Benefits of risk management ppt

business risk and financial risk ppt and absolute and relative risk aversion
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Prof.EvanBaros,United Kingdom,Teacher
Published Date:26-07-2017
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Risk Neutral Pricing Black-Scholes Formula Lecture 19 Dr. Vasily Strela (Morgan Stanley and MIT)Risk Neutral Valuation: Two-Horse Race Example  One horse has 20% chance to win another has 80% chance  10000 is put on the first one and 50000 on the second If odds are set 4-1: • Bookie may gain 10000 (if first horse wins) • Bookie may loose 2500 (if second horse wins) • Bookie expects to make 0.2 (10000) + 0.8 (-2500) = 0 If odds are set 5-1: • Bookie will not lose or gain money no matter which horse wins 2Risk Neutral Valuation : Introduction We are interested in finding prices of various derivatives. Forward contract pays S-K at time T : Forward Contract 150 100 50 F(t,S) F(T,S) 0 0 20 40 60 80 100 120 140 160 180 200 -50 -100 S(t)=80, K=88.41, T=2 (years) 3Risk Neutral Valuation: Introduction European Call option pays max(S-K,0) at time T European Call Option 140 120 100 80 C(t,S) 60 C(T,S) 40 20 0 0 20 40 60 80 100 120 140 160 180 200 -20 S(t)=80, K=80, T=2 (years) 4Risk Neutral Valuation: Introduction European Put option pays max(K-S,0) at time T European Put Option 90 80 70 60 50 P(t,S) 40 P(T,S) 30 20 10 0 0 20 40 60 80 100 120 140 160 180 200 -10 S(t)=80, K=80, T=2 (years) 5Risk Neutral Valuation: Introduction  Given current price of the stock and assumptions on the dynamics of stock price, there is no uncertainty about the price of a derivative  The price is defined only by the price of the stock and not by the risk preferences of the market participants  Mathematical apparatus allows to compute current price of a derivative and its risks, given certain assumptions about the market 6Risk Neutral Valuation: Replicating Portfolio Consider Forward contract which pays S-K in time dt. One could think that its strike K should be defined by the “real world” transition probability p: p(S -K)+(1-p)(S -K)=pS +(1-p)S -K 1 2 1 2 K = pS +(1-p)S 0 1 2 If p=1/2, K =(S +S )/2 0 1 2 7Risk Neutral Valuation: Replicating Portfolio Consider the following strategy: 1. Borrow S to buy the stock. Enter Forward 0 contract with strike K 0 2. In time dt deliver stock in exchange for K and 0 rdt repay S e 0 rdt • If K S e we made riskless profit rdt 0 0  K S e rdt 0 0 •If K S e we definitely lost money 0 0 Current price of a derivative claim is determined by current price of a portfolio which exactly replicates the payoff of the derivative at the maturity 8Risk Neutral Valuation: One step binomial tree Suppose our economy includes stock S, riskless money market account B with interest rate r and derivative claim f. Assume that only two outcomes are possible in time dt: rdt S , B e , f 1 0 1 p S , B , f 0 0 0 1-p rdt S , B e , f 2 0 2 9Risk Neutral Valuation: One step binomial tree For a general derivative claim f, find a and b such that rdt f =aS +bB e 1 1 0 rdt f =aS +bB e 2 2 0 Then f =aS +bB 0 0 0 Easy to see that f f S f S f 1 2 1 2 2 1 a , b rdt S S (S S )B e 1 2 1 2 0  f f S f S f rdt rdt 1 2 1 2 2 1 f e S e  0 0 S S S S  1 2 1 2 10Risk Neutral Valuation: One step binomial tree One should notice that rdt rdt  S e S S S e rdt 0 2 1 0 f e f f  0 1 2 S S S S  1 2 1 2 -rdt f = e (f q + f (1 - q)) 0 1 2 where rdt q=(S e -S )/(S -S ), 0q1 0 2 1 2 Moreover rdt S q+S (1-q)= e S 1 2 0 11Risk Neutral Valuation: Continuous case -r(T-t) f =e E f t Q T Q is the risk neutral (martingale) measure under which -rt S =e E S 0 Q t 12Black-Scholes equation Assume that the stock has log-normal dynamics: dS = Sdt + SdW Where dW is normally distributed with mean 0 and standard deviation (i.e. W is a Brownian Motion) dt We want to find a replicating portfolio such that df = adS + bdB 13Black-Scholes equation Use Ito’s formula: 2 ff 1 f 2 df (S,t) dt dS (dS) 2 tS 2S 2 2 2 (dS) S dt (analogous to first order Taylor expansion, up to dt term) 14Black-Scholes equation df=adS+bdB 2 Substitute dS, df, dB=rBdt and (dS) 2  ff 1 ff 2 2 S S dt SdW (aS brB)dt a SdW  2 tS 2SS  Compare terms 2 ff 1 f 2 2 a , brB S 2 St 2S 15Black-Scholes equation bB=f-aS is deterministic and as dB=rBdt d(f-aS)=r(f-aS)dt 2 f ff 1 f 2 2 and a Substituting once again df dt dS S dt 2 S tS 2S we obtain the Black-Scholes equation 2 f 1 ff 2 2  S rS rf 0 2 t 2SS Fisher Black, Myron Scholes – paper 1973 Myron Scholes, Robert Merton – Nobel Prize 1997 16Black-Scholes equation • Any tradable derivative satisfies the equation • There is no dependence on actual drift  • We have a hedging strategy (replicating portfolio) • By a change of variables Black-Scholes equation transforms into heat equation 2  u u  2 x 17Black-Scholes equation Boundary and final conditions are determined by the pay-off of a specific derivative For European Call C(S,T)=max(S-K,0) C(0,t) 0,C(,t) S For European Put P(S,T)=max(K-S,0) r(Tt) P(0,t) Ke , P(,t) 0 18Black-Scholes equation For European Call/Put the equation can be solved analytically r(Tt) r(Tt) C e e SN(d ) KN(d )  t 1 2 r(Tt) r(Tt) P e KN(d ) e SN(d )  t 2 1 where 2 ln(S / K) (r /2)(T t) d 1  T t 2 ln(S / K) (r /2)(T t) d 2  T t x 2 1 u /2 N(x) e du  2  19Black-Scholes: Risk Neutral Valuation -r(T-t) f =e E f t Q T Q is the risk neutral measure under which dS=rSdt+SdW 2 2  1 (ln(S / S ) (r /2)(T t)) T t PDF(S ) exp  T 2 2 (T t)  S 2T  20

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