Part V: Option Pricing Basics


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1 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 to the Blackcholes pricing model binomial trees & riskneutral valuation 1 Part V: Option Pricing Basics 2 Option Pricing Principles Option Pricing Principles: Notation Fundamentals time value vs. intrinsic value key determinants of option values American vs. European options Early exercise Putcall parity nondividend paying stocks dividend adjustment Option pricing Blackcholes formula : trike price = exercise price c : European call option price p : European put option price C : American call option price P : American put option price t : Current time T : Maturity = time when option expires t : pot price at time t σ: Volatility of the underlying s price : PV of ividends r : Relevant riskfree rate (continuous compounding) 3 4 Option Pricing Principles 2 intrinsic value vs. time value intrinsic value» calls: Max[0, t  ]» put: Max[0,  t ] time value = option premium minus intrinsic value» at worst, equal to 0? (nope: European vs. American)» strictly positive for outof the money options» usually positive for inthemoney options Option Pricing Principles 3 Key determinants of option prices American options vs. European options» at least as valuable (C c, P p)» equal values at maturity time to maturity» American options: T P and C» European options? strike price» p&p but c&c 5 6 1
2 Option Pricing Principles 4 Key determinants of option prices (continued) price of underlying asset» t p&p but c&c ividends» c&c but p&p» IV (European options) vs. TV effect (American options) volatility of underlying asset» σ p&p and c&c (intuition?) hard floors vs. soft floors 7 Option Pricing Principles 5 H7 Table 9.1; H8 Table 10.1 Variable T t σ r c p C P ?? Option Pricing Principles 6 Option Pricing Principles 7 Hard and oft Floors hard floor (American calls)» C t Max[0, t ]» if not satisfied, arbitrage exists (buy call & strike now) soft floor (all calls, but only on nondividend paying stocks)» t c t = C t Max[0, t  /(1+r) T ]» if not satisfied, arbitrage exists (buy call & riskfree bond)» consequence: early exercise of American calls is not optimal if the underlying asset pays no dividends 9 Early exercise (American calls) nondividend paying stocks» never optimal to exercise early» intuition: C t Max[0, t  /(1+r) T ] > Max[0, t  ]» corollary: same bound for European calls on such assets dividend paying stocks?» early exercise may be optimal» but only if stock pays large dividend prior to maturity 10 Option Pricing Principles 8 Option Pricing Principles 9 Hard and oft Floors (continued) Question: uppose an American call option is written on Nortel stock. The exercise price is $105 (J ) and the present value of the exercise price is $100. (a) What is the hard floor price of the option if Nortel stock sells for $160? ketch a graph of the hard floor option prices against (i.e., in terms of) the Nortel stock s price. (b) At a stock price of $125, you notice the option selling for $18. Would this option price be an equilibrium price? Explain. Hard and oft Floors (continued) Answer: (a) Hard floor price = V = $160  $105 = $55. (b)an option price of $18 is below the hard floor price of $. In this case, everyone would want the call option. You could then acquire a share of Nortel stock for less than the current market price. imply buy the option (for $18), exercise it (paying $105), and you would then own a share of Nortel for a total price of $
3 Option Pricing Principles 10 Hard and oft Floors (American puts) hard floor» Max[0,  t ]» if not satisfied, arbitrage exists soft floor?» Max[0, /(1+r) T  t ]» BKM4 Fig Option Pricing Principles 11 Early exercise (American puts) can be optimal to exercise early» intuition 1: stock price cannot fall below 0» intuition 2: T /(1+r) T impact of dividend payments» dividends probability of early exercise Options: Early Exercise (Recap) Calls Puts often not optimal» never optimal for nondividend paying stocks importance of capturing dividends can be optimal to exercise early impact of dividend payments» dividends probability of early exercise Putcall parity PutCall Parity European options only» applicability to American options? Intuition Formally noarbitrage» if the payoffs of 2 portfolios are equal» then the costs of both portfolios must be equal reverse engineer the prices examples Intuition Profit PutCall Parity 2 T T Profit (a: covered call) (b) Profit Profit T T Putcall parity PutCall Parity 3 < T > T cash now 1. buy a call T  0 c OR 2a. buy a put 0  T p 2b. sell disc. bond   /(1+r) T 2c. buy stock T T Total T  0 (c: protect. put) (d) 17 hence: c = p + /(1+r)T  0 and thus c = p /(1+r) T 18 3
4 PutCall Parity 4 PutCall Parity 5 Question: European put and a European call on the same stock exercise price = $ 75 same expiration dates The current price of the stock is $68. The put s current price is $6.50 higher than the call s price A riskless investment over the time until expiration yields 3 percent. Answer: According to the parity equation: VP VC = [/(1 + rf) t ] V => VP VC = [$75/( )] $68 = $4.82. Thus, with the put being priced $6.50 higher than the call, the two options are out of parity. A riskless arbitrage opportunity would exist: Given this information, is there any riskless profit opportunities available? 19 Answer: A riskless arbitrage opportunity exists: PutCall Parity 6 ell the stock short..$68.00 ell the put option.. & Buy the call option. $ 6.50 Proceeds $74.50 Invest the proceeds at the riskless rate of 3%. At maturity, you will have the value at expiration of $76.74 = [$74.50 x 1.03]. Also, you can acquire a share of stock (to cover the short sale) for $75, no matter what happens to the stock price. You are assured $1.74 without putting any of your own money at risk PutCall Parity 7 Putcall parity (continued) continuoustime version dividends» c = p + t e r (T t )» c p = t e r (T t )» adjustment needed» c = p + PV( T )  PV(dividend)  /(1+r) T PutCall Parity 8 Extensions (NOT Exam Material) American options; = 0 < C P < e r (T t ) (H8 eq. 10.4) European options; > 0 c p = e r (T t ) (H8 eq. 10.7) American options; > 0 < C P < e r (T t ) (H7, 9.8 p. 215) (H8 eq. e ) Analytical Option Pricing Methods Blackcholes pluses (quick) & minuses (European calls, assumptions) Numerical Binomial Trees Monte Carlo Methods Finite difference Methods Analytical Approximation 24 4
5 Option Pricing Key Problem Uncertainty olution we don t know future stock prices Assume a distribution for periodic returns Assume a stochastic process for stock prices Option Pricing in Practice Blackcholes gives price of European call c = e where r( T t) r( T [ N( d )e 1 t) N( d )] 2 ln( / ) + ( r + σ / 2)( T t) d1 = σ T t 2 ln( / ) + ( r σ / 2)( T t) d2 = σ T t 2 = d σ T t 1 interpretation? Option Pricing in Practice 2 r( T t) r( T t) c = e [ N( d1)e N( d2)] N(z) = Prob(Z<z)» Z is standard normal N(d 2 )» = probability of exercise. N(d 2 )» = expected payout at exercise N(d 1 )exp(r(tt))» = expected value of the stock price, if exercised. Option Pricing in Practice 3 Blackcholes (continued) gives price of European call price of European put?» use putcall parity Profit» intuition: American options?» optimality of early exercise T Option Pricing in Practice 4 Binomial option pricing problem» assumptions needed for B& are not always realistic» example: interest rates are deterministic?! solutions: 1. numerical approximations» grid, riskneutral probabilities» as accurate as needed/desired 2. PB = Practitioners B& (e.g., Berkowitz) 29 Numerical Pricing Methods RiskNeutral valuation Methods Binomial Trees Early Exercise Possible Monte Carlo Methods everal Underlying Variables Possible Finite difference Methods Early Exercise Possible Analytical Approximation American Options 30 5
6 Approach RiskNeutral Valuation introduce binomial trees now to start thinking about Applicability riskneutral valuation of derivatives and dynamic hedging strategies use riskneutral valuation throughout the course return to binomial trees in Parts III & IV Example Call Option example (H7 Fig. 11.1; H8 12.1): 3month call option with strike price = 21 tock price = $ Call Price =? price of the call today? use riskneutral valuation tock Price = $22 Call Price = $1 tock Price = $18 Call Price = $ Example 2 Riskless Portfolio Portfolio LONG Δ shares HORT 1 call option =$ $22Δ $1 $18Δ =$18Δ  $0 Portfolio is riskless» if $22Δ $1 = $18Δ i.e. if Δ = 0.25» LONG 0.25 shares and HORT 1 call option 33 Example 3 Value of the riskless portfolio in 3 months if the stock price moves up:» $ = $4.50 if the stock price moves down:» $ = $4.50 today PV of $4.50 at the riskfree rate (why?) if annual continuouslycompounded riskfree rate is 12%, portfolio is worth: $4.50 e = $ Example 4 Value of the Option Today entire portfolio» worth $4.367 shares» worth Δ = 0.25 $ = $5 Value of the option» is therefore: $5 $4.367 = $ Binomial Option Pricing Fundamentals Why? approximate the movements in an asset s price to simplify the pricing of derivatives on the asset What? discretize underlying asset s price movements and value options as if in a riskneutral world How? asset price at the BEGINNING of any period can lead to only 2 stock prices at the EN of that period 36 6
7 Binomial Trees Asset Price Movements divide the time from t to T into small intervals Δt in each time interval, assume the asset s price can move UP by a proportional amount u or move OWN by a proportional amount d Binomial Trees 2 Movements in time interval Δt (H7 Fig.19.1; H8 12.2) u ƒ p (1 p ) ƒ u d ƒ d What? p, u and d Parameter values? Tree Parameters tree must give correct values for the mean & standard deviation of the stock price changes in a riskneutral world implification tree is recombining: u = 1/ d 39 Tree Parameters 2 Complete Tree (Fig. 19.2) u 3 u 2 u u d d d 2 d 3 u 4 u 2 d 2 d 4 40 RiskNeutral Valuation Assumption no arbitrage opportunity exists Basic idea assume a binomial tree for asset price movements create a riskless portfolio stock plus option riskless portfolio always possible with binomial tree value the portfolio if riskless, then riskneutral valuation is OK Reference CoxRossRubinstein (Journal of Financial Economics 79) 41 RiskNeutral Valuation 2 European Put example (u=1.1; d=1/u): 3month put with strike price = 21 tock price = $ Put price =? price of the put? use riskneutral valuation tock Price = $22 Put Price = $0 tock Price = $18.18 Put Price = $
8 RiskNeutral Valuation 3 Riskless Portfolio Portfolio LONG Δ shares LONG 1 put option (why?) =$ $22Δ $18.18Δ +$2.82 Portfolio is riskless if 22Δ = 18.18Δ+2.82 i.e if Δ = LONG shares and LONG 1 put option 43 RiskNeutral Valuation 4 Value of the entire (riskless) portfolio in 3 months if the stock price moves up:» $ $0 = $16.24 if the stock price moves down:» $ $2.82 = $13.42+$2.82 = $16.24 today PV of $16.24 at the riskfree rate (why?) if annual continuouslycompounded riskfree rate is 12%, portfolio is worth: $16.24 e = $ RiskNeutral Valuation 5 RiskNeutral Valuation 6 Value of the Option Today Entire portfolio is worth $15.76 hares are worth $ = $14.76 Value of the put option is therefore $ $14.76 = $ Generalization (H7 Fig. 11.2; H8 Fig. 12.2) derivative value f expires at time T is dependent on a stock u ƒ ƒ u d ƒ d 46 RiskNeutral Valuation 7 RiskNeutral Valuation 8 Riskless portfolio LONG Δ shares and HORT 1 derivative Δ  f u Δ ƒ u d Δ ƒ d riskless ƒu ƒd Δ =» if u Δ ƒ u = d Δ ƒ d or u d Value of the portfolio at time T :» (up state) u Δ ƒ u = d Δ ƒ d (down state) Value of the portfolio today :» (u Δ ƒ u )e rt» and also» Δ ƒ Hence» ƒ = Δ (u Δ ƒ u )e rt
9 RiskNeutral Valuation 9 Thus: ƒ = Δ (u Δ ƒ u )e rt ƒu ƒd Δ = u d ubstituting for Δ, we obtain ƒ = [ p ƒ u + (1 p )ƒ d ]e rt where rt d p = e u d 49 RiskNeutral Valuation 10 Interpretation ƒ = [ p ƒ u + (1 p ) ƒ d ]e rt p and (1 p ) can be interpreted as the riskneutral probabilities of up & down movements Value of a derivative = its expected payoff in a riskneutral world discounted at the riskfree rate ƒ p (1 p ) u ƒ u d ƒ d 50 Irrelevance of tock s Expected Return When valuing an option in terms of the underlying stock, the expected return on the stock is irrelevant 51 Original Example Revisited Call, H8 Fig.12.1 ( = ; = 21; Δt = T = 3 months) u = 22 ƒ u = 1 ƒ p (1 p ) riskneutral probabilities: d = 18 ƒ d = 0 rt e d e 0.9 p = = = u d Original Example Revisited 2 H8 Fig ƒ u = 22 ƒ u = 1 d = 18 ƒ d = 0 Value of the option c = e [ $ $0] = $ Original Example Revisited 3 Key Result riskneutral valuation ( revisited ) coincides with the ( original ) noarbitrage valuation. Generalization in general when pricing derivatives using riskneutral valuation is ok 54 9
10 Original Example Revisited 4 Valuing the tock in a riskneutral world stock must also earn the riskfree rate consequence ince p is a riskneutral probability $ x e = $22 x p +$18 x (1 p ) p = A Twotep Call Option Example H8 Fig (=21; u=1.1; d=0.9; T=6 months) Each time step is Δt=3 months A Twotep Call Option Example 2 Call value (Fig. 12.4; =21) 22 B A C Value at node B = e 0.12x0.25 ( x x 0) = Value at node A = e 0.12x0.25 ( x x 0) = No difference between American & European calls E F A Twotep Put Option Example Fig.12.7 (=52; u=1.2, d=0.8, T=2 years) 60 B 50 A E F Each time step is Δt=1 year; annual riskfree rate = r =5% r. Δt e d e 0.8 p = = = u d C 58 A Twotep Put Option Example 2 European put value (Fig. 12.7; =52) A C Value at node B = F e 0.05x1 ( x $ x $4) = $ Value at node C = e 0.05x1 ( x $ x $) = $ > Value at node A = e 0.05x1 ( x $ x $9.4636) = $ B E 59 A Twotep Put Option Example 3 American put value (Fig. 12.8; =52) or 2 Value at node B A 60 B or 0 E or = Max[  B, ] = $ Value at node C = Max[  C, ] = Max[12, ] = $12 > Value at node A = Max [2, e 0.05x1 ( x $ x $12)] = $ C F 10
11 efinition elta (Δ) is the ratio ynamic hedging elta» of the change in the price of a stock option» to the change in the price of the underlying stock The value of Δ varies from node to node» ynamic hedging needed! 61 elta 2 Riskless portfolio at a given node: LONG Δ shares and HORT 1 derivative u Δ ƒ u Δ  f d Δ ƒ d ƒu ƒd Δ = riskless u d» if u Δ ƒ u = d Δ ƒ d or 62 elta 3 European put (=50; =52) Δ= A 60 B , Δ=1/6 E , Δ=1 F Value of Δ at node B = (04)/(7248) = 1/6 (i.e., short 1 put and short 1/6 share) Value at node C = (4)/(4832) = 1 (i.e., short 1 put and short 1 share) Value at node A = ( )/(6040) = (short 1 put and share) C 63 11
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