I am working on a homework problem and am stumped on part B.

Kim hotels is interested in developing a new hotel in seoul. The company estimates that the hotel would require an initial investment of 20 million. Kim expects the hotel will produce positive cash flows of 3 million a year at the end of each year for the next 20 years. The projects cost of capital is 13%

A what is the projects net present value

B Kim expects the cash flows to be 3 million a year but recognizes that the cash flows could actually be much higher or lower depending on whether the Korean government imposes a large hotel tax. One year from now, Kim will know whether the tax will be imposed. There is a 50% chance the tax will be imposed, in which case the yearly cash flows will only be 2.2 million. At the same time the there is a 50% chance that the tax will not be imposed, in which case the yearly cash flows will be 3.8 million. Kim is deciding whether to proceed with the hotel today of to wait a year to find out whether the tax will be imposed. If Kim waits a year, the initial investment will remain at 20 million. Assume that all cash flows are discounted at 13%. Use the decision tree analysis to determine whether Kim should proceed with the project today or wait a year before deciding.

SECTION 8-1 SOLUTIONS TO SELF-TEST Brighton Memory's stock is currently trading at $50 a share. A call option on the stock with a $35 strike price currently sells for $21. What is the exercise value of the call option? What is the time value? Stock price Strike price Market price of option $30 $25 $7 Exercise value of option $5.00 Time value of option $2.00 SECTION 8-5 SOLUTIONS TO SELF-TEST What is the value of a call option with these data: P = $15, X = $15, r RF = 6%, t = 0.5 (6 months), and variance of stock 0.12? P X rRF t s $15 $15 6.0% 0.50 35% (d1) 0.245 (d2) 0.000 N(d1) 0.5968 N(d2) 0.5000 V= $1.67 SECTION 8-6 SOLUTIONS TO SELF-TEST P= X= rRF = $33.00 $32.00 6.00% 1.00 $6.56 t= V (call price) = Put = $3.70 SECTION 8-2 SOLUTIONS TO SELF-TEST Inputs: Current stock price Strike price u d Risk-free rate Time to exercise $20 $21 1.30 0.80 5% 1.00 Stock price if u $26.00 Stock price if d $16.00 Option payoff if u $5.00 Option payoff if d $0.00 N 0.50 Hedge portfolio payoff if u $8.00 Hedge portfolio payoff if d $8.00 Portfolio value today (PV of payoff) $7.61 Current option value $2.39 26-1 A 0 -20,000,000.00 1 3,000,000.00 2 3,000,000.00 3 3,000,000.00 4 3,000,000.00 5 3,000,000.00 6 3,000,000.00 7 3,000,000.00 8 3,000,000.00 9 3,000,000.00 10 3,000,000.00 11 3,000,000.00 12 3,000,000.00 13 3,000,000.00 14 3,000,000.00 15 3,000,000.00 16 3,000,000.00 17 3,000,000.00 18 3,000,000.00 19 3,000,000.00 20 3,000,000.00 21 3,000,000.00 1,074,254.73 50% 0% 50% cost of capital -20,000,000.00 -20,000,000.00 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 2,200,000.00 3,800,000 -4,376,589.84801697 13% 6,985,890.26 -4545546.53 1,074,254.73 6,694,056.00 1074254.735 26-1B Kim should wait another year to see if the tax will be imposed. Figures above show that if the tax is imposed then it will be a negative amount. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 B C Tool Kit D E F G H 12/8/2012 Chapter 8 Financial Options and Applications in Corporate Finance 8-1 Overview of Financial Options An option is a contract which gives its holder the right to buy (or sell) an asset at a predetermined price within a specified period of time. Option contracts, though often quoted in terms of single shares, usually are contracts for a 100 shares. A call option describes a situation in which one investor may sell to someone the right to buy his/her shares of a stock over some interval of time. In this scenario, the writer of the call option (the party that surrenders the right to exercise) is said to hold a short position on the option. Meanwhile, the party that has purchased this right to buy is said to hold a long position on the option. The predetermined price that the stock may be purchased for is called the strike, or exercise, price. When an investor "writes" call options against stock held in his/her portfolio, this is called a "covered call". When the call options are written without the stock to back them up, they are they are called "naked calls". When the strike price is below the current market price, the call option is said to be "in-the-money". Likewise, when the strike price exceeds the current market price, the call option is said to be "out-of-the-money". For instance, if you believed that the price of stock was primed to rise, a call option would allow you to capture a profit off of the rise in price. 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 A put option allows you to buy the right to sell a stock at a specified price within some future period. If you happened to believe that the price of a stock was ready to fall, a put option would allow you to turn a profit out of that decline. In the cases of both call and put options, the profit or loss made on an options transaction is determined by the value of the underlying asset, the strike price of the option, and the price of the option. FOR A CALL, AT EXPIRATION If the value of the underlying asset exceeds the strike price, the profit/loss from the call transaction would be equal to the difference between the value of the asset and the strike price less the price of the call. In this case there could be either a net profit or loss depending upon the exercise value and the price of the call. If the value of the underlying asset equals the strike price, the profit/loss from the call transaction would be equal to the price of the call, because whether exercised or unexercised the call value would be zero. In this case there is a loss equal to the price of the call. If the value of the underlying asset is less than that of the strike price, the profit/loss from the call transaction would be equal to the price of the call, because the option would not be exercised if the strike price was greater than the market price. In this case there is a loss equal to the price of the call. FOR A PUT, AT EXPIRATION If the value of the underlying asset is less than the strike price, the profit/loss from the put transaction would be equal to the difference between the strike price and value of the asset less the price of the put. In this case there could be either a profit or loss depending upon the exercise value and the price of the put. If the value of the underlying asset equals that of the strike price, the profit/loss from the put transaction would be equal to the price of the call, because whether exercised or unexercised the put value would be zero. In this case there is a loss equal to the price of the put. If the value of the underlying asset exceeds that of the strike price, the profit/loss from the put transaction would be equal to the price of the put, because the option would not be exercised if the market price was greater than the strike price. In this case there is a loss equal to the price of the put. Table 8-1 January 7, 2013, Listed Options Quotations CALLSLAST QUOTE Closing Strike Price February March May Price PUTSLAST QUOTE February March May General Computer Corporation (GCC) 53.50 53.50 53.50 50 55 60 U.S. Biotec 56.65 55 Food World 56.65 55 Note: r means not traded. 4.25 1.30 0.30 4.75 2.05 0.70 5.50 3.15 1.50 0.65 2.65 6.65 1.40 r r 2.20 4.50 8.00 5.25 6.10 8.00 2.25 3.75 r 3.50 4.10 r 0.70 r r Suppose you purchase GCC's May call option with a strike price of $50 and the stock price goes to $60. What is the rate of return on the stock? What is the rate of return on the option? Stock Return Intital stock price Final stock price Rate of return on stock $53.50 $60.00 12.1% Call Option Return Intital cost of option Market price of stock Strike price Profit from exercise Rate of Return $5.50 $60.00 $50.00 $10.00 81.8% Suppose you purchase GCC's May put option with a strike price of $50 and the stock price goes to $45. What is the rate of return on the stock? What is the rate of return on the option? Stock Return Intital stock price Final stock price Rate of return on stock $53.50 $33.00 -38.3% Put Option Return Intital cost of option Market price of stock Strike price Profit from exercise Rate of Return $2.20 $33.00 $50.00 $17.00 672.7% What is the exercise value of GCC's May call option with a strike price of $50? What is the exercise value of GCC's May call option with a strike price of $55? I J K L M A 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 B C D Exercise Value Stock price Strike price Exercise value F G H $53.50 $50.00 $3.50 Stock price Strike price Exercise value E $53.50 $55.00 $0.00 8-2 The Single-Period Binomial Option Pricing Approach Consider a call option on a stock. The stock's current price, denoted by P, is $40 and the strike price, denoted by X, is $35. The option expires in 6 months. The nominal annual risk-free rate is 8%. PAYOFFS IN A SINGLE-PERIOD BINOMIAL MODEL At expiration, the stock can take on only one of two possible values. It can either go up in price by a factor of 1.25, or down in price by a factor of 0.80. Inputs: Key output: Current stock price, P = Nominal annual risk-free rate, rRF = Strike price, X = Up factor for stock price, u = Down factor for stock price, d = Years to expiration, t = Number of periods until expiration, n = VC = $40.00 $7.71 8% $35.00 1.25 0.80 0.50 1 Consider the value of the stock and the payoff of the option. Figure 8-1 Binomial Payoffs Strike price: X = Current stock price: P = Up factor for stock price: u = Down factor for stock price: d = $35.00 $40.00 1.25 0.80 Cu, Ending up stock price P (u) = =A146*D135 = $50.00 ending up option payoff MAX[P(u) X, 0] = =MAX[D141 D133,0] = $15.00 VC, P, current stock price $40 current option price ? Cd, Ending down stock price P (d) = =A146*D136 = $32.00 ending down option payoff MAX[P(d) X, 0] = =MAX[D151 D133,0] = $0.00 THE HEDGE PORTFOLIO APPROACH We can form a portfolio by writing 1 call option and purchasing N s shares of stock. We want to choose Ns such that the payoff of the portfolio if the stock price goes up is the same as if the stock price goes down. This is a hedge portfolio because it has a riskless payoff. Step 1. Find the number of shares of stock in the hedge portfolio. Ns = Cu - Cd = 0.83333 P(u - d) Step 2. Find the hedge portfolio's payoff. If the stock price goes up: Portoflio payoff = Ns (P)(u) - Cu = $26.6667 If the stock price goes down: Portoflio payoff = Ns (P)(d) - Cd = $26.6667 Figure 8-2 The Hedge Portfolio with Riskless Payoffs Strike price: X = Current stock price: P = Up factor for stock price: u = Down factor for stock price: d = Up option payoff: Cu = MAX[0,P(u)-X] = $35.00 $40.00 1.25 0.80 $15.00 Down option payoff: Cd =MAX[0,P(d)-X] = $0.00 Number of shares of stock in portfolio: Ns = (Cu - Cd) / P(u-d) = 0.83333 Stock price = P (u) = $50.00 Portfolio's stock payoff: = P(u)(Ns) = $41.67 Subtract option's payoff: Cu = $15.00 Portfolio's net payoff = P(u)Ns - Cu = $26.67 Stock price = P (d) = $32.00 Portfolio's stock payoff: = P(d)(Ns) = $26.67 Subtract option's payoff: Cd = $0.00 Portfolio's net payoff = P(d)Ns - Cd = $26.67 P, current stock price $40 Step 3. Find the present value of the hedge portfolio's riskless payoff. The present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding) is: I J K L M A 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 B C D Frequency compounded in year: Nominal risk-free rate, rRF: Years to end of binomial period: Payoff of hedged portfolio: E F G H I J 365 8% 0.5 $26.6667 Inputs to PV function: N= 182.5 = t(Frequency) I/YR = 0.02192% = Rrf / Frequency PMT = 0 FV = -$26.6667 Negative because the PV is the amount we want in exchange for payoff of hedged portfolio PV of payoff = $25.6212 Use the PV function: =PV(I/YR,N,PMT,FV) Alternatively, you can use the present value equation and adjust for the frequency of compounding: f payoff = Payoff (1 + rRF/365)365*(t) = $26.6667 = $25.6212 1.04081 Step 4. Find the option's current value. The current value of the hedge portolio is the the stock value (Ns x P) less the call value (VC). But the hedge portfolio has a riskless payoff, so the hedge portfolio's value must also be equal to the present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding). With a little algebra, we get: VC = Ns (P) - Present value of riskless payoff VC = $7.71 THE REPLICATING PORTFOLIO If a portfolio can be formed such that is has the same cash flows as an option, the the option value must equal the value of this replicating portfolio. It is possible to replicate an option's cash flows with a portfolio of stock and risk-free bonds, as we show in the next section. Suppose we form a portfolio with Ns shares of stock (as determined by the formula for the number of shares of stock in the hedge portfolio). How much could we borrow so that the net payoff from the stock and the repayment of the loan (and its interest) has the same payoff as the option? Inputs: Current stock price, P = Risk-free rate, rRF = Strike price, X = Up factor for stock price, u = Down factor for stock price, d = Years to expiration, t = Number of periods until expiration, n = $40.00 8% $35.00 1.25 0.80 0.50 1 Intermediate calculations: Up payoff for stock, Pu = Down payoff for stock, Pd = Cu = $50.00 $32.00 Cd = $0.00 Ns = Cu - C d $15.00 = 0.8333 P(u - d) If we form a portfolio with Ns shares of stock, how much can we afford to borrow so that the portfolio's net payoff is equal to the option's payoff? Value of stock in portfolio if up = = Cu = t of borrowing (plus interest) that can be repaid = Value of stock in portfolio if down = = Cd = t of borrowing (plus interest) that can be repaid = Ns P u $41.67 $15.00 $26.6667 Ns P d $26.67 $0.00 $26.6667 up or down. To find the amount we can borrow, we find the present value fo the amount we can repay. Option pricing assumes that interest rates are compounded very frequently. We will assume daily compounding (which is a good approximation for continuous compounding). Amount borrowed = Amount repaid (1 + rRF/365)365*(t) = 25.6212 A summary of the replicating portfolio value and payoff's is shown below: Replicating Portfolio Payoffs Number of shares of stock: Ns = Current stock price: P = Up factor for stock price: u = Up stock price: P(u) = Down factor for stock price: d = Down stock price: P(d) = Risk-free rate: rRF = Years to expiration: t = Number of periods until expiration: n = Amount of principal and interest repaid = Amount borrowed = 0.8333 $40.00 1.2500 $50.00 0.8000 $32.00 8.00% 0.50 1 $26.67 $25.62 (Ns) x (Pu) = $41.67 Loan repayment = Net portfolio payoff = $26.67 $15.00 (Ns) x (Pd) = $26.67 Loan repayment = Net portfolio payoff = $26.67 $0.00 Current value of portfolio: (Ns) x (P) = $33.33 Amount borrowed = Total portfolio net cost = $25.62 $7.71 K L M A 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 B C D E F G H The call option has the same cash flows as the replicating portfolio, so the call's price must be equal to the value of the replicating portfolio: VC = Total portfolio value = $7.71 8.3 The Single-Period Binomial Option Pricing Formula The step-by-step hedge portfolio approach works fine, but for problems in which you want to change the inputs, it is easier to use the binomial option pricing formula shown below. e rRF ( t / n ) d u e rRF ( t / n ) Cu Cd u d u d VC rRF ( t / n ) e Inputs: P= X= u= d= Cu = Cd = $40.00 $35.00 1.25 0.80 $15.00 $0.00 Risk-free rate, rRF = 8% 0.50 1 Years to expiration, t = Number of periods until expiration, n = VC = $7.71 The Simplified Binomial Option Pricing Formula P= X= u= d= Cu = Cd = $40.00 $35.00 1.25 0.80 $15.00 $0.00 Risk-free rate, rRF = 8% 0.50 1 Years to expiration, t = Number of periods until expiration, n = We can simplify the model by defining p u and pd as: e rRF ( t / n ) d u d pu e rRF ( t / n ) u e rRF ( t / n ) u d pd rRF ( t / n ) e The binomial option pricing model then simplifies to: VC = Cu p u + Cd p d For Western's 6-month options, we have: pu = 0.5141 pd = 0.4466 We can find the value of Western's 6-month call with a $35 strike price: VC = Cu VC = VC = $15.00 $7.71 pu x x + + 0.5141 Cd $0.00 x x pd 0.4466 Find the value of a 6-month call option with a $30 strike price: x= $30.00 Cu = MAX[0,Pu-X] = Cd = MAX[0,Pd-X] = VC = Cu VC = VC = $20.00 $11.18 x x $20.00 $2.00 pu + + 0.5141 Cd $2.00 x x pd 0.4466 In fact, we can use the p's to find the value of any security with payoffs that depend on Western's 6-month stock price. 8-4 The Multi-Period Binomial Option Pricing Model Suppose we divide the year into two 6-month periods. We will allow the stock to only go up or down each period, but because there are more periods there will be more possible stock prices. The key is to keep the standard deviation of the stock's return the same as we divide the year into smaller periods. If we know the standard deviation of the stock's return and the number of periods, there is a formula that will show us what u and d must be. s is the standard deviation of stock return. Here are the formulas relating s to to u and d: s t u e d 1 u The standard deviation of Western's stock return is shown below. Notice that this provides the values for u and d that we used in the single-period model. Annual standard deviation of stock return, = Years to expiration, t = Number of periods prior to expiration, n = u= d= Multi-period 12.0000% 0.5 2 1.0618 0.9418 Single-period 31.5573% 0.5 1 1.250 0.8000 I J K L M A 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 B C D E F G H Here are the other data for Western, taken from the original problem: Current stock price, P = Risk-free rate, rRF = Strike price, X = $40.00 40.000 8% $35.00 35.000 8% Because we are going to solve a binomial problem repeatedly, it will be easier if we go ahead and calculate the p's now. pu = 0.64031 pd = 0.33989 Applying these values of u and d to the intital stock price gives the possible stock prices after 3 months. We can then apply u and d to these 3-month values to get the stock values at the end of 6 months, as shown below. Notice that because d = 1/u, the "middle" stock value at the end of the year is the same whether the stock initially went up and then went down, or whether it went down and then went up. I J K L M A 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 B C D E F G H I Notice that the range of final outcomes at 6 months is wider than the previous problem. However, the standard deviation of stock returns is the same as before, because most of the time the stock price will end up at the middle outcome rather than at the top or bottom outcomes. Figure 8-3 The 2-Period Binomial Lattice and Option Valuation Standard deviation of stock return: = Current stock price: P = Up factor for stock price: u = Down factor for stock price: d = Strike price: X = Risk-free rate: rRF = 12.0000% $40.00 1.0618 0.9418 $35.00 8.00% 0.50 2 Years to expiration: t = Number of periods until expiration: n = Price of $1 payoff if stock goes up: u = 0.64031 Price of $1 payoff if stock goes down: d = 0.33989 Now 3 months 6 months Stock = P (u) (u) = $45.10 Cuu = Max[P(u)(u) X, 0] Cuu = $10.10 Stock = P (u) = $42.47 Cu = Cuuu + Cudd Cu = $8.17 P = $40.00 VC=Cuu+Cdd VC = $6.37 Stock = P (u) (d) = P (d) (u) Stock = $40.00 Cud = Cdu = Max[P(u)(d) X, 0] Cud = $5.00 Stock = P (d) = $37.67 Cd = Cudu + Cddd Cu = $3.36 Stock = P (d) (d) = $35.48 Cdd = Max[P(d)(d) X, 0] Cdd = $0.48 To find the current value of the option, we can break the binomial lattice into three problems. Problem #1 is to find the option value at the end of six months, given that the stock moved upward from its initial value. Problem #2 is to find the option value at the end of six months, given that the stock moved downward from its intitial value. Finally, problem #3 is to find the current value of the option, given its two possible values at the end of six months. In this example, we divided time into two periods. If we were to divide time into more periods, we would get a distribution of stock prices in the last period that would be very realistic, which would give a very accurate option price. It is true that dividing time into more periods would create more binominal problems to solve, but each problem is very easy and computers can solve them very quickly. J K L M A 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 B C D E F G H 8-5 The Black-Scholes Option Pricing Model (OPM) In deriving this option pricing model, Black and Scholes made the following assumptions: 1. The stock underlying the call option provides no dividends or other distributions during the life of the option. 2. There are no transaction costs for buying or selling either the stock or the option. 3. The short-term, risk-free interest rate is known and is constant during the life of the option. 4. Any purchaser of a security may borrow any fraction of the purchase price at the short-term, risk-free interest rate. 5. Short selling is permitted, and the short seller will receive immediately the full cash proceeds of today's price for a security sold short. 6. The call option can be exercised only on its expiration date. 7. Trading in all securities takes place continuously, and the stock price moves randomly. The derivation of the Black-Scholes model rests on the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, an investor can create a risk-free investment position, where gains on the stock are exactly offset by losses on the option. Ultimately, the Black-Scholes model utilizes these three formulas: VC = P[ N (d1) ] - X e-r t [ N (d2) ] Note: r is the risk free rate, rRF. d1 = { ln (P/X) + [rRF + s2 /2) ] t } / (s t1/2) d2 = d1 - s (t 1 / 2) In these equations, V is the value of the option. P is the current price of the stock. N(d 1) is the area beneath the standard normal distribution corresponding to (d 1). X is the strike price. rRF is the risk-free rate. t is the time to maturity. N(d2) is the area beneath the standard normal distribution corresponding to (d2). is the volatility of the stock price, as measured by the standard deviation. Looking at these equations we see that you must first solve d1 and d2 before you can proceed to value the option. First, we will lay out the input data given earlier for Western Cellular's call option. Inputs: P= X= rRF = t= s Key Output: VC = $15 $15 $0.75 6.00% 0.5 12.000% Now, we will use the formula from above to solve for d 1. d1 = 0.3960 Having solved for d1, we will now use this value to find d2. d2 = 0.3111 At this point, we have all of the necessary inputs for solving for the value of the call option. We will use the formula for V from above to find the value. The only complication arises when entering N(d1) and N(d2). Remember, these are the areas under the normal distribution. Luckily, Excel is equipped with a function that can determine cumulative probabilities of the standard normal distribution. This function is located in the list of statistical functions, as "NORMSDIST". For both N(d1) and N(d2), we will follow the same procedure of using this function in the value formula. The data entries for N(d1) are shown below. I J K L M A 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 B C D N(d1) = 0.6539 N(d2) = E F G H I J K L M 0.6221 By applying this method for cumulative distributions, we can solve for the option value using the formula above. VC = $0.75 EFFECTS OF OPM FACTORS ON THE VALUE OF A CALL OPTION The figure below shows 3 of Westerns's call options, each with a $35 strike price. One option has 1 year until expiration, 1 has 6 months (0.5 years), and 1 has 3 months (0.25 years). Figure 8-5 Western Cellular's Call Options with a Strike Price of $35 50 Data for the figure. Option Price ($) t= 1 t = 0.5 t = 0.25 45 40 35 30 25 20 15 Exercise Value 10 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 St ock Price ($ ) The figure shows that: 1. Option prices increase as the stock price increases relative to the strike price. 2. Option prices increase as time to expiration increases. 3. Obviousy, an increase in the strike price will cause the option price to fall. The impact of changes in . We keep all inputs constant except the standard deviation: Standard Call option deviation price 0.001% $0.44 10.000% $0.68 31.557% $1.55 40.000% $1.89 60.000% $2.71 90.000% $3.92 The impact of changes in the risk-free rate. We keep all inputs constant except the risk-free rate: Risk-free rate (rRF) 0% 4% 8% 12% 20% Call option price $0.51 $0.66 $0.85 $1.05 $1.50 Stock Pric $0.00 $2.50 $5.00 $7.50 $10.00 $12.50 $15.00 $17.50 $20.00 $22.50 $25.00 $27.50 $30.00 $32.50 $35.00 $37.50 $40.00 $42.50 $45.00 $47.50 $50.00 $52.50 $55.00 $57.50 Time until expiration 1 0.5 $0 $1.22 $0.75 $0.00 $0.00 $0.00 $0.00 $0.00 $0.02 $0.75 $2.95 $5.44 $7.94 $10.44 $12.94 $15.44 $17.94 $20.44 $22.94 $25.44 $27.94 $30.44 $32.94 $35.44 $37.94 $40.44 $42.94 0.25 $0.48 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.48 $2.72 $5.22 $7.72 $10.22 $12.72 $15.22 $17.72 $20.22 $22.72 $25.22 $27.72 $30.22 $32.72 $35.22 $37.72 $40.22 $42.72 N 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 O P Q R S T N 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 O P Q R S T N 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 O P Q R S T N 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 O P Q R S T