Pricing and Hedging of Japan Equity-linked Power Reverse Dual Note Using Least Squares Monte Carlo Simulation Frank Fung Shaoyu Hou Nicholas Lee Mayank Rao Abstract We computed the price of a structured product, the Japan Equity-linked Power Reverse Dual note (hereafter referred to as JRD), using a Monte Carlo approach with Least Squares to account for early exercise. We benchmarked the pricing model on a number of simpler derivatives with closed form solutions or highly accurate pricing from FD schemes. The sensitivity of JRD with respect to various parameters and performance from different basis functions is investigated. We employed quasi-random sequences, latin hypercubes and Brownian bridges as Variance Reduction techniques. We also hedge the JRD using static and dynamic delta hedging.
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Pricing and Hedging of Japan Equity-Linked Power Reverse Dual Note
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Pricing and Hedging of Japan Equity-linked Power Reverse Dual Note Using Least Squares
Monte Carlo Simulation
Frank Fung
Shaoyu Hou
Nicholas Lee
Mayank Rao
Abstract
We computed the price of a structured product, the Japan Equity-linked Power Reverse Dual
note (hereafter referred to as JRD), using a Monte Carlo approach with Least Squares to
account for early exercise. We benchmarked the pricing model on a number of simpler
derivatives with closed form solutions or highly accurate pricing from FD schemes. The
sensitivity of JRD with respect to various parameters and performance from different basis
functions is investigated. We employed quasi-random sequences, latin hypercubes and
Brownian bridges as Variance Reduction techniques. We also hedge the JRD using static and
dynamic delta hedging.
Nomenclature
For illustrative purpose we consider an instance of the product with 8 months until maturity,
2 coupon payments in total (including that at maturity) and monthly observations:
T: Date of maturity
τ: Time interval between two consecutive observation dates t�: The date of the payment of the nth
coupon (n=1 in this case) S�: Stock price of the ith
stock in the basket B�: Knock-out barrier of the ith
stock in the basket
Problem Description
Japan Equity-linked Power Reverse Dual Target Accrual Note (JRD): The JRD is a note targeted
towards Japanese investors wishing to gain exposure to US equity markets. Investors can also
gain exposure to FX via the “power reverse” feature. The note pays off a coupon based on
the best performing stock in a basket of 5 stocks. A detailed definition of terms and product
specifications can be found in the following sections.
Definition of terms:
Power Reverse Dual: Reverse dual notes offer coupons in domestic currency based on a
foreign interest rate. Power reverse dual notes offer FX exposure and are suitable when you
expect the domestic/foreign rate to decline.
The JRD note offers a twist on this standard product by allowing investors to gain equity
exposure instead of just interest rates (which are currently at historic lows and not attractive).
Moreover, the JPY is currently very strong. However given the state of the Japanese economy
relative to the US economy, we expect the JPY to decline - thereby the “power” feature in
the note, which exposes the investor to the FX rate, enhances the yield if the yen declines.
Equity Linked Note: This note pays a coupon on a principle of ¥100 based on the best
performing stock from a basket of 5 stocks.
Up and Out Barrier: If any of the stocks in the underlying basket breach their preset barriers,
the note terminates immediately, and investors a paid a bonus coupon to compensate them
for the early termination.
τ
T t�
Product Specification:
The diagram above illustrates the basic setup of the JRD note. The mechanics are as follows:
• Investor pays ¥X to the issuing bank
• Principal is protected, but subject to FX risk.
• On an annual basis, the investor receives ¥ Y*JPYUSD from the bank. Here Y is
calculated as:
� � ��� − �� ��� , 0� ∗ 100 ∀� Here ��� is the stock prices of stock i on each observation day; ��� is the stock
price of stock i at the beginning of the coupon interval and �� is the strike price for
stock i
• The JPYUSD exchange rate is calculated on the day of any cashflow.
• If any of the stocks ��� in the basket hits their “knock out barrier” ��, the note is
terminated, and the investor is paid a bonus coupon
• �� is checked on each observation day.
• After 1 year, if the note has not been redeemed or called, the note terminates and
the investor received his final coupon on a notional of ¥100 (the basic pricing unit for
the JRD)
• On any of the observation dates, the investor can call the product. If the product is
called the investor will be paid a coupon linked to the highest performing asset as of
the exercise date (only in the Bermudan version of the JRD)
Pricing a product with early exercise features is a complicated exercise when using Monte
Carlo simulation. A simple way to do this would be to simulate entire trajectories, and then
at each point consider exercising versus holding on. However the price obtained by following
this procedure will be biased upwards, as it is a “perfect foresight” estimate of the price. In
reality we do not know what the optimal exercise decision will be. Longstaff and Schwartz[1]
present a technique based on a least squares regression that allows you to make a decision
for early exercise based on the conditional expectation of the underlying asset price. This
technique requires the use of basis functions that are capable of expressing information
Investor
Issuer
Basket of
5 stocks
X JPY
Y*JPYUSD Y=Best of 5
Equity exposure
about the underlying prices that can then be used in an ordinary least squares regression to
generate conditional expectations. There is no hard and fast rule for the choice of basis
functions, and we need to experiment with various kinds of functions to determine a good fit.
In this paper we experiment with various kinds of functions (Monomials, Laguerre, Bessel,
Hermite and Chebyshev). We pick a function that best captures the behavior of the JRD
under various scenarios.
To focus on the simulation and the analysis of our product, we decide to use a simple
geometric Brownian motion model for the stock and FX dynamics, namely dS� = �r − y�"S�dt + σ�S�dW� �1" dQ = �r& − r "Qdt + σ'QdW' �2"
with the correlation matrix for the Brownian motions (5 stocks, and 1 FX rate which is
assumed to be uncorrelated to any of the stocks hence has zero entries at all places except a
‘correlation’ of 1 with itself)
ρ) =*++,
1ρ�.ρ./ρ/0ρ010
ρ.�1ρ./ρ.0ρ.10
ρ/�ρ/.1ρ/0ρ010
ρ0�ρ0.ρ0/1ρ010
ρ1�ρ1.ρ1/ρ1010
0000012334 �3"
here r is the foreign risk-free rate, y� is the dividend yield for the ith
stock, σ� is the
volatility of the ith
stock, r& is the domestic risk-free rate and σ' is the volatility of the
6789:;<= =>??9@=AB7?9<C@ =>??9@=A FX rate. Considering the complexity in the structure, we will assume a
constant flat yield curve throughout.
In this paper we consider we have a basket of 5 stocks whose correlation matrix we derived
from historical data. The correlation matrix along with the other parameters can be found in
Appendix A.
Methodology
Here we briefly describe how we priced the product including the least squares regression.
Equity Linked option: Using cholesky decomposition, we generate correlated weiner
processes. These processes are used to simulate stock trajectories. At each observation date
we consider a payout based on the best performing stock if the exercise cashflow is greater
than the conditional expectation of the stock prices at that point (from the Least Squares
regression)
Barrier Option: through matrices “stillAlive” and “breachingPoint” we keep of track of when
the barrier was breached by any stock
Power Reverse Dual (PRDC): Each cashflow is multiplied by the corresponding FX rate which
is also simulated at each point
Pricing on intermediate dates: This feature allows us to dynamically hedge the product. To
handle this, we divide the maturity into intervals and calculate the number of intervals
remaining in each coupon.
Least Squares Regression: At each observation date we determine the conditional
expectation of future prices by regressing the current stock prices against some basis
functions. This is further discussed in the “Basis Functions” section.
Benchmarking
We benchmarked our pricing function against several sources.
1) The basket call from Tavella [2] page 200: This example has a benchmark that was
obtained by running a high quality finite differencing scheme on a basket of 3 assets.
There are no barriers or other exotic features. The results obtained are shown in
Table 1
Exercise
Opportunities
Benchmark JRD
Pricer
Error Diff against
Benchmark
10 17.844 17.782 0.18 -0.062
15 17.969 17.891 0.13 -0.078
30 18.082 17.949 0.18 -0.133
Table 1: Benchmarking against basket of 3 assets from FD scheme[2]
2) Benchmarking against options with analytical results: To validate the JRD’s rainbow,
barrier and FX features, we benchmark our product with simpler derivatives where
analytical solutions are available.
We implemented general closed form pricing models for options contingent to the
maximum or the minimum of any number of assets using the result by Johnson[4].
With respect to the barrier feature, since our product has a discrete barrier which
does not have a closed form solution, we benchmarked our result to a continuous
barrier option’s analytical solution, with displaced barrier as suggested by Broadie
and Glasserman[5].
Theoretical value JRD Pricer
Rainbow option 13.243 13.288
Barrier option (pricing from adjusted
continuous barrier)
3.4371 3.4512
Quanto option 4.1075 4.1409
Table 2: Benchmarking against Analytical option prices
3) Benchmarking against European structures: Another way to benchmark a pricing
function is to consider the European version of the product and the Bermudan
premium. This premium is discussed more extensively in the “European versus
Bermudan structures” section.
Results and Discussion
For pricing the Bermudan JRD note, we utilized Monte Carlo simulations with the Least
Squares approach for early exercise. We also experimented with several variance reduction
techniques to accelerate the pricing over the naïve MC approach. We discuss these
techniques briefly and then present our results.
Quasi-Random Sequences: We implemented the Sobol sequence. In the naïve approach we
sample randomly from a normal distribution. However this approach does not cover the
sample space uniformly and as a result we experience a clustering effect. QRS numbers
rectify this problem. In lower dimensions, QRS numbers cover the sample space relatively
well and therefore lead to more accurate pricing. A problem with QRS numbers is that unlike
the naive approach which has a clearly defined convergence law (and therefore a well
defined termination criterion), convergence is not as well defined. This is primarily because
the theoretical error can be drastically different from the numerical error obtained. It is
therefore good practice to calibrate to the naïve MC. The reader is referred to [2] for further
details on this.
Quasi-Random Sequences with Brownian Bridges: numbers generated from a QRS tend to
work well in low dimensions. As the dimensionality of the problem increases however, the
numbers generated from quasi-random sequences tend to leave gaps. This reduces the
acceleration obtained in higher dimensions. This can be exploited by using Brownian bridges.
The numbers from the lower dimensions can be used at the more important points (end of
the trajectory, mid point of trajectory etc). Each time we add a point using the Brownian
bridge technique, the variance of the new point is reduced by half. We can therefore obtain
significant acceleration in this way. To control the random numbers used in the construction
of the bridge we developed our own implementation of the bridge (rather than use the
default one implemented in MATLAB).
Table 3 shows the results from under different scenarios. The parameters used to obtain
these results are shown in Appendix A. For the Least Squares regression we used 10 Hermite
polynomials on 3 cross products ( �., .. & � .) on the two best performing stocks in the
Figure 3 demonstrates an important point. When the rebate level is set higher than
the barrier (i.e. it is advantageous to hit the barrier), the premium is low (negative in
Figure 3). We know that the Bermudan premium cannot be below the European
value, as we can always choose note to early exercise. We conjecture that the
negative premiums observed are due to the regression function that we implement.
In our implementation we only consider the top two stocks at each observation
dates in order to reduce computational costs. If we early exercise based on these
stocks, it is possible that the European version of the JRD becomes more valuable
due to the barrier being breached by some other stock in the basket.
To correct for this, we experimented with different cross products and also tried
including all stocks prices in the regression. We were unable to determine a
regression function that worked well for this particular case. It was also observed
that including all stocks actually deteriorates the regression, and the interested
reader is referred to Appendix C for a figure that demonstrates the premium
obtained in this case. An added complexity is the FX dynamics which can also
influence the exercise decision.
For our case, we can simply set the price equal to the price of a European version
when the rebate is greater than the barrier (we do not expect the premium to be
very high in this case anyway). The uncertainty introduced by the barrier here should
be dealt with through more sophisticated schemes such as Finite Differences.
Basis Functions
The quality of the Least Squares Monte Carlo can be greatly impacted by the choice of basis
functions. Simple polynomials may not be adequate for use in the least squares regression.
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65-0.05
0
0.05
0.1
0.15
0.2
Rebate
Bermudan Premium as fraction of European Price
Laguerre
Monomial
Hermite
We therefore experimented with a number of polynomials including Laguerre, Bessel
(varying orders), Chebyshev (1st and 2
nd kinds) and Hermite functions. Figure 4 shows the
rebate generated through different basis functions.
Fig 4: Bermudan premium versus number of basis functions for the JRD
For the JRD, we have a basket of 5 correlated stocks. So our choice of basis functions should
also allow for the correlation between these stocks. For computational economy we only
considered running regressions on the two best performing stocks and taking various cross
products of these stocks. A definition of the cross products can be found in Appendix B. We
found that the Chebyshev polynomials and the Bessel functions did not work well with a low
number of cross products (negative premiums generated). Figure 5 demonstrates the impact
of changing the number of cross products on Laguerre and Hermite polynomials.
Note that the “monomial” basis function was taken from [2], and we do not change the
number of cross products in this, therefore it is simply one value.
1 2 3 4 5 6 7-0.05
0
0.05
0.1
0.15
0.2
Number of Basis Functions
Bermudan Premium as fraction of European Price
Laguerre
Monomial
Hermite
Chebyshev1st
Chebyshev2nd
Bessel
Fig 5: Bermudan premium versus number of cross products for the JRD
Another check on the use of different basis functions is to consider the impact of changing
the strike. We do not expect big differences in premiums depending on whether the JRD is in
the money or out of the money. Basis functions that describe this behavior are more suited
for pricing the JRD dynamically. Figure 6 shows premium for Laguerre, Monomials and
Hermite functions (the others do not produce valid results).
Fig 6: Bermudan premium versus normalized strike for the JRD
Figure 6 demonstrates that Monomials and Hermite functions produce the most consistent
results overall. In the pricing numbers presented in tables 3 and 4, we use 10 Hermite
polynomials.
1 2 3 4 5 6 7-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Number of cross products
Bermudan Premium as fraction of European Price
Laguerre
Monomial
Hermite
0.94 0.96 0.98 1 1.02 1.04 1.060.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Normalized Strike
Bermudan Premium as fraction of European Price
Laguerre
Monomial
Hermite
Hedging
In this section we present two techniques for hedging the JRD 1) Dynamic Hedging and 2)
Static Hedging. In dynamic hedging we simulate various trajectories for the underlying stocks,
and at each point compute the delta. Based on the delta we can form a replicating portfolio
that can hedge our short position in the product (assuming we sold the product). By contrast
static hedging assumes that we do not change our hedging portfolio over the lifetime of the
product. We therefore need to mimic the JRD’s early exercise and FX features, and we
achieve this by using a combination of barrier and quanto options.
Dynamic Delta Hedging
We follow a dynamic delta hedging strategy and compute the cost of hedging based on the
change in delta at each time step. This procedure is outlined in greater detail in Hull [3].
To test this strategy we first simulate a set of asset trajectories (without loss of generality we
assume the FX rate is fixed). We will test the delta hedging scheme as if these simulated
trajectories are out of sample realized prices. We choose to carry out the hedging using
Monte Carlo, and in particular using the CV approach. The other two alternatives (i.e.,
Likelihood Ratio and Pathwise Derivative) require us to compute partial derivatives, which is
outside the scope of our investigation.
Figure 7 shows some sample trajectories. Note that the prices here are the actual stock
prices from Appendix A (i.e not the normalized stock prices).
Fig 7: Bermudan premium versus number of cross products for the JRD
Starting from time zero and moving forward in time, we calculate the unperturbed derivative
price PF and five perturbed prices PG�,.,/,0,1. The unperturbed price is calculated simply by
using the MC pricing function with maturity and spot prices updated for each time step. For
0 2 4 6 8 10 12 14 16 180
50
100
150
200
250
300
350Sample Realized Trajectories
each of the perturbed prices, we change the corresponding spot price by multiplying it by �1 + ϵ" where ϵ is small. Note that in calculating the 6 prices (perturbed or unperturbed),
we have used the same set of Wiener processes for the Monte Carlo simulations. This
ensures that the error converges. The individual deltas are then approximated by taking the
ratio of
Δ� = PG� − PFϵPF , ∀i We rebalance the replicating portfolio one period later by changing our unit of stock holdings
to these one-period-lagged deltas. The rebalancing is self-financed by changing the position
in a USD, and in turn a JPY, cash account (recall that in this case the FX rate is fixed). By
repeating this procedure until knock-out or expiry, we obtained a cost (or profit) of hedging
in JPY. If the hedging has been effective, we would expect that the discounted cost should
equal the derivative price.
Figure 8 shows a histogram with the distribution of hedging costs for multiple simulations.
The histogram is close to a normal distribution. However, we notice that the average cost of
hedging is as high as JPY2700.
Fig 8: Cost of hedging (JPY)
To investigate why delta hedging does not work as expected, we first repeat the experiment
using one tenth of the previous volatility levels. Figure 9 shows a histogram of the cost in this
case.
-8 -6 -4 -2 0 2 4
x 104
0
5
10
15
20
25
Cost in JPY
Histogram of Cost of Delta Hedging
Fig 9: Cost of hedging (One Tenth volatility)
The average cost of hedging drops from JPY2700 to JPY800. Furthermore, if we look at the
change in the deltas in different time steps, as shown in the following table, the value of
delta can change abruptly at different points in time.
Time Step KL KM KN KO KP
1 0.0326 0.1617 0.7020 0.1110 3.7821
2 0.0348 0.1829 5.7690 3.7583 1.0930
3 0.0410 0.1817 0.4511 0.1464 -16.9050
4 0.0378 0.2077 1.4777 0.1937 1.4602
Table 5: Delta with respect to individual stocks
Taking the two pieces of evidence into account, we make the following postulation regarding
the poor performance of delta hedging for JRD. The payoff of JRD depends on the return of
the best performer in the basket. By this nature, the value of the JRD is a highly non-linear
function of the underlying asset prices. As a didactic example, suppose that by the CV
approach we perturb the price of Asset 1 and found that Δ� = 0.03 and suppose that at
this price level Asset 1 is the best performer. When the price of Asset 1 changes further,
however, it ceases to be the best performer and the JRD value is determined by the return of
Asset 4. As a consequence Δ� is not a smooth function of S1. This agrees with the abruptly
changing deltas we saw in the table. In other words, the gamma of JRD is so high that delta
hedging simply cannot capture the instantaneous risk. This also agrees with the fact that
under lower volatility the hedging performance is marginally improved, since under lower
volatility a best performer has a higher probability to remain the best performer, and vice
versa.
-1.5 -1 -0.5 0 0.5 1 1.5
x 104
0
5
10
15
20
25
Cost in JPY
Histogram of Cost of Delta Hedging
This analysis suggests that hedging may better be achieved through a static approach, where
we do not need to rebalance and incur hedging costs at each observation date. This is
discussed in the next section.
Static Hedging
The exotic features of this product include:
Rainbow option (best performance of the five underlying)
Exchange option (paid in JPY)
Knock out Barrier option (can be triggered by any of the five assets)
First touch option (rebate)
Due to the nature of complexity of the product, it is quite difficult to achieve perfect hedging
by implementing a static hedging strategy. Therefore, we employ an approximate static
hedging using Mean Monte Carlo (MMC). This procedure is equivalent to partially hedging
the claim using a portfolio of simpler assets. If the residual risk is deemed appropriate or can
be reduced to a suitable level, the static portfolio can be employed in place of complex
dynamic hedging procedures (for further details refer to Pellizzari 2001 [6]).
Here we denote the final payoff of the coupon at R = S as T� �U , .U … WU" . We then