matrix Σ = 1 ρ ρ 1 It can be verified that the Cholesky Matrix is M = 1 0 ρ (1 ρ 2 ). Hence, to simulate bidimensional correlated Wiener Process, we will create two independent standard normal variates Z 1 and Z 2 and use: x 1 = Z 1 and x 2 = ρz 1 + (1 ρ 2 )Z 2 Copyright Arkus Financial Services Mountain Range options Page 10

11 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. Copyright Arkus Financial Services Mountain Range options Page 11

12 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: ds t = μs t d t + σs t dw t Using Ito s Lemma, we may transform the above equation into the following form: dlogs t = (μ 1 2 σ2 )dt + σdwt The last equation is particularly useful, as it can be integrated exactly and discretized, yielding to: S t = S 0 e (νδt+σ δtε ) Copyright Arkus Financial Services Mountain Range options Page 12

13 Level of the Underlying Altiplano con Memoria Options Case A: All coupons paid T1 T2 T3 T4 T5 T6 S1 S2 S3 S4 B1 B2 B3 B4 Barrier Observation Dates Copyright Arkus Financial Services Mountain Range options Page 13

14 Level of the Underlying Altiplano con Memoria Options Case B Some coupons paid (C1,C2 & C3) and some not (C4,C5 & C6) T1 T2 T3 T4 T5 T6 S1 S2 S4 B1 B2 B3 B4 S3 Barrier Observation Dates Copyright Arkus Financial Services Mountain Range options Page 14

15 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(s 0) S 0 f S 0 + δs 0 = lim δs 0 0 δs 0 f S 0 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 S 0 + δs 0, ω f S 0 δs 0, ω 2δS 0 Copyright Arkus Financial Services Mountain Range options Page 15

16 Should you have any questions Paolo Pirruccio Risk Analyst Copyright Arkus Financial Services Mountain Range options Page 16

17 Copyright Arkus Financial Services Mountain Range options Page 17