28C00400_lecture_3 - Aalto University 2013 Derivatives LECTURE 3 Matti Suominen MULTIPERIOD CASE Extend basic case(EX = 50 rf = 5 by 1 period t=0 t=1

# 28C00400_lecture_3 - Aalto University 2013 Derivatives...

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Aalto University 2013 Derivatives LECTURE 3 Matti Suominen
2 MULTIPERIOD CASE Extend basic case (EX = 50, r f = 5%) by 1 period: t = 0 t = 1 t=2 t = 0 t = 1 t=2 60.5 C uu = 10.5 55 C u = 7.5 50 49.5 C 0 = 5.36 C ud = 0 45 C d = 0 40.5 C dd = 0 At time t=1, how much money is needed to purchase the duplicating portfolio? If S u = 55 h u = 0.955 B u = 45.0 ð C u = 7.5 If S d = 45 C d = 0 How about at time 0? S 0 = 50 h 0 = 0.75 B 0 = 32.14 C 0 = 5.36
3 Dynamic Hedging: # of shares amount in shares amount borrowed Value of portfolio At t=0, S 0 =50 At t=1, if S 1 =55 before rebalancing after rebalancing rebalancing trades: Buy: Borrow: At t=1, if S 1 =45 before rebalancing after rebalancing rebalancing trades: Sell: Borrow:
4 Note that “risk neutral pricing” still works. We just move down the “option tree”: 0.75 = 0.9 - 1.1 0.9 - 1.05 = d - u d - ) r + (1 = p f 7.5 = 1.05 0.25x0 + 0.75x10.5 = r + 1 p)C - (1 + pC = C f ud uu u 0 = C d ð 5.36 = 1.05 0.25x0 + 0.75x7.5 = r + 1 p)C - (1 + pC = C f d u 0
5 ALLOWING FOR MORE FREQUENT PRICE CHANGES: Keep T fixed and reduce Δ t: u ² S uS S udS dS d ² S Δ t T How to choose u and d? Take u = e t σ Δ and d = 1/ u, where σ is the empirically observed volatility of the return on the stock.
6 What is the final distribution of stock prices? As Δ t 0 this approaches a continuous model of stock prices, where the stock return (between time t = 0 and t = T) is distributed normally with some mean μ T (that depends on the true probability of stock price going up) and variance σ 2 T. This implies that the stock prices are distributed lognormally :
7 What is the limiting value of the call option? Black-Scholes Option Pricing Formula Let: σ = volatility of returns r c = interest rate (continuously compounded) T = time to expiration N( d ) = Pr {z d }, where z is distributed according to a standard normal distribution. As Δ t 0 the duplicating portfolio at time 0 approaches: C 0 = h 0 S 0 - B 0 where: h 0 = N( d 1 ) B 0 = N( d 2 ) PV(EX) d S EX r T T T c 1 0 2 = + + ln ( / ) σ σ d 2 = d 1 - σ T