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Option Pricing GSB 732 Spring 2009 4/22/09 (See original in 2006 Fall GSB 732)

Option Pricing GSB 732 Spring 2009 4/22/09 (See original in 2006 Fall GSB 732). Option Pricing Relationship Put Call Parity for Stock Options. > What is the price of a call? > To find out, form the following riskless portfolio * :

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Option Pricing GSB 732 Spring 2009 4/22/09 (See original in 2006 Fall GSB 732)

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  1. Option PricingGSB 732Spring 20094/22/09(See original in 2006 Fall GSB 732)

  2. Option Pricing RelationshipPut Call Parity for Stock Options > What is the price of a call? > To find out, form the following riskless portfolio*: Write one call struck at K for C dollars Buy one put struck at K for P dollars Buy one share of stock for S0 dollars Borrow the Present Value of K Denote the riskless interest rate as r > The price of the call depends on P, S0, K, and r! > This price web called Put-Call Parity! * This combo is called a conversion; see Strong p 73 fn #5 for other names.

  3. Option Pricing RelationshipPut Call Parity for Stock Options Let’s examine the cash flows of this portfolio.

  4. Option Pricing RelationshipPut Call Parity for Stock Options * ** * If one knows P, S0, K, & r, then one knows C! * Note to compare futures opts:

  5. ZZOption Pricing RelationshipPut Call Parity for Futures OptionsZZ

  6. ZZOption Pricing RelationshipPut Call Parity for Futures OptionsZZ * * * * vs. * Recall for stocks we had:

  7. YYOption Pricing RelationshipPut Call Parity for Futures OptionsYY

  8. YYOption Pricing RelationshipPut Call Parity for Futures OptionsYY * * * * vs. * Recall for stocks we had:

  9. Call is a Leveraged Stock Purchase But what is the price of a call? The Pricing logic underling option pricing is: “…a riskless investment should earn the riskless rate of interest. If this is not the case, arbitrageurs will quickly transact so as to move prices to their equilibrium relationship.” Strong, pp 115-16.

  10. Call is a Leveraged Stock Purchase What is the price of a call? Facts: > Current stock price S0 is $75 > Strike price K is $75 > Interest rate is 10% > Time horizon 1 year > Stock price either goes up to $100 or falls to $50 $100 S0 = $75 $50

  11. Call is a Leveraged Stock Purchase Consider the cash flows of two alternate strategies #1: Buy 2 calls each struck @ $75 for premium of C$ each #2a: Buy 1 share of stock at $75 = S0 b: Borrow @ 10%

  12. Call is a Leveraged Stock Purchase Can we fill in [ ] so plan 2 payoff = plan 1’s $0,$50?

  13. Call is a Leveraged Stock Purchase Can we fill in [ ] so plan 2 payoff = plan 1’s $0,$50?

  14. Call is a Leveraged Stock Purchase Yes! Fill in [ ] with -$50! Both plan’s payoff $0,$50!

  15. Call is a Leveraged Stock Purchase Yes! Fill in [ ] with -$50! Both plan’s payoff $0,$50!

  16. Call is a Leveraged Stock Purchase -$50 at T means borrow $50/(1+10%) at t0 = $45.46!

  17. Call is a Leveraged Stock Purchase Since the two strategies have the same payoffs, the price paid at t0 must be the same for the strategies. 2C$ = $75 – $45.46 C$ = $29.54/2 C$ = $14.77

  18. Call is a Leveraged Stock Purchase Since the two strategies have the same cash inflows at time T (0,50) and cash outflow at time t0 (= $29.54 = 75-45.46) A Call is a Leveraged Stock Purchase!

  19. Hedged PortfolioSell Call, Buy Stock, Borrow * If Call is equivalent to a leveraged stock purchase: 1. Call price = price of leveraged stock in Equilib. 2. Riskless to: write calls & buy stock on leverage * If Call price > or < price of leverage stock then ***** ARBITRAGE POSSIBILITIES!* Earn more than the riskless rate of interest by Buying under priced & selling over priced

  20. Hedged PortfolioSell Call, Buy Stock, Borrow > Price call using hedged portfolio > Hedged portfolio combines call & its equivalent leveraged stock purchase. > Hedged means KNOW FV for SURE! So we can use PV*(1+r) = FV. > Strong uses this approach to pricing a call.> Benefits * See, rather than assume, number of calls = N. * See option price flowing from arbitrage. * Extendable framework  more real world.

  21. Hedged PortfolioSell Call, Buy Stock, Borrow > Consider the hedged portfolio composed of: Write N calls struck @$75, premium of C$ each Buy 1 share of stock @ $75 = S0 Borrow B$ > 1 year time horizon (again) > 10% riskless interest (again) > Price either goes down to $50 or up to $100

  22. Adjust Hedged Portfolio FactsVolatility: S1 goes $50/$100 t0 T r = 10% $100 S0 = $75 $50

  23. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values

  24. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value!

  25. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value! $50 = $100-$25N $50-$100 = -$25N -$50/(-$25) = N 2 = N

  26. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value! $50 = $100-$25N $50-$100 = -$25N -$50/(-$25) = N 2 = N

  27. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values [$75-2*C$]grows to $50 either way! RISKLESS!

  28. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values [$75-2*C$]grows to $50 either way! [$75-2*C$] * [1+r] = $50 either way PV * [1+r] = FV PV = FV/[1+r] [$75 –2C$] = $50/[1+r]

  29. Hedged PortfolioSell Call, Buy Stock, Borrow Portfolio Values So solve for C$ PV* (1+r) = FV($75-2C$)*(1.10)1 = $50 ($75-2C$) = $50/(1.10)1 = $45.56 -2C$ = $45.46 - $75 = -$29.54C$ = -$29.54/(-2) = $14.77 PV0 is[$75-2*C$]&grows to FV of $50 either way……… so RISKLESS!

  30. Hedged PortfolioSell Call, Buy Stock, Borrow Cash Flows > Note, along the way to finding N & C$, we found B$ =$45.46. > At T, portfolio worth $50 either way. So bank would lend $50/(1.1)=$45.46 at t0. > If your own $45.46 finances this hedged portfolio rather than borrow, you would earn 10% riskless rate as your $45.46 grows to $50.> Earn riskless rate only  option is FAIRLY PRICED, NO ARB Profit! >>> NOW what if C$ was $15 rather than $14.77? What would you do? <<<

  31. Hedged PortfolioSell Call, Buy Stock, Borrow Cash Flows NOW what would you do if C$ was $15 rather than $14.77?  Arbitrage! At time t0 you could take in $0.46 on net and pay out $0 on net at T! $30.00 in from writing calls > $29.54 out on net $75 out from buying stock $45.46 in from borrowing @ 10%Implicit: rebalance hedge each period; but only one period so rebalance hidden!

  32. Adjust Hedged Portfolio FactsMore Volatility: S1 goes $40/$110 vs $50/100 t0 T r = 10% $110 $100 S0 = $75 $50 $40

  33. Adjust Hedged Portfolio FactsMore Volatility: S1 goes $40/$110 vs $50/100 Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value! $40 = $110-$35N $40-$110 = -$35N -$70/(-$35) = N 2 = N

  34. Adjust Hedged Portfolio FactsMore Volatility: S1 goes $40/$110 vs $50/100 Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value! $40 = $110-$35N $40-$110 = -$35N -$70/(-$35) = N 2 = N

  35. Adjust Hedged Portfolio FactsMore Volatility: S1 goes $40/$110 vs $50/100 Portfolio Values So solve for C$ ($75-2C$)*(1.10)1 = $40 ($75-2C$) = $40/(1.10)1 = $36.36 -2C$ = $36.36 - $75 = -$38.64C$ = -$38.64/(-2) = $19.32 > $14.77 with lower volatility! [$75-2*C$] grows to $40 either way so RISKLESS! Ceteris paribus, higher volatility causes higher option price!

  36. Adjust Hedged Portfolio FactsMore Time: 2 Years vs. 1 Year t0 T 2T r = 10% $110 $110 $100 S0 = $75 $50 $40 $40

  37. Adjust Hedged Portfolio FactsMore Time: 2 Years vs. 1 Year Portfolio Values So solve for C$ ($75-2C$)*(1.10)2 = $40 ($75-2C$) = $40/(1.10)2= $33.06 -2C$ = $33.06 - $75 = -$41.94C$ = -$38.64/(-2) = $20.97 > $19.32 with one year! [$75-2*C$] grows to $40 either way so RISKLESS! Ceteris paribus, more time causes higher option price!

  38. Hedged PortfolioAsymmetric Down & Up: $60/$100 vs $50/$100 Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value!

  39. Hedged PortfolioAsymmetric Down & Up: $60/$100 vs $50/$100 Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value! $60 = $100-$25N $60-$100 = -$25N -$40/(-$25) = N 1.6 = NThe relation between #shares & #options is important. One can not do 1.6 options. So just increase position size 10x to 10 shares & 16 options.

  40. Hedged PortfolioAsymmetric Down & Up: $60/$100 vs $50/$100 Portfolio Values Pick N so the portfolio is riskless: That is so the T down value = T up value! $60 = $100-$25N $60-$100 = -$25N -$40/(-$25) = N 1.6 = N

  41. Hedged PortfolioAsymmetric Down & Up: $60/$100 vs $50/$100 Portfolio Values [$75-1.6*C$]grows to $60 either way so RISKLESS!

  42. Hedged PortfolioAsymmetric Down & Up: $60/$100 vs $50/$100 Portfolio Values [$75-1.6*C$]grows to $60 either way so      RISKLESS! So solve for C$ ($75-1.6C$)*(1.10)1 = $60 ($75-1.6C$) = $60/(1.10)1 = $54.54 -1.6C$ = $54.54 - $75 = -$20.45C$ = -$20.45/(-1.6) = $12.78 vs. $14.77 for symmetric case!

  43. Option Pricing via Binomial > Binomial Bi is two and Nomial is name > Assumed stock price either goes up or down to known values > Results: get C$ & get C$ directly related to volatility & to time > Buts: More than two possibilities in real world More often than once per year > Eliminate these Buts by Extending Binomial MULTIPLE periods

  44. Option Pricing via Binomial > Extending Binomial to MULTIPLE periods > Two technical tools 1. Continuous Compounding 2. Implied Branch Probabilities > Method 1. Build value tree to find terminal period stock values 2. Convert to Intrinsic values 3. Find implied branch probabilities 4. Work backwards to expected present value of call

  45. Option Pricing via BinomialTool: Continuous Compounding > Discrete vs continuous compounding: T= 1 & r = 10% > Discrete: $10(1+r)t = $10(1.1)1 = $10(1.1) = $11.00 > Continuous: $10ert = $10(e10%*1) = $10(1.1052) = $11.05 > Discrete Compounding when solving for C$ ($75-2C$)*(1.10)1 = $50 ($75-2C$) = $50/(1.10)1 = $45.56 -2C$ = $45.46 - $75 = -$29.54C$ = -$29.54/(-2) = $14.77 > Continuous Compounding when solving forC$ ($75-2C$)*e10%*1 = $50 ($75-2C$) = $50/e10%*1 = $50/1.1052 = $45.24 -2C$ = $45.24 - $75 = -$29.76C$ = -$29.76/(-2) = $14.88

  46. Option Pricing via BinomialTool: Implied Branch Probabilities > Suppose non-dividend stock expected to return 10.52% riskless rate r = 10%, compounded continuously, 1 year. ert => e10%*1 for 10.52% > Then $100 invested in the stock should return ert $100(Pup*U) + $100(Pdown*D) = $100ert (Pup*U) + (Pdown*D) = ert > Where U = 1+ % increase if stock goes up D = 1- % decrease if stock goes down Pup = implied probability that stock goes up Pdown = implied probability that stock goes down

  47. Option Pricing via BinomialTool: Implied Branch Probabilities [1] Given (Pup * U) + (Pdown * D) = ert [2] Since Pup + Pdown = 1, we know Pdown = (1-Pup) Solve for the implied probabilities by sub [2] into [1] gives [3] [3] Pup = [ert –D] / [U – D] > If U is 1.3333 for a stock rising 33.33% D is 0.6667 for a stock falling 33.33% r = 10% for ert = 1.1052 >Then Pup = [1.1052-1.3333] / [1.3333-0.6667] = 65.78% Pdown = (1-Pup) = 34.22%

  48. Option Pricing via BinomialTool: Implied Branch Probabilities Method 1. Build value tree to find terminal period stock values 2. Convert to intrinsic values 3. Assign implied branch probabilities 4. Work backwards to expected present value for C$

  49. Option Pricing via BinomialStep 1. Build Value Tree/Find Terminal Stock Values t0 T r = 10% $75*U = $75*(1.3333) =$100 S0 = $75 $75*D = $75*0.6667 = $50

  50. Option Pricing via BinomialStep 2. Convert to Intrinsic Values t0 T r = 10% S1 IntrinsicValue $100  $25 = ($100-K) S0 = $75 Strike K = $75 $50  $0 ↔ ($50-K)<0

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