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
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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
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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 = - ´ = - = = - ´ = + - = = - - = - - =
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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!
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# 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
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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
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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
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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 :
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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
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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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