This document describes using a Monte Carlo simulation to price a foreign exchange (FX) target redemption note. The note pays annual coupons based on the exchange rate between two currencies, with the first coupon fixed and subsequent coupons varying. It terminates if accumulated coupons reach a cap. The simulation models the FX rate and domestic/foreign interest rates as correlated stochastic processes. It runs trials simulating the rates over time, calculates coupons, and discounts cash flows to value the note. The Hull-White model is used to simulate the interest rate processes.

1. Pricing Of FX Target Redemption Note By
Simulation
Shing Hing Man
http://lombok.demon.co.uk/financialTap/
October 23, 2012
Abstract
A Monte Carlo method to price a FX target Redemption note is de-
scribed.
1 Introduction And Notation
A FX target redemption note (FX TARN) is a note that pays coupon at regular
intervals on a

2. xed notional. The

3. rst coupon is

4. xed. The coupons after the

5. rst one depend on the exchange rate of two currencies. Also, if the accumulated
coupon reaches a pre-de

6. ned cap, the note is terminated immediately. On early
termination, the notional is also paid. Below is an example of FX TARN.
Notional : 1,000,000 JPY
Maturity : 30 years
Coupon is paid annually
Year 1 coupon is 6%
Year 2 onwards : max(1:0% (FX 80); 0) where FX is AUD/JPY rate
on coupon date.
Accumulated coupon is capped at 12%
In the above example, when the accumulated coupon reaches 12% or more,
the note is terminated. Suppose in year 2, coupon is 5% and year 3 coupon is
2%. Then the note will be terminated at year 3 after the last coupon of 2%
(and the notional) is paid.
In the coupon formula
max(1:0% (FX 80); 0)
1:0% is the scale factor and 80 is the barrier. Also AUD and JPY are the
domestic and foreign currency respectively.
1

7. 2 Assumption On Market Data
The following notation will be used.
rd(t) the short rate of the domestic currency.
rf (t) the short rate of the foriegn currency.
x(t) the FX Spot. At t, x(t) foreign currency is equal to one domestic
currency.
The short rates of the domestic and foreign currency are to be modelled by
the one factor Hull-White model (please see (Section 4)). It is assumed that
x(t); rd(t); rf (t) satisfy the following stochastic equations.
dx(t)
x(t)
= (rd(t) rf (t))dt + x dW (1)
drd(t) = (d(t) ad r(t))dt + d dWd (2)
drf (t) = (f (t) af r(t))dt + f dWf (3)
where W;Wd;Wf are correlated Brownian motion with correlation matrix
0
@
1 d f
d 1 df
f df 1
1
A
For simplicity sake, it is assumed ad; af ; d; f ; x are constant.
3 Simulation
Let t 0 be given. Suppose the initial FX spot, initial zero curve of domestic
and foreign currency, and ad; af ; d; f ; x are given.
3.1 One Trial
Start with t = 0.
1. Draw three correlated standard normal rvs Z1;Z2;Z3 with the given cor-
relation matrix. Use equations (1), (2), (3) to deduce x(t + t); rd(t +
t); rf (t + t).
2. If t+t is a coupon date, then compute the coupon at t+t and move to
step 3. Otherwise, repeat step 1 with t = t + t.
3. If accumlated coupon reaches or exceeds the cap, or maturity date is
reached, then the note is terminated. (Note that notional is also repaid.)
End of trial.
4. Repeat step 1 with t = t + t.
When the note is terminated, the cash
ows (coupons and notional on actual
maturity date) are discounted by the initial foreign currency yield curve. This
is the pv from one trial. For m trials, the price of the FX TARN is the sum of
pv from m trials divided by m.
2

8. 4 Appendix One Factor Hull-White Model
Let r(t) be the short rate. The Hull-White (one factor) interest rate model
describes r(t) by the following stochastic equation ([1, Section 23.9].
dr = ((t) a r(t))dt + dW p
Continuous version (4)
r = ((t) a r(t))t + Z
t Discrete version (5)
r(t + t) = r(t) + ((t) a r(t))t + Z
p
t (6)
where
1. (t) =
dF(0; t)
dt
+aF(0; t)+
2
2a
(1e
2at) and F(0; t) is the initial forward
curve.
2. Z is a random variable from the standard normal distribution.
3. a and (volatility) are parameters. These are usually estimated from
market data.
References
[1] John C. Hull, Options, Futures and Other Derivatives, Prentice Hall, Fifth
Edition
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