FINANCIAL OPTION ANALYSIS HANDOUTS


 Jonas Collins
 5 years ago
 Views:
Transcription
1 FINANCIAL OPTION ANALYSIS HANDOUTS 1
2 2
3 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any number. A person buying the object S today must pay 100 now, and a person selling the object S will receive 100 now. It is common knowledge that the prevailing price tomorrow S 1 will either be S 1 (u) = 125 if the market for S is up or S 1 (d) = 80 if the market for S is down. A person who purchases a unit of S today will receive either 125 or 80 tomorrow (depending on the outcome), and the person who sells a unit of S today is obligated to buy back a unit of S at the new prevailing market price (125 or 80). There is also a market for an object called C. It may be bought or sold by anyone for any number. A person who buys the object C today must pay $C 0 now, and a person who sells the object C will receive $C 0 now. A person who buys C today has the option but not the obligation to buy 1 unit of the object S tomorrow at today s market price of 100. A person who sells C today has the obligation of supplying a unit of S tomorrow for the price of 100 if a holder of a C wishes to exercise his option. There is also a market for dollars, the socalled money market M. A person who buys m dollars today receives m dollars today and owes (1.10)m tomorrow; that is, the person is taking out a loan. A person who sells m dollars today must give m dollars today (to the person buying), and will receive (1.10)m the next period; that is, when a person sells dollars today, they are acting like a banker in that they are loaning money to the person buying. QUESTION: How much is a unit of C worth to you? How do you assess its value? 3
4 OPTION EXAMPLE 1 In each period the stock price may go up by a factor of u = 1.25 or down by a factor of d = The stock price at time 0 is 100. Riskfree rate each period is a constant 10%. Price a oneperiod call option with strike price 100. Determine the selffinanced replicating portfolio. 2/ S 40.40M 1/ NOTES = ΔC/ΔS = [25 0] [125 80] 2. The value of the call option at time 0 = (1/1.11)[ 2/3*25 + 1/3*0] = The money market amount = [100] = State Equations: h*s 1 (u) + m*r = C 1 (u) 125h + 1.1m = 25 h*s 1 (d) + m*r = C 1 (d) 80h + 1.1m = 0 4
5 OPTION EXAMPLE 2 In each period the stock price may go up by a factor of u = 1.25 or down by a factor of d = The stock price at time 0 is 100. Riskfree rate each period is a constant 10%. Price a oneperiod put option with strike price 100. Determine the selffinanced replicating portfolio. 2/ S M 1/ NOTES = ΔC/ΔS = [0 20] [125 80] 6. The value of the put option at time 0 = (1/1.11)[ 2/3*0 + 1/3*20] = The money market amount = *[100] = State Equations: h*s 1 (u) + m*r = P 1 (u) 125h + 1.1m = 0 h*s 1 (d) + m*r = P 1 (d) 80h + 1.1m = 25 5
6 OPTION EXAMPLE 3 In each period the stock price may go up by a factor of u = 1.25 or down by a factor of d = The stock price at time 0 is 100. Riskfree rate each period is a constant 10%. Price a 2period call option with strike price 100. Determine the selffinanced replicating portfolio. 2/ /3 1.00S 90.90M 100 1/ S 55.10M 2/3 1/ / NOTES 9. a = max( , 0). b. 0 = max( , 0) c. 0 = max(64 100, 0) d = (1/1.1)[2/3* /3*0] e = ΔC/ΔS = [ ] [ ] f = *125 g = (1/1.1)[2/3* /3*0] h = ΔC/ΔS = [ ] [125 80] i = * The value of the call option at time 0 = (1/1.21)[ 4/9* /9*0 + 1/9*0 ] 6
7 OPTION EXAMPLE 4 In each period the stock price may go up by a factor of u = 1.25 or down by a factor of d = The stock price at time 0 is 100. Riskfree rate each period is a constant 10%. Price a 2period put option with strike price 100. Determine the selffinanced replicating portfolio. 2/ / / S M 2/3 1/ S M 1/ NOTES 11. a. 0 = max( , 0). b. 0 = max( , 0) c. 36 = max(100 64, 0) d = (1/1.1)[2/3*0 + 1/3*36] e = ΔC/ΔS = [0 36] [100 64] f = *80 g = (1/1.1)[2/3*0 + 1/3*10.90] h = ΔC/ΔS = [ ] [125 80] i = * The value of the put option at time 0 = (1/1.21)[ 4/9*0 + 4/9*0 + 1/9*36 ] 13. European PutCall Parity: S T + P T C T = K at time T, which implies that S 0 + P 0 C 0 = K/(1.1) 2. Verification: = 100/1.21 =
8 OPTION EXAMPLE 5 In each period the stock price may go up by a factor of u = 1.25 or down by a factor of d = The stock price at time 0 is 100. Riskfree rate each period is a constant 10%. Price a 2period call option with strike price 80. Determine the selffinanced replicating portfolio. 2/ /3 1.00S 72.72M 100 1/ S 53.87M 2/3 1/ S 32.32M 1/ NOTES 14. a = max( , 0). b. 20 = max(100 80, 0) c. 0 = max(64 80, 0) d = (1/1.1)[2/3* /3*20] e = ΔC/ΔS = [ ] [ ] f = *125 g = (1/1.1)[2/3*20 + 1/3*0] h = ΔC/ΔS = [20 0] [100 64] i = *80 j = (1/1.1)[2/3* /3*12.12] k = ΔC/ΔS = [ ] [125 80] l = * The value of the call option at time 0 = (1/1.21)[ 4/9* /9*20 + 1/9*0 ] 8
9 OPTION EXAMPLE 6 In each period the stock price may go up by a factor of u = 1.25 or down by a factor of d = The stock price at time 0 is 100. Riskfree rate each period is a constant 10%. Price a 2period put option with strike price 80. Determine the selffinanced replicating portfolio. 2/ / / S M 2/3 1/ S M 1/ NOTES 16. a. 0 = max( , 0). b. 0 = max(80 100, 0) c. 16 = max(80 64, 0) d = (1/1.1)[2/3*0 + 1/3*16] e = ΔC/ΔS = [0 16] [100 64] f = *80 g = (1/1.1)[2/3*0 + 1/3*4.84] h = ΔC/ΔS = [0 4.84] [125 80] i = * The value of the put option at time 0 = (1/1.21)[ 4/9*0 + 4/9*0 + 1/9*16 ] 18. European PutCall Parity: S T + P T C T = K at time T, which implies that S 0 + P 0 C 0 = K/(1.1) 2. Verification: = 80/1.21 =
10 10
11 CHOOSER OPTION Consider a nondividend paying stock whose initial stock price is 62 and which has a logvolatility of σ = The interest rate r = 2.5% continuously compounded. Consider a 5month option with a strike price of 60 in which after exactly 3 months the purchaser may declare this option to be a (European) call or put option. QUESTIONS: 1. Determine the value of u and d for the binomial lattice. The value for U = exp{σ(δt) 1/2 } = exp{0.20(1/12) 1/2 } = Note that D = 1/U = Determine the values for the binomial lattice for 5 1month periods Stock Price Determine the appropriate riskfree rate. The interest rate per month R = exp(0.025*1/12) = Determine the riskneutral probability q of going UP. The value for q satisfies q(us 0 ) + (1q)(DS 0 ) = RS 0, which implies that q = (RD)/(UD) =
12 5. Determine the values for the call option and put option along the lattice Call Option Put Option Find the value of this Chooser Option. Compute the terminal value of the Chooser Option at t = 3 as the maximum of the call and put options at t = 3. From there we work backwards in the usual manner Chooser Option
13 Table 1: A stochastic volatility, random interest rate model t = 0 t = 1 t = 2 S 0 = 4, r 0 = 25% S 1 (U) = 8, r 1 (U) = 25% S 2 (UU) = 12 S 1 (D) = 2, r 1 (D) = 50% S 2 (UD) = S 2 (DU) = 8 S 2 (DD) = 2 Option Analysis with Stochastic Interest Rates In this problem we consider a twoperiod, stochastic volatility, random interest rate model. The stock prices and interest rates are provided in Table 1. Consider the European option whose final payoffs are V 2 = max{s 2 7, 0}. Determine the value of this option at times 0 and 1. 13
14 Table 2: Solution to the stochastic volatility, random interest rate model t = 0 t = 1 t = = [0.5(2.4) + 0.5(0. 1)] 2.4 = [0.5(5) + 0.5(1)] = [(1/6)(1) + (5/6)(0)] 1 0 The riskneutral q changes along the tree since the riskfree rate is stochastic. Otherwise, all the calculations are the same, since at each node the replicating portfolio and corresponding riskneutral discounted expectation ideas still apply. The solution is provided in Table 2. 14
15 Asian Option Consider a nondividend paying stock S whose price process follows a binomial lattice with U = 2 and D = 0.5. R = 1.25 and S 0 = 4. Define Y t := t S k, t = 0, 1, 2, 3 k=0 to be the sum of the stock prices between times zero and t. 1. Consider a (European) Asian call option that expires at time three and has a strike price K = 4; that is, its payoff at time three is { Y3 } max 4 4, 0. This is like a European call option, except the payoff of the option is based on the average stock price rather than the final stock price. Let V t (s, y) denote the price of this option at time n if S t = s and Y t = y. In particular, { y } V 3 (s, y) = max 4 4, 0. (a) Develop an algorithm for computing V t recursively. In particular, write a formula for V t in terms of V t+1. (b) Apply the algorithm developed in (a) to compute V 0 (4, 4), the price of the Asian option at time zero. (c) Provide a formula for δ t (s, y), the number of shares of stock that should be held by the replicating portfolio at time t if S t = s and Y t = y. 2. What is the value of a (European) Asian put option that expires at time three and has a strike price K = 4; that is, its payoff at time three is { max 4 Y } 3 4, 0. 15
16 1. Riskneutral q = 0.5. ( ) (a) V t (s, y) = (1/1.25) 0.5V t+1 (us, y + us) + 0.5V t+1 (ds, y + ds). (b) V 3 (32, 60) = 11; V 3 (8, 36) = 5; V 3 (8, 24) = 2; V 3 (2, 18) = 0.5; V 3 (8, 18) = 0.5; V 3 (2, 12) = V 3 (2, 9) = V 3 (0.5, 7.5) = 0. V 2 (16, 28) = (1/1.25)[0.5(11) + 0.5(5)] = V 2 (4, 16) = (1/1.25)[0.5(2) + 0.5(0.5)] = 1.0. V 2 (4, 10) = (1/1.25)[0.5(0.5) + 0.5(0)] = V 2 (1, 7) = (1/1.25)[0.5(0) + 0.5(0)] = 0. V 1 (8, 12) = (1/1.25)[0.5(6.4) + 0.5(1.0)] = V 1 (2, 6) = (1/1.25)[0.5(0.2) + 0.5(0)] = (c) V 0 (4, 4) = (1/1.25)[0.5(2.96) + 0.5(0.08)] = Remark. The value of this Asian option, 1.216, equals the discounted expectation of the final payoffs using the riskfree rate and the riskneutral probability, i.e., (1.25) 3. 8 Simulation is an especially useful computational approach for valuing pathdependent, Europeanstyle derivative securities, since their value can be obtained as a (discounted) sample average of the value along a sample path. δ t (s, y) = V t+1(us, y + us) V t+1 (ds, y + ds). (u d)s 16
17 No Arbitrage Bounds Consider a family of call options on a nondividend paying stock, each option being identical except for its strike price. The value of the call with strike price K is denoted by C(K). Prove the following two general relations using arbitrage arguments: 1. If K 2 > K 1, then K 2 K 1 C(K 1 ) C(K 2 ). 2. If K 3 > K 2 > K 1, then C(K 2 ) ( K3 K ) ( 2 K2 K ) 1 C(K 1 ) + C(K 3 ). K 3 K 1 K 3 K 1 Hint: For both parts find a portfolio that is guaranteed to have no negative but sometimes positive final payoffs. If there is to be no arbitrage, such a portfolio must cost something to acquire today. In each part plot the final payoffs for each portfolio, otherwise known as the payoff diagram. 17
18 Solution: For both parts below one finds a portfolio that is guaranteed to have no negative but sometimes positive final payoffs. If there is to be no arbitrage, such a portfolio must cost something to acquire today. In each case below it will be instructive to plot the final payoffs for each portfolio, otherwise known as the payoff diagram. 1. Verify that the portfolio C(K 1 ) + C(K 2 ) + (K 2 K 1 ) has no negative but sometimes positive final payoffs. Hence, its cost today must be nonnegative, which establishes the result. 2. Consider the portfolio mc(k 1 ) C(K 2 ) + nc(k 3 ) with m, n > 0 and m < 1. This portfolio s final payoffs are zero if S T K 1 and will be positive on the interval K 1 S T K 2. Since m < 1 the payoffs decline on the interval K 2 S T K 3. If we set m so that m(k 3 K 1 ) = (K 3 K 2 ), then the payoffs will remain nonnegative in the interval K 2 S T K 3. In particular, the payoff when S T = K 3 will be zero. If we further set n so that m + n = 1, the final payoffs will be zero on the interval S T K 3. We conclude that the portfolio ( K3 K ) ( 2 K2 K ) 1 C(K 1 ) C(K 2 ) + C(K 3 ) K 3 K 1 K 3 K 1 has no negative but sometimes positive final payoffs. nonnegative, which establishes the result. Hence, its cost today must be 18
19 FINANCIAL OPTION ANALYSIS PROBLEMS A. We consider a single period binomial lattice with S 0 = 50, u = 1.20 (d = 1/1.20) and R = a. What is the objective probability of an upward movement in the stock price, p, if the market s required expected percentage return on the stock, r S, is 8%? [p = ] b. Suppose the objective probability of an upward movement in the stock price p is What is the expected percentage return, r S, on the stock? [r S = 9%] 2. A derivative security C has final payoffs given by C 1 = (max [S 1 50, 0]) 2, where S 1 is the final stock price. Assume an objective probability of an upward movement in the stock price p = a. Determine the noarbitrage value C 0 for C. [ ] b. Determine the market s expected percentage return on C (using objective probabilities). [29.16%] c. Determine the replicating portfolio hs + mm. [5.4545S M] d. Determine the portfolio weights, w S and w M, on the stock and the money market in the replicating portfolio. [w S = 5.032, w M = ] e. Determine the portfolio s expected percentage return using the portfolio weights and the expected percentage returns on the stock and bond. [29.16%] f. Compare your answers to (b) and (e). 19
20 B. A nondividend paying stock has an initial price = 100. Its price path is modeled as a binomial lattice with U = 1.3 and D = 1/1.3. Period length = 1 year. The riskfree rate is 6% per year. 1. Determine the value of a 2 year European put option with strike price K = An American put permits the holder to exercise at any date t, and receive the intrinsic value max (0, K S t ). Hence, at each time t, the holder can take one of two possible actions: no exercise, as in a European option, or exercise. Determine the value of a 2 year American put option with strike price K =
21 C. Smith knows that the value of a six month European call option with strike price K = 24 on a nondividend paying stock is The current value of the stock is 26. However, he wishes to price a European put option on the same stock with the same strike price and maturity. He knows that the riskfree rate is 2.50% per year, but he is stuck because he does not know the logvolatility σ upon which he would calculate the value of U. He comes to you for help. What say you? 21
22 D. Do the following markets exhibit arbitrage? If so, demonstrate. If not, state why not Securities S1 S2 price at time up state at time down state at time Securities S1 S2 S3 price at time up state at time down state at time Securities S1 S2 S3 S4 S5 S6 S7 S8 S9 Price at time Up state at time Down state at time
23 E. The evolution of the stock price over 2 periods is shown in the figure below. Let S 2 denote the (random) value of the stock price at t = 2. The appropriate riskadjusted rate of return (cost of capital) is 20% per period. The riskfree rate is 4% per period. In this problem we shall consider pricing a European square root derivative security with strike price 140 that pays off ( S ) 1/2 at time t = Determine the objective probabilities along the lattice. 2. Determine the final period payoffs for the square root option. (Enter them in the figure.) 3. Determine the DecisionTree value of the square root option by (1) first computing the expectation of the final period payoffs using the objective probabilities, and then (2) discounting using the riskadjusted rate of return. 4. Fill out the above figure by placing the correct value of the square root option as the 2 nd entry, and recording the selffinanced replicating portfolio as the 3 rd entry. Determine the riskneutral probabilities along the lattice and mark them with an asterisk. 5. Suppose the current market value of the square root option is the DecisionTree value. Conceptually explain how you would use your answer to (d) to obtain a riskfree profit from the incorrect pricing. Be specific about your dynamic trading strategy. (No further calculations are required.) 23
24 Sample Worksheet 1: Sample Worksheet 2:
25 /** FOA_VanillaOptionAnalysis.hava */ import Lattice; import FOA_ReplicatingPortfolio; /* PROBLEM CLASSIFICATION */ token EUROPEAN, AMERICAN, CALL, PUT; /** */ type = AMERICAN; kind = PUT; strike = 60; r = 0.10; p0 = 62; maturity = 5/12; sigma = 0.20; dt = 1/12; dividendrate = 0.00; /** */ /* PROBLEM SPECIFICATION */ numperiods = round(maturity/dt); initialstate = State(numPeriods, p0, false); private SOLUTION = sdp_result(initialstate); /** */ import SDP_AO; 25
26 /* STATE DEFINITION */ // n = date, measured in periods // p = asset price on this date // estate = TRUE (if exercised) struct State(n, p, estate); /* ACTION SET DEFINITION */ // Actions = exercise or continue (depending on option type) token CONT, EX; function sdp_actionset(state)= if (state.n == 0) {collect(ex)} else if (type == AMERICAN) {(CONT, EX)} else {collect(cont)}; /* NEXT STATE DEFINITION */ function sdp_nextstate(state, action, event) = if (action == CONT) {State(state.n1, (1dividendRate*dt)*event.outcome, false)} else {State(state.n, state.p, true)}; /* CASH FLOW DEFINITION */ table sdp_cashflow(state, action) = if (action == EX) { if (kind == CALL) {foa_callpayoff(state)} else if (kind == PUT) {foa_putpayoff(state)} else {IGNORE} } else {0}; function foa_callpayoff(state) = max(state.p  strike, 0); function foa_putpayoff(state) = max(strike  state.p, 0); /** */ 26
27 /* DISCOUNT DEFINITION */ discount = exp(r*dt); private sdp_discount(state, action, event) = discount; /* TERMINAL CONDITION DEFINITION */ function sdp_terminalcondition(state) = (state.estate == true); /* SAMPLE SPACE GENERATION */ private lattice = BinomialLattice(sigma, r, dt); function sdp_generatesamplespace(state, action) = BL_GenerateSampleSpace(state, action, lattice); table latticedata = BL_LatticeData(lattice); /* private lattice = UpDownLattice(U, D, r, dt); function sdp_generatesamplespace(state, action) = UD_GenerateSampleSpace(state, action, lattice); table latticedata = UD_LatticeData(lattice); */ /* PROBLEM REPORTING */ function sdp_information(state, action) = rp_replicatingportfolio(state, action); 27
28 28
29 FOA_VanillaOptionAnalysis.hava type AMERICAN kind PUT strike 60 r 0.1 p0 62 maturity sigma 0.2 dt dividendrate 0.0 numperiods 5 initialstate State(5, 62, false) SDP_AO sdp_result state value n p estate policy information action value hratio cash assetw true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false EX true IGNORE false CONT true IGNORE
30 false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false EX true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT true IGNORE false CONT SDP_AO sdp_expvalue state action value n p estate false EX false EX false EX false EX false EX false EX false EX false EX false EX false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX
31 false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX 0.0 SDP_AO sdp_exppresentvalue state action value n p estate false EX false EX false EX false EX false EX false EX false EX false EX false EX false EX false CONT false EX false CONT false EX false CONT false EX false CONT
32 false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX 0.0 sdp_cashflow state action value n p estate false EX false EX false EX false EX false EX false EX false EX false EX false EX false EX false CONT false EX false CONT false EX false CONT false EX
33 discount latticedata UP DOWN RETURN RN_PROB false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX false CONT false EX 0 33
34 34
35 FINANICAL OPTION ANALYSIS SAMPLE PROBLEMS 1. The price of asset S at time t = 0 is S 0 = 20. The asset s value at time t = 1 is S 1 (u) = 28 in the up state is S 1 (d) = 14 in the down state. The riskfree rate is 5%. The objective probability of the up state is A derivative security V has final payoffs at time t = 1 of V 1 (u) = 126 in the up state and V 1 (d) = 84 in the down state. a. Determine the expected return, r S, on the asset S. b. Determine the noarbitrage value for V. c. Determine the replicating portfolio for V. d. Determine the portfolio weights of S and M in the replicating portfolio for V. e. Determine the expected return, r V, on the derivative security V. 2. A nondividend paying stock has an initial price S 0 = 100. It s price path is modeled as a binomial lattice with u = 1.25 and d = Period length is 1 year. The riskfree rate is 10% per year. a. Determine the value of a 2 year American put option with strike price K = 90. b. Determine the value of a 2 year European put option with strike price K = 90. c. Use PutCall Parity to value a 2 year European call option with strike price K = Does the following market exhibit arbitrage? If so, provide a concrete example of arbitrage. If not, state why not. Securities S1 S2 S3 S4 S5 S6 S7 S8 S9 Price at time Up state at time Down state at time The price process of a nondividend paying stock over the next 2 years is shown in the following table: t = 0 t = 1 t = The riskfree rate is 5% per year. Define Y 2 = S 0 + S 1 + S 2 denote the sum of the stock prices between times zero and 2. A European Asian put option that expires at time t = 2 and has strike price K = 104 has its payoff at time t = 2 equal to max{104  Y 2 /3, 0}. (This is like a European put option, except that the payoff of this option is based on the average stock price rather than the final stock price.) Determine the noarbitrage value of this pathdependent option. 35
36 5. The evolution of the stock price over 2 periods is shown in Figure 1 below. Let S 2 denote the (random) value of the stock price at t = 2. The appropriate riskadjusted rate of return (cost of capital) is 20% per period. The riskfree rate is 4% per period. In this problem we wish to price a European derivative security that pays off ( S S S ) at time t =2. a. For each box in Figure 1 place the correct value of the derivative security as the 2 nd entry and record the selffinanced replicating portfolio as the 3 rd entry. Figure b. Determine the value at time t = 0 of this derivative security according to DTA. c. Suppose the current market value of the derivative security is 150. Exactly explain how you could guarantee a riskfree profit from the incorrect pricing. Be specific with numbers. 6. Consider dividendprice data for a complete, noarbitrage market with the following three securities: Security 1 Security 2 Security 3 Payoff Vector V Price at t = ? Payoff in state 1 at t = Payoff in state 2 at t = Payoff in state 3 at t = a. Use the replicating portfolio approach to determine the correct value of the payoff vector V at time 0. b. Use the riskneutral approach to determine the correct value of the payoff vector V at time 0. 36
37 7. a. Does the following market exhibit arbitrage? If so, provide a concrete example of arbitrage. If not, state why not. Securities S1 S2 S3 S4 S5 Price at time State State State State b. Determine the riskfree rate. 8. The evolution of the stock price over 2 periods is shown in Figure 1 below. Let S 2 denote the (random) value of the stock price at t = 2. The appropriate riskadjusted rate of return (cost of capital) is 20% per period. The riskfree rate is 5% per period. In this problem we wish to price a European derivative security that pays off S /2 at time t =2. a. Determine the noarbitrage value of this derivative security. b. Determine the replicating portfolio at time 0. Figure Consider a binomial lattice with S 0 = 4, u = 2 and d = The riskfree rate if 25%. A 3year, lookback option is a European pathdependent derivative security that pays off at time three. V 3 = max {0 t 3} S t S 3 a. Determine the noarbitrage value of this lookback option. b. Determine the replicating portfolio at time t = 0. 37
38 FINANCIAL OPTION ANALYSIS SAMPLE PROBLEM SOLUTIONS 1. a. r S = [0.6(28) + 0.4(14)]/201 = b. Since [0.5(28) + 0.5(14)]/1.05 = 20, q = 0.5. Thus, V 0 = [0.5(126) + 0.5(84)]/1.05 = 100. c. (12684)/(2814) = 3. So h = 3. Thus, replicating portfolio is 3S + 40M. d. 60/100 is invested in S, so w S = 0.6 and w M = 0.4. e. r V = [0.6(126) + 0.4(84)]/1001 = a. t = 0 t = 1 t = [0] [0] 3.03 = 1/3(10)/1.1 80* [10] 100 [0] since 10 > 1/3(26)/ [26] b. (1/9)(26)/(1.1) 2 = c. 90/(1.1) 2 = S + P C = C. C = First four securities show that q = 0.5. To be consistent, the value S9 = [0.5(385) + 0.5(245)]/1.05 = 300, which is less than 305. The portfolio 10S S2 replicates S9 and costs 300. So sell S9 for 305 and buy this replicating portfolio for 300, and pocket the difference of There are four possible paths. The average values along the uu, ud, du, and dd paths are, respectively, , , 94, and 79. The values of the Asian put option for these paths are, respectively, 0, 0, 10, and 25. The q probability of path du is (0.5)(0.6) = 0.3, and the q probability of path dd is (0.5)(0.4) = 0.2. Therefore, the value of this option is [(0.3)(10) + (0.2)(25)]/(1.05) 2 = a. Figure 1. Objective probabilities are in parentheses (0.622) (0.80) S+3.36M (0.377) S+151.3M (0.64) (0.20) S M (0.36)
39 b. [185(0.80*0.622) (0.80* *0.64) (0.20*0.36)] / (1.2) 2 = c. Derivative security is overpriced, so you would want to sell it. Collect 150 and use of it to purchase the replicating portfolio of S B. Invest the in the bank or buy lunch. If the price goes up next period to 130, rebalance the portfolio to S B, which you can afford to do since it will cost and this is precisely what the portfolio S B equals when the price = 130. If the price goes down next period to 80, rebalance the portfolio to S , which you can afford to do by the same reasoning as before. Finally, when period 2 comes around, your updated portfolio will exactly match the final payoffs of the derivative security (185, or 167.5) regardless of the final state and so you will be able to meet your obligations. (If you were forced to buy back the derivative security at time 1, you would have the exact money to do so.) 6. a. Since S2 does not payoff in either state 1 or 2 only S1 and S3 can be used to replicate the payoffs of the Vector V in these two states. (S3 is of course our old friend M.) We re back to the hs1 + M : here, h = ( )/( ) = 1 and M = S3 = 200. Now use state 3 to pin down the number of units of S2 to hold: 176(S2) + 1.1(200) = 44 S2 = 1. Thus, the replicating portfolio is 1S1 1S for a cost today of 220. b. Let q = (q 1, q 2, q 3 ) denote the riskneutral probability vector. Obviously R = 1.1. Discounted expectation using q and R applied to S2 implies that (1/1.1)176q 3 = 80 q 3 = 0.5. Thus, q 1 + q 2 = 0.5. Discounted expectation using q and R applied to S1 implies that (1/1.1)[200q q 2 ] = 100 q 1 =q 2 = Riskneutral valuation says that (1/R)E q [V] = (1/R) q T V = p for any vector V. Thus, the value of V is (1/1.1)[0.25(420) (460) + 0.5(44)] = 220, which coincides with the answer in part (a) as it should. 7. a. We can use the first four assets to determine the stateprices. The equations are: 125y y 2 = y y 2 = 60. These equations imply that y 1 = 4/11, y 2 = 2/11. 28y y 4 = y y 4 = 40. These equations imply that y 3 = 4/21, y 4 = 4/21. Now we can use these stateprices to determine the noarbitrage value of asset 5 as: (4/11)22 + (2/11)11 + (4/21)21 + (4/21)42 = 22. Since this IS the price of asset 5, this market does NOT exhibit arbitrage. b. Recall that the reciprocal of the sum of the stateprices equals R = 1 + riskfree rate. Thus, R = 1/(4/11 + 2/11 + 4/21 + 4/21) = 231/214 = r = 7.94%. Alternatively, one can easily combine assets 3 and 4 to obtain a constant payoff vector. For example, a purchase of 1 unit of asset 3 and 110/105 units of asset 4 yields a constant payoff of 110. The cost of this portfolio is 60(1) + 40(110/105) = Thus, the total return on the riskfree security is 110/ = , same as above, as it should. 39
40 8. Figure S M S+6.559M S M Stock price paths Lookback option value over time uuu: = 0 uud: = 8 4 udu: = 0 udd: = duu: = 0 dud: = 2 1 ddu: = 2 ddd: = 3.5 Riskneutral probability = = [( )/8]/(1.25) = [( )/4](1.25) = [( )/4](1.25) 2. h = ( )/(82) = = 13/75 and M = =
Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationOne Period Binomial Model
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationOptions. + Concepts and Buzzwords. Readings. PutCall Parity Volatility Effects
+ Options + Concepts and Buzzwords PutCall Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options
More informationOverview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies
Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies Noarbitrage bounds on option prices Binomial option pricing BlackScholesMerton
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationLecture 3: Put Options and DistributionFree Results
OPTIONS and FUTURES Lecture 3: Put Options and DistributionFree Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distributionfree results? option
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationCall and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options
Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder
More informationLecture 5: Put  Call Parity
Lecture 5: Put  Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible
More information2. Exercising the option  buying or selling asset by using option. 3. Strike (or exercise) price  price at which asset may be bought or sold
Chapter 21 : Options1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. BlackScholes
More informationOption Basics. c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153
Option Basics c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153 The shift toward options as the center of gravity of finance [... ] Merton H. Miller (1923 2000) c 2012 Prof. YuhDauh Lyuu,
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationOptions Markets: Introduction
Options Markets: Introduction Chapter 20 Option Contracts call option = contract that gives the holder the right to purchase an asset at a specified price, on or before a certain date put option = contract
More informationS 1 S 2. Options and Other Derivatives
Options and Other Derivatives The OnePeriod Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating
More informationOption Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration
CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value  profit that could be made if the option was immediately exercised Call: stock price  exercise price Put:
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationIntroduction to Binomial Trees
11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationFinance 436 Futures and Options Review Notes for Final Exam. Chapter 9
Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying
More informationUCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:
UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,
More informationAdditional questions for chapter 4
Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two sixmonth periods it is expected to go up by 1% or go down by 1%. The riskfree interest rate is 8% per annum with
More informationFigure S9.1 Profit from long position in Problem 9.9
Problem 9.9 Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances
More informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 PutCall Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Putcall
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. YuhDauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More informationChapter 21: Options and Corporate Finance
Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the
More informationLecture 9. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 8
Lecture 9 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 8 Lecture 9 1 RiskNeutral Valuation 2 RiskNeutral World 3 TwoSteps Binomial
More informationLecture 17/18/19 Options II
1 Lecture 17/18/19 Options II Alexander K. Koch Department of Economics, Royal Holloway, University of London February 25, February 29, and March 10 2008 In addition to learning the material covered in
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationComputational Finance Options
1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationLectures. Sergei Fedotov. 20912  Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912  Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationPart V: Option Pricing Basics
erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, putcall parity introduction
More informationBinomial trees and risk neutral valuation
Binomial trees and risk neutral valuation Moty Katzman September 19, 2014 Derivatives in a simple world A derivative is an asset whose value depends on the value of another asset. Call/Put European/American
More informationAmerican Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options
American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the putcall parity theorem as follows: P = C S + PV(X) + PV(Dividends)
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More informationDynamic Trading Strategies
Dynamic Trading Strategies Concepts and Buzzwords MultiPeriod Bond Model Replication and Pricing Using Dynamic Trading Strategies Pricing Using Risk eutral Probabilities Onefactor model, noarbitrage
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.
More information1 The BlackScholes model: extensions and hedging
1 The BlackScholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationConsider a European call option maturing at time T
Lecture 10: Multiperiod Model Options BlackScholesMerton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T
More informationa. What is the portfolio of the stock and the bond that replicates the option?
Practice problems for Lecture 2. Answers. 1. A Simple Option Pricing Problem in One Period Riskless bond (interest rate is 5%): 1 15 Stock: 5 125 5 Derivative security (call option with a strike of 8):?
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationLecture 11. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7
Lecture 11 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating
More informationCurrency Options (2): Hedging and Valuation
Overview Chapter 9 (2): Hedging and Overview Overview The Replication Approach The Hedging Approach The Riskadjusted Probabilities Notation Discussion Binomial Option Pricing Backward Pricing, Dynamic
More informationOptions 1 OPTIONS. Introduction
Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or
More informationFinance 400 A. Penati  G. Pennacchi. Option Pricing
Finance 400 A. Penati  G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (riskneutral)
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
More informationInvesco Great Wall Fund Management Co. Shenzhen: June 14, 2008
: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of
More informationUnderlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)
INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the PutCallForward
More informationTABLE OF CONTENTS. A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13
TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationUse the option quote information shown below to answer the following questions. The underlying stock is currently selling for $83.
Problems on the Basics of Options used in Finance 2. Understanding Option Quotes Use the option quote information shown below to answer the following questions. The underlying stock is currently selling
More informationTwoState Model of Option Pricing
Rendleman and Bartter [1] put forward a simple twostate model of option pricing. As in the BlackScholes model, to buy the stock and to sell the call in the hedge ratio obtains a riskfree portfolio.
More informationACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)
Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A fouryear dollardenominated European put option
More informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More informationDERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options
DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis review of pricing formulas assets versus futures practical issues call options
More informationPractice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?
Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions
More informationChapter 21 Valuing Options
Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
More informationModelFree Boundaries of Option Time Value and Early Exercise Premium
ModelFree Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 331246552 Phone: 3052841885 Fax: 3052844800
More informationEXP 481  Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481  Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the
More information1 Introduction to Option Pricing
ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of
More informationOptions. Moty Katzman. September 19, 2014
Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to
More information10 Binomial Trees. 10.1 Onestep model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1
ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 10 Binomial Trees 10.1 Onestep model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t
More informationExpected payoff = 1 2 0 + 1 20 = 10.
Chapter 2 Options 1 European Call Options To consolidate our concept on European call options, let us consider how one can calculate the price of an option under very simple assumptions. Recall that the
More informationOption Premium = Intrinsic. Speculative Value. Value
Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An OptionPricing Formula Investment in
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationArbitrageFree Pricing Models
ArbitrageFree Pricing Models Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models 15.450, Fall 2010 1 / 48 Outline 1 Introduction 2 Arbitrage and SPD 3
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationOption Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)
Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.
More informationOption Markets 232D. 1. Introduction. Daniel Andrei. Fall 2012 1 / 67
Option Markets 232D 1. Introduction Daniel Andrei Fall 2012 1 / 67 My Background MScF 2006, PhD 2012. Lausanne, Switzerland Since July 2012: assistant professor of finance at UCLA Anderson I conduct research
More informationFinancial Options: Pricing and Hedging
Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equitylinked securities requires an understanding of financial
More information1.1 Some General Relations (for the no dividend case)
1 American Options Most traded stock options and futures options are of Americantype while most index options are of Europeantype. The central issue is when to exercise? From the holder point of view,
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X
More information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationLecture 12. Options Strategies
Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same
More informationCHAPTER 7: PROPERTIES OF STOCK OPTION PRICES
CHAPER 7: PROPERIES OF SOCK OPION PRICES 7.1 Factors Affecting Option Prices able 7.1 Summary of the Effect on the Price of a Stock Option of Increasing One Variable While Keeping All Other Fixed Variable
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationForward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.
Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero
More informationCHAPTER 20 Understanding Options
CHAPTER 20 Understanding Options Answers to Practice Questions 1. a. The put places a floor on value of investment, i.e., less risky than buying stock. The risk reduction comes at the cost of the option
More informationHow To Value Real Options
FIN 673 Pricing Real Options Professor Robert B.H. Hauswald Kogod School of Business, AU From Financial to Real Options Option pricing: a reminder messy and intuitive: lattices (trees) elegant and mysterious:
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
More informationChapter 5 Financial Forwards and Futures
Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment
More informationOption Payoffs. Problems 11 through 16: Describe (as I have in 110) the strategy depicted by each payoff diagram. #11 #12 #13 #14 #15 #16
Option s Problems 1 through 1: Assume that the stock is currently trading at $2 per share and options and bonds have the prices given in the table below. Depending on the strike price (X) of the option
More information