1. 2000 년 4 월 19 일 On Financial Derivatives 김 용 환 국제금융센타 ( KCIF )

2. Assume IBM price on 4/19/2000 to be x: 4/19/2000 => Long IBM @100 4/19/2001 => Short IBM @x 4/19/2001 => Profit and Loss = x – 100

3. [ Agreement ] 4/19/2000 => Long IBM @100 4/19/2001 => Short IBM @x 4/19/2001 => Finance Charge for 1-year = 10 4/19/2001 => Profit and Loss = x – (100+10)

4. [ Agreement ] 4/19/2000 => Agreed to be Long IBM @y on 4/19/2001 ( y is fixed on 4/19/2000 ) 4/19/2001 => Profit and Loss = x – y ( x is IBM price on 4/19/2001 ) Note: if y = 110, P&L of Scenario II = P&L of Scenario III. (profit and loss)

5. Assume the finance charge to be 10% per annum: 4/19/2000 => Borrow Cash y/1.1 4/19/2000 => Long IBM @100 4/19/2000 => Value = 100 – (y/1.1) 4/19/2001 => Pay back Cash y 4/19/2001 => Short IBM @x 4/19/2001 => Profit and Loss = x – y – [100 – (y/1.1)] Note: if y = 110, P&L of Scenario II = P&L of Scenario IV.

6. Kolmogrov’s Strong Law of Large Numbers Let X i ’s be independent random numbers sampled from the same distribution with mean (expectation)  and let S(n) = (  X i )/n (arithmetic average). Then lim S(n) =  with probability one. Example Consider a game - tossing a coin and receiving $1 for heads and nothing for tails. Fifty cents is a fair price of the game in the Strong Law sense.

7. Time Value of Money Let t < T. Then the value at time t of a dollar promised at time T is given by exp{– r(T– t)} for some constant r > 0. The rate r is then the continuously compounded interest rate for this period. Stock Model Assume X  N ( ,  ). Then the natural logarithm of the stock price over some time period is given by X, i.e., ln (S T ) – ln (S t ) = X, or S T = S t exp(X)

8. Forward Two parties enter into a contract whereby one agrees to give the other the stock at some agreed time(T) in the future in exchange for an amount(K) agreed now. By the strong law, E[ exp (–rT) (S T – K) ] = 0 => K = E[S T ] => K = S 0 exp(  +0.5  2 ) Now: Borrow S 0 and buy the stock with it and short forward at K At T: Pay back S 0 exp(rT) and settle forward at K => K = S 0 exp(rT)

9. Call Option The buyer of a call option has the right to buy the underlying security on or before a specific date (T) at a specific price (K). Then the payoff = Max[ S T – K, 0 ]. Let C = C( t, S(t) ) denote the value of the option at time t. dS =  S dt +  S dW dB = rB dt dC = C t dt + C S dS + 0.5 C SS dS dS = [ C t +  SC S + 0.5  2 S 2 C SS ] dt +  SC S dW Let  = C –  S with  = C S, then d  = [ C t + 0.5  2 S 2 C SS ] dt = r  dt

10. We have the Black-Sholes partial differential equation C t + rSC S + 0.5  2 S 2 C SS = rC with the boundary condition, C( T, S(T) ) = payoff dS =  S dt +  S dW dB = rB dt, letting Z = S/B, then dZ = (  – r)Z dt +  Z dW ( by Ito’s lemma ) letting = (  – r)/ , then dW = dt + dW ( by Girsanov’s Theorem ) dZ =  Z dW Z T = Z 0 exp(–0.5  2 t +  dW T ) C( 0, S(0) ) = E[ exp(– rT) payoff ]

11. There is a risk-neutral probability or an equivalent Martingale measure if and only if there is no arbitrage. Also, there is a unique risk-neutral probability if and only if any European derivative asset can be hedged. In this case, the model is said to be complete.