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Motivation
A Barrier Reverse Convertible (BRC) is an investment product with the following specifications:
1. The investor pays a certain nominal value and receives fix coupons payment up to maturity of the instrument. The coupon rate is typically much higher than the market rate for an investment in an ordinary bond with same maturity.
2. The repayment of the nominal value at maturity is conditioned on the price evolution of one or more underlying assets, usually stocks.
i) If during the lifetime of the instrument none of the assets falls below a certain barrier, which is usually 50% to 80% of the asset prices when the BRC is issued, then the nominal value is repaid.
ii) If at least one of the assets falls below the barrier and at the maturity date the worst performing stock …
Is lower than at issue, then the product is redeemed in the asset with the worst performance.
Finishes above the initial price, then the nominal value is repaid.
BRCs mainly address retail clients, who are interested in attractive coupon payments, as well as certain institutional investors like smaller pension funds, who want to realize specific payoff structures. The holder of a BRC gives up the upside potential of the underlying asset(s) in exchange for the enhanced coupon, while they are at the same time partially protected against downside risk as long as the barriers are not crossed. Thus, the product is suitable for investors expecting a sideways trending market.
General approach
Several methods have been proposed in the literature for the pricing of BRCs. A straightforward approach is the generation of paths with (Quasi-) Monte Carlo. Let n be the number of assets in the basket of underlyings. Each individual stock follows a Geometric Brownian motion (GBM):
Programming assignment
Write a function in Matlab that determines the value of the option component of a non-callable BRC on an arbitrary number of stocks nn when the product is issued. The valuation is based on (Quasi-) Monte Carlo. The function should receive the following parameters:
Barrier level (single value between 0 and 1). The absolute barriers HH1, … , HHnn for each stock are obtained by multiplying this level with the reference prices.
Nominal value of the product.
Risk-free interest rate rr, continuously compounded.
Lifetime of the instrument TT, specified in years.
The covariance matrix Σ of the joint GBM for the asset prices.
Step size Δtt in years, i.e., Δtt = 1/250 would represent daily steps (250 is the assumed number of trading days per year).
Sample size MM (number of paths).
The following parameters are optional (use name-value pair arguments):
'Dividends' - followed by a cell array. The elements of this cell array are vectors with the (possibly several) dividend amounts for each stock. Since the number of payments (and, thus, the dimension of the vectors) might differ among the stocks, a cell array is used to hold the vectors of different lengths.
'DividendDates' - followed by a cell array. The elements are vectors with the corresponding time points of the dividend payment. The sizes must be consistent with the vectors for the dividend amounts and the times are again measured in years (e.g., 0.25 represents 3 months from the time of the valuation). Must be specified if 'Dividends' are given.
'Antithetic' - followed either by true or false if variance reduction with the antithetic variable method should be used (default is false).
'BBAdjust'- followed either by true or false if an adjustment should be used to reduce the bias of time-discretization (see below, default is false)
Detail
For the scenario generation with QMC, use a multi-dimensional Sobol sequence in combination with the Brownian bridge approach. The latter generates first the realizations of the Wiener process increments, which are then resorted. You can use the function "BrownianBridge" from Studynet (Module 14) and apply it to each individual component of the multidimensional process. In this way, you obtain trajectories of nn independent Wiener processes (or their increments, respectively). See for example the program "qmc test.m" in Module 14. In the next step, you impose the correlation structure to obtain correlated Wiener process increments. Then you would use the realizations of the Wiener process increments to obtain the stock price paths.
Recall that the Brownian bridge allows to overcome the problem of the loss of uniformity in the higher dimensions of QMC sequences when trajectories are generated. Think if it could also be advantageous to order the trajectories from higher volatility processes to smaller volatility processes...
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