Lecture 8. Spring Semester, Rutgers University. Lecture 8. Options Markets and Pricing. Prof. Paczkowski
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1 Rutgers University Spring Semester, 2009 (Rutgers University) Spring Semester, / 31
2 Part I Assignment (Rutgers University) Spring Semester, / 31
3 Assignment (Rutgers University) Spring Semester, / 31
4 Part II (Rutgers University) Spring Semester, / 31
5 Concept An option is the right to do something, without the obligation to do it. A call option is the right to buy an asset at a fixed price, within a fixed time period. A put option is the right to sell an asset at a fixed price, within a fixed time period. (Rutgers University) Spring Semester, / 31
6 Concept The price at which the exchange is made is called the strike or exercise price. Exercising the option involves exchanging cash for the underlying asset. When a call option is exercised by the holder, the underlying asset is purchased by paying the exercise price. When a put option is exercised by the holder, the underlying asset is sold for the exercise price. (Rutgers University) Spring Semester, / 31
7 Concept We will develop the concept of options first without and then with a premium The premium is the amount paid for the contract itself Later, we ll focus on this premium for a call option, also called the price of the call or P C It offsets the cost and risk of writing the contract Someone could walk away from the options contract so the writer does not receive any compensation This differs from the futures contract which has no premium since the contract will technically be settled (Rutgers University) Spring Semester, / 31
8 Concept have a finite life. Upon expiration, the option contract is null and void. An American option can be exercised at any time prior to expiration. A European option can only be exercised at maturity, not before. We will only consider European options (Rutgers University) Spring Semester, / 31
9 Concept Let s consider a call option first. Scenario Suppose Carla buys some land for $60,000 and immediately sells a 1-year European call option on it to Alex. Carla sells a call perhaps because she expects the price of land to fall, thus incurring a loss. The call protects her from this event. Assume the option has an exercise price of $65,000. (Rutgers University) Spring Semester, / 31
10 Concept There are many possible results depending on the States-of-the World (SOW) that materialize. Consider just two... 1 The land value is $68,000 1 year later Alex would gain $3,000 by exercising his option. 2 The land value is $62,000 1 year later Alex s option would expire worthless. The value of the call to Alex is... V C = max [Market Value $65, 000, 0] (Rutgers University) Spring Semester, / 31
11 Concept In general, the value of the call is... where... V C = max [P 0 X, 0] P 0 = the price of the underlying today X = the exercise price (Rutgers University) Spring Semester, / 31
12 Concept Suppose Alex bought the call option from Carla for $2,000, the premium he pays Carla to cover her costs of writing the contract. Compute the dollar profit (or loss) for Carla and Alex at various land values at option maturity. The premium, $2,000, is the price of the call since Alex paid this The price is nonrefundable (Rutgers University) Spring Semester, / 31
13 Concept Suppose the market value of the land is $55,000 when the option matures. What would Alex do? What is the result? Alex would not exercise the option. Thus, he would lose all the $2,000 he invested in the option for a 100% rate of return. (Rutgers University) Spring Semester, / 31
14 Concept Suppose the market value of the land is $68,000 when the option matures. Alex would exercise the option: He would buy the land from Carla for $65,000 and sell it in the open market for $68,000. His net profit would be $3,000 - $2,000 or $1,000. His rate of return would be $1,000 $2,000 = 50% (Rutgers University) Spring Semester, / 31
15 Model There are different formulations depending on whether we are analyzing stocks or bonds We will discuss stocks first and then bonds Stocks are like the land Carla purchased in our example: the land itself has no fixed maturity date (Rutgers University) Spring Semester, / 31
16 Model Assumptions... Frictionless and competitive capital markets Riskless arbitrage opportunities exist Two possible SOWs so that S = 2 and s = {1, 2} s = 1 is an upward movement in stock prices s = 2 is a downward movement in stock prices The two states suggests a binomial approach (Rutgers University) Spring Semester, / 31
17 Model Assume, for illustration, that... P S0 = $20 = current spot market price p = 0.5 = Pr(s = 1) q = 1 p = Pr(s = 2) r f = 0.10 u = 1.2 = stock price multiplier if s = 1 d = 0.67 = stock price multiplier if s = 2 (Rutgers University) Spring Semester, / 31
18 Model At the end of one period, the stock price could rise from P S0 = $20.00 to up S0 = $24.00 if s = 1 with probability p, or fall to $13.40 if s = 2 with probability q The call values are... Up World V cu = max [0; up S0 X ] = $3 Down World V cd = max [0; dp S0 X ] = $0 (Rutgers University) Spring Semester, / 31
19 Today p = 0.50 Tomorrow SOW: s = 1 $24.00 = $20.00*1.2 P S 0 $20.00 q = 0.50 SOW: s = 2 $13.40 = $20.00*0.67 Time
20 P S 0 $20.00 Today p = 0.50 Tomorrow SOW: s = 1 $24.00 = $20.00*1.2 S Max 0, up - X $3 0 q = 0.50 SOW: s = 2 $13.40 = $20.00*0.67 S Max 0, dp - X $0 0 Time
21 Model To find the call price, create a hedging portfolio a portfolio that mimics the returns on the call using stocks and bonds... Up World $24S + $110B = $3 Down World $13.40S + $110B = $0 and solve for S and B simultaneously... B = $13.40 S = S $110 S = (Rutgers University) Spring Semester, / 31
22 Model Sanity check that these numbers work... Up World Stocks $ = $6.79 Bonds $110 ( ) = $3.79 Down World Stocks $ = $3.79 Bonds $110 ( ) = $3.79 (Rutgers University) Spring Semester, / 31
23 Model The cost of this hedging portfolio today is... P HP,0 = $ $100( ) = $2.21 Therefore, by arbitrage... P C0 = P HP,0 So,... P C0 = P S0 S + P B0 B = P HP,0 (Rutgers University) Spring Semester, / 31
24 Model Notice that... S = = $3 $24 $13.40 $3 $0 $24 $13.40 = V cu V cd P S0 P Sd (Rutgers University) Spring Semester, / 31
25 Model And... B = $13.40S $110 P B0 B = $13.40S P B0 $110 $13.40S = $100(1 + r f ) P B0 = $13.40S 1 + r f = P SdS r f (Rutgers University) Spring Semester, / 31
26 Model Therefore,... P C0 = SP S0 + BP [ B0 Vcu V cd = P Su P Sd [ ] Vcu Vcd P Su P Sd = ] P S0 P SdS V cd 1 + r f P S0 (1 + r f ) P Sd S V cd P Sd + V cd 1 + r f (Rutgers University) Spring Semester, / 31
27 Model Simplifying notation with... we get... p = (1 + r f ) d u d q = 1 p = u (1 + r f ) u d P C0 = pv cu + qv cd 1 + r f (Rutgers University) Spring Semester, / 31
28 Model From basic number theory, we have where... nc r = n! r!(n r)! 0C 0 1C 0 1 C 1 2C 0 2 C 1 2 C 2 3C 0 3 C 1 3 C 2 3 C 3 (Rutgers University) Spring Semester, / 31
29 Model We can now write... P C0 = pv cu + qv cd 1 + r f = 1 pv cu + 1 qv cd 1 + r f = 1 C 0 pv cu + 1 C 1 qv cd 1 + r f 1 r=0 1C r p r q 1 r max [ 0; u r d 1 r P S0 X ] = 1 + r f (Rutgers University) Spring Semester, / 31
30 p 2 = 0.25 Max 2 S 0, u P - X $ P S 0 $20.00 p = 0.50 q = 0.50 p(1-p) = 0.25 q(1-q) = 0.25 Max S 0, udp - X $0 0 (1-q) 2 = 0.25 Max 2 S 0, d P - X $0 0
31 Model We can now have... 2 r=0 2C r p r q 2 r max [ 0; u r d 2 r P S0 X ] P C0 = (1 + r f ) 2 (Rutgers University) Spring Semester, / 31
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