Pricing of fx target redemption note by simulation
- 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
- 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 3