DERI L3 slides R.pdf - DERIVATIVES AND RISK MANAGEMENT Option Valuation II(Lecture 3 Matti Suominen January 2018 Two-Period Binomial Tree Model S0=50

# DERI L3 slides R.pdf - DERIVATIVES AND RISK MANAGEMENT...

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DERIVATIVES AND RISK MANAGEMENT Option Valuation II (Lecture 3) Matti Suominen January 2018
Two-Period Binomial Tree Model S 0 =50, u =1.1, d =0.9, (1+r f )=1.05, EX=50, T=2 S 0 = S u = 55 S d = 45 S uu = 60.5 = 50 x (1.1) 2 S ud = 49.5 S dd = 40.5 C 0 = C u = C d = C uu = C ud = C dd = Stock price tree: Call price tree: max(10.5;0) =10.5 0 0
Two-Period Binomial Tree Model Start at t=2 and work backwards, solving one-period trees, using either Replicating portfolio method, or Risk-neutral pricing. Using replicating portfolio method First, at S u node (time t=1) (replicating portfolio: h u shares, borrow B u ) 50 . 7 00 . 45 55 955 . 0 B S h C 00 . 45 05 . 1 0 5 . 49 955 . 0 ) r 1 ( C S h B 955 . 0 5 . 49 5 . 60 0 5 . 10 S S C C h u u u u f ud ud u u ud uu ud uu u = - ´ = - = = - ´ = + - = = - - = - - =
Two-Period Binomial Tree Model This we already saw but just to check, at S d node (time t=1) (replicating portfolio: h d shares, borrow B d ) Now finally at S 0 node (time t=0) (replicating portfolio: h 0 shares, borrow B 0 ) 36 . 5 14 . 32 50 75 . 0 B S h C 14 . 32 05 . 1 0 45 75 . 0 ) r 1 ( C S h B 75 . 0 45 55 0 5 . 7 S S C C h 0 0 0 0 f d d 0 0 d u d u 0 = - ´ = - = = - ´ = + - = = - - = - - = 0 C , 0 B , 0 h d d d = = = What if call trades at 7,36? Arbitrage!
# of shares € amount in shares € amount borrowed Value of portfolio At t=0, S 0 =50 0.75 37.5 (=0.75x50) -32.14 5.36 At t=1, if S 1 =55 before rebalancing 0.75 41,25 -33,75 7.5 after rebalancing 0.955 52.5 -45 7.5 rebalancing trades: Buy: 0,955-0,75 shares @ 55 11,25 Borrow: -(45 - 33,75) -11,25 NET 0 At t=1, if S 1 =45 before rebalancing 0,75 33,75 -33,75 0 after rebalancing 0 0 0 0 Dynamic Hedging: rebalancing trades: Sell: 0,75 shares @ 45 33,75 Pay back loan: -33,75 - 33,75 NET 0
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 5.36 = 1.05 0.25x0 + 0.75x7.5 = r + 1 p)C - (1 + pC = C f d u 0
ALLOWING FOR MORE FREQUENT PRICE CHANGES: Keep T fixed and reduce Δt: u ² S uS S udS dS d ² S D t T How to choose u and d? Take u = e t s D and d = 1/ u, where σ is the empirically observed volatility of the return on the stock. Annual volatility Time during which prices change once as fraction of year
What is the final distribution of stock prices? As D 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 s 2 T. And the stock prices are distributed lognormally :
What is the limiting value of the call option? Black-Scholes Option Pricing Formula Let: s = 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 D 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 ( / ) s s d 2 = d 1 - s T
Example: Call option on a stock: S 0 = 50 EX = 45 s = 18% (annual) T = 3 months = .25 r A = 6% (annual) ( ) d r T c 1 50 45 18 25 18 25 2 = + + ln / . .

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