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# RNV - UGBA 103 Options Pricing and Risk-neutral Valuation 1...

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UGBA103 Avinash Verma 1 UGBA 103 Options Pricing and Risk-neutral Valuation

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2 Fruit seller's Problem Price #(Apples) #(Bananas) Basket 1 \$19 30 10 Basket 2 \$13 10 20 Given the observed market prices of the following : Price the following new basket: Price #(Apples) #(Bananas) Basket 3 ? 70 90 UGBA103 Avinash Verma
3 First Approach to the Problem   In  Replicating Portfolio  (RP) approach, we  replicate   the new basket as a “portfolio” of the two existing  baskets  Suppose  x 1  units of basket 1  and  x 2  units of basket 2   replicate the new basket. Then, because #(apples) in RP  must equal those in basket 3 : 30 x 1  +10 x 2  = 70…………………… ………………… ( 1 ), Similarly equating #(Bananas) in RP and basket 3  gives  us: 10 x 1  +20 x 2  = 90 ……………………………………… ( 2 ) UGBA103 Avinash Verma

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4 RP Approach (2)  The solution is  x 1  = 1;  x 2  = 4   If the price of basket 3  differs from that  of RP, there will be opportunities for risk  free arbitrage (assuming no  transactions costs)  Therefore,  P 3  =  x P 1  +  x P = 1*19 + 4*13  = \$71 UGBA103 Avinash Verma
5 Second Approach to the Problem  Solve for  q A , the (implied) price of an apple,  and  q B , the (implied) price of a banana  The equations for these implied prices are:  30 q A   + 10 q B   = 19 ……………………………… ( 3 ),    10 q A    + 20 q B   = 13 … …………….…………    ( 4 )  The solution is  q A  = 0.5;  q B   = 0.4, and       P 3  = 70 q A   + 90 q B   = \$71 UGBA103 Avinash Verma

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6 Investment Banker's Problem Price at t=0 Future (t=1) Price in Event A Event B Security 1 \$19 \$30 \$10 Security 2 \$13 \$10 \$20 Given the observed market prices of the following : Price the following new security or IPO: UGBA103 Avinash Verma Price at t=0 Future (t=1) Price in Event A Event B Security 3 ? 70 90
7 Investment Banker's Problem (2)  The mathematics of the solution is not affected by the  change in context from fruit-seller’s problem of pricing a  new basket to the Investment Banker’s (IB) problem of  pricing an IPO  Both approaches remain valid  Approach 1 is known as Replicating Portfolio (RP)  Approach    Approach 2 is known as “Pure Securities” (PS) Approach  Approach 1 and 2 are mathematically equivalent * , and  together make up  risk-neutral valuation (RNV)    *They are known as duals of each other in linear algebra UGBA103 Avinash Verma

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8 Investment Banker's Problem (3)    In contrast to RNV, Standard [fundamental  or DCF] approach to IB problem would call for:  Estimating Expected Future Cash flows  (Since we are in a one-period framework,  assuming no dividends, expected future cash  flows are expected prices of the two securities
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RNV - UGBA 103 Options Pricing and Risk-neutral Valuation 1...

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