Ass 23 Portfolio Manager Big Bank Toronto Tdesk

Ass #2/3 - Portfolio. Manager Big. Bank Toronto Tdesk 1 Tokyo Tdesk 2 New York Jdesk 1 Ndesk 1 10 6 7 8 100 4 9 20 13 bond 1 bond 2 stock 1 option 1 fxfut 2

Portfolio. Mangaer % java Portfolio. Manager infile instruments mark-to-market: portfolio/positions CADdown: $230, 942, 340 -$3, 456, 333 Irdown: +$2, 456 market data scenarios … ASCII, XML l Zero coupon bonds l FX futures l Equities l European equity call options

Interest Rates - Pricing a Bond Zero Coupon Bond l Face Value: $1000 l Matures: April 19, 2001 l Interest: 5% simple $1050 6 months 19/10/2000 19/04/2001 $? , ? ? ?

Interest Rates - Relative Pricing It depends on what other investments are available. Assume only other investment is a US T-Bill returning 7% each half-year. $(1. 07 x. P) T T + 6 mo. $P

Interest Rates - Pricing the First Coupon $(1. 07 x. P) T T + 6 mo. $1050 $P 19/10/2000 Alternate investment. 19/04/2000 $? ? $1050 = 1. 07 x P P = $981. 31 Supply and Demand will bring prices in-line

Interest Rates - Adding More Realism l Actually, – T-Bills are priced by the market like anything else. – There alternative investments at all sorts of maturities out to 30 years.

Interest Rates - The Spot Zero Curve l The “spot zero curve” captures these rates of return in one concise curve. l Gives YTM (yield-to-maturity) for non coupon bearing bonds of various maturities. l Better to use a concept called “discount factors”

Interest Rates - Units l Discount Factors – converts future dollars to present dollars l Can express equivalently as interest rates which are considerably more intuitive. Say 5 yr. discount factor is 0. 50835 Bond worth $1000 five years from now costs $508. 35 today. Can express YTM of bond in units of annualised interest compounded annually. Can also express in units of annualised interest compounded semi-annually. All the same!

Interest Rates - Compounded Units 98/01/01 98/06/01 99/01/01 99/06/01 00/01/01 00/06/01 01/01/01 01/06/01 02/01/01 02/06/01 03/01/01 $1000 x 0. 50835 $508. 35 x. Y 508. 35 x Y $1000 Y = 1. 07 x. Y 10 x. Y x. Y = YTM = 7% semi-annual, semi-annually compounded YTM = 14% annualised, semi-annually compounded

Interest Rates - Daycount Basis l Glossed over issue of units of time. l Actually, all units are in days, although they seem to be quoted in years! l Missing bit of information is the “daycount basis”. Time in days = Time in years Daycount l Examples of daycount bases: ACT/360, ACT/365, ACT/ACT

Interest Rates - Years in Daycount of Bond $50 98/01/01 98/06/01 $46. 73 l Years between 98/01/01 and 98/06/01 are computed as follows: l Days between = 31 + 28 + 31 + 30 = 181 For a ACT/360 daycount, Time in years = 181/360 = 0. 50278 l For a ACT/365 daycount, Time in years = 181/365 = 0. 49589 l For a ACT/ACT daycount, Time in years = 181/365 = 0. 49589 l

Interest Rates - Converting using Daycount 98/01/01 98/06/01 99/01/01 99/06/01 00/01/01 00/06/01 01/01/01 01/06/01 02/01/01 02/06/01 03/01/01 $1000 x 0. 50835 $508. 38 = days = 365 + 366 + 365 = 1826 1 x $1000 2*days/360 (1 + YTM 2 ) 5 ACT/360 daycount basis annualised rates w/ semi-annual compounding

Interest Rates - Same Rate, Different Units YTM annualised rates, semi-annually compounded, ACT/360 daycount 13. 793% annualised rates, semi-annually compounded, ACT/365 daycount 13. 991% annualised rates, semi-annually compounded, ACT/ACT daycount 13. 999% annualised rates, annually compounded, ACT/ACT daycount 14. 489% annualised rates, daily compounded, ACT/365 daycount 13. 526% annualised rates, continuously compounded, ACT/365 daycount 13. 523%

Interest Rates - Continuous Compounding 1 $508. 38 = x $1000 m*days/365 (1 + YTM m ) 5 In the limit as ‘m’ (number of compounding periods in a year) goes to infinity: $508. 38 = e -YTM x days/365 x $1000 YTM = -ln(508. 38/1000)*365/1826 = 13. 523% cont. ACT/365

Interest Rates - Bond Pricing w/ a Zero Curve Bond pricing using a “real” spot zero curve $1, 05 0 98/01/01 03/01/01 Units are annualised rates, continuously compounded, on an ACT/365 daycount basis P = $1050 x e- 0. 06 x 5. 03 P = $776. 46

Interest Rates - Parity l Each distinct currency has its own zero curve. l No reason borrowing in USD should be the same rate as borrowing in CAD. USD 1 yr. rate = 10% ANNU ACT/ACT CAD 1 yr. rate = 5% ANNU ACT/ACT l Q. Why not convert into USD and invest there? A. Because exchange rates could move in 1 yr. and kill you. l But, by using FX Futures contracts, I can lock in a rate today and know exactly what the exchange rate will be in 1 yr. ’s time.

Interest Rates - Parity l This leads to a relationship between – the CAD-USD spot fx rate, – the USD 1 yr. spot IR rate, – the CAD-USD 1 yr. forward fx rate. l If this relationship is broken, arbitrageurs working at large banks will trade and make instantaneous risk-free profits. l Forces of supply and demand will force the prices back into alignment.

Interest Rates - Parity $137 CAD 1 yr. rate = 5% ANNU ACT/ACT Borrow in Canada $143. 85 CAD spot fx = 1. 37 CAD/USD 1 yr. forward fx must be = 1. 31 CAD/USD $110 USD Lend in U. S. 1 yr. rate = 10% ANNU ACT/ACT $100 USD

Interest Rates - IR Parity Arbitrage Say 1 yr. future fx rate was 1. 37 and not 1. 31. – Borrow $100 CAD at 5% (owe $105 CAD in 1 yr. ’s time) – – Buy $4. 55 CAD worth of candy bars. Convert $95. 45 CAD at 1. 37 to $69. 67 USD Loan $69. 67 USD at 10% Enter into 1 yr. fx forward contract at 1. 37 CAD/USD – In 1 yr. ’s time • Get back $76. 64 USD • Use forward contract to convert to $105 CAD at 1. 37 CAD/USD • Pay back $105 CAD dept in its entirety – Net result: Ahead 4 candy bars! No risk taken!

Option Pricing l Deals with the valuation of risky securities. 50% +$1, 000 -$1, 000 50% +$2, 000 -$? ? 50% +$1, 000 50% $0 Q. How much would you pay? A. It depends.

Option Pricing - Stock Call Option to purchase 100 shares of IBM stock l l On Feb. 17, 1998 l At a strike of $65 per share l Current price is $62 $? ? 97/10/22 98/01/17 $? ?

Option Pricing - Call Option Payout 100 x (St - X) X = $65 97/10/22 So = $62 $0 Option Payout St-X X Stock Price: St 98/01/17

Option Pricing - Computing Option Value X St Sto S 0 ck P ric e + e m i T

Option Pricing - Model of Stock Prices l To compute distribution of stock price in the future, we need a model of how stock prices will change through time. l Model used is geometric Brownian motion.

Option Pricing - Markov Process l A stochastic process where only the current value is relevant for predicting the next value l Past history is not taken into account.

Option Pricing - Wiener Process l Also called Brownian motion Used in Physics to describe the motion of a particle that is subject to a large number of small molecular shocks. l dz = n. sqrt(dt) where n is drawn from a standardised normal distribution N(0, 1) z t

Option Pricing - The Generalised Wiener Process dz = n. sqrt(dt) expected drift rate mean of change in x = a. T dx = a. dt + b. dz variance of change in x = b 2. T variance rate where a and b are constants. x = a. t x T t

Option Pricing - Constant 14% Rate of Return

Option Pricing - Ito Process dz = n. sqrt(dt) dx = a(x, t). dt + b(x, t). dz where a and b are functions of x and t.

Option Pricing - Stock Process S 0=60, m=14%, s=20% d. S/S = m. dt + s. dz d. S = S. m. dt + S. s. dz constant rate of return drift constant rate of return variance T

Option Pricing - Lattices Can model this process as a lattice on stock prices. l dt S. u u=e 5 s. sqrt(dt) 4 S. u. d p S. u 3 2 2 3 S. u. d S 1 -p S. d S. u. d S. d 5 d = 1/u 4 p = (e m. dt - d)/(u-d)

Option Pricing - Example Lattice S 0=100, m=12%, s=30% $119. 7 $112. 7 $106. 2 $100 0. 525 $100 0. 475 $94. 2 $88. 7 dt = 0. 04 yr. $83. 6 $127. 1 (p = 0. 076) $112. 7 (p = 0. 275) $100 (p = 0. 373) $88. 7 (p = 0. 225) $78. 7 (p = 0. 051)

Option Pricing - Pricing an Option S 0=100, m=12%, s=30% $119. 7 $112. 7 X = $110 0. 525 $100 0. 475 $106. 2 $100 $106. 2 $94. 2 $88. 7 dt = 0. 04 yr. $83. 6 $127. 1 (p = 0. 076) $17. 1 $112. 7 (p = 0. 275) $2. 7 $100 (p = 0. 373) $0 $88. 7 (p = 0. 225) $0 $78. 7 (p = 0. 051) $0 Option expected value = 0. 076 x $17. 10 + 0. 275 x $2. 70 = $2.

Option-Pricing - Black-Scholes l In the limit as dt ® 0, can derive a closed-form solution for the expected value of a European option. l Black-Scholes equation. c = S. N(d 1) - X. e -r. (T-t) . N(d 2) d 2 = d 1 - s. sqrt(T-t) 2 d 1 = (ln(S/X) + (r+s /2). (T-t)) / s. sqrt(T-t)