Contents Learning Outcome Statements (LOS) Reading 32: The Term Structure and Interest Rate Dynamics Exam Focus Module 32.1: Spot and Forward Rates, Part Module 32.2: Spot and Forward Rates, Part Module 32.3: The Swap Rate Curve Module 32.4: Spread Measures Module 32.5: Term Structure Theory Module 32.6: Interest Rate Models Key Concepts Answer Key for Module Quizzes Reading 33: The Arbitrage-Free Valuation Framework Exam Focus Module 33.1: Binomial Trees, Part Module 33.2: Binomial Trees, Part Key Concepts Answer Key for Module Quizzes Reading 34: Valuation and Analysis of Bonds With Embedded Options Exam Focus Module 34.1: Types of Embedded Options Module 34.2: Valuing Bonds With Embedded Options, Part Module 34.3: Valuing Bonds With Embedded Options, Part Module 34.4: Option-Adjusted Spread Module 34.5: Duration Module 34.6: Key Rate Duration Module 34.7: Capped and Floored Floaters Module 34.8: Convertible Bonds 10 Key Concepts 11 Answer Key for Module Quizzes Reading 35:Credit Analysis Models Exam Focus Module 35.1: Credit Risk Measures Module 35.2: Analysis of Credit Risk Module 35.3: Credit Scores and Credit Ratings Module 35.4: Structural and Reduced Form Models Module 35.5: Credit Spread Analysis Module 35.6: Credit Spread Module 35.7: Credit Analysis of Securitized Debt Key Concepts 10 Answer Key for Module Quizzes Reading 36: Credit Default Swaps Exam Focus Module 36.1: CDS Features and Terms Module 36.2: Factors Affecting CDS Pricing Module 36.3: CDS Usage Key Concepts Answer Key for Module Quiz Topic Assessment Answers: Fixed Income Topic Assessment: Fixed Income Reading 37: Pricing and Valuation of Forward Commitments Exam Focus Module 37.1: Pricing and Valuation Concepts Module 37.2: Pricing and Valuation of Equity Forwards Module 37.3: Pricing and Valuation of Fixed Income Forwards Module 37.4: Pricing Forward Rate Agreements Module 37.5: Valuation of Forward Rate Agreements Module 37.6: Pricing and Valuation of Currency Contracts Module 37.7: Pricing and Valuation of Interest Rate Swaps Module 37.8: Currency Swaps 10 Module 37.9: Equity Swaps 11 Key Concepts 12 Answer Key for Module Quizzes 10 Reading 38: Valuation of Contingent Claims Exam Focus Module 38.1: The Binomial Model Module 38.2: Two Period Binomial Model and Put-Call Parity Module 38.3: American Options Module 38.4: Hedge Ratio Module 38.5: Interest Rate Options Module 38.6: Black-Scholes-Merton and Swaptions Module 38.7: Option Greeks and Dynamic Hedging Key Concepts 10 Answer Key for Module Quizzes 11 Topic Assessment: Derivatives 12 Topic Assessment Answers: Derivatives 13 Formulas 14 Copyright List of Pages 10 11 12 13 14 15 16 17 18 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 vi vii viii ix 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 33 34 35 36 37 38 39 40 41 42 43 44 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 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203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 LEARNING OUTCOME STATEMENTS (LOS) STUDY SESSION 12 The topical coverage corresponds with the following CFA Institute assigned reading: 32 The Term Structure and Interest Rate Dynamics The candidate should be able to: a describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve (page 2) b describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models (page 4) c describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping (page 5) d describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management (page 7) e describe the strategy of riding the yield curve (page 10) f explain the swap rate curve and why and how market participants use it in valuation (page 12) g calculate and interpret the swap spread for a given maturity (page 14) h describe the Z-spread (page 15) i describe the TED and Libor–OIS spreads (page 16) j explain traditional theories of the term structure of interest rates and describe the implications of each theory for forward rates and the shape of the yield curve (page 17) k describe modern term structure models and how they are used (page 20) l explain how a bond’s exposure to each of the factors driving the yield curve can be measured and how these exposures can be used to manage yield curve risks (page 22) m explain the maturity structure of yield volatilities and their effect on price volatility (page 24) The topical coverage corresponds with the following CFA Institute assigned reading: 33 The Arbitrage-Free Valuation Framework The candidate should be able to: a explain what is meant by arbitrage-free valuation of a fixed-income instrument (page 33) b calculate the arbitrage-free value of an option-free, fixed-rate coupon bond (page 34) c describe a binomial interest rate tree framework (page 35) d describe the backward induction valuation methodology and calculate the value of a fixed-income instrument given its cash flow at each node (page 36) e describe the process of calibrating a binomial interest rate tree to match a specific term structure (page 40) f compare pricing using the zero-coupon yield curve with pricing using an arbitrage-free binomial lattice (page 41) g describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixed-income instrument given its cash flows along each path (page 43) h describe a Monte Carlo forward-rate simulation and its application (page 44) KEY CONCEPTS LOS 38.a To value an option using a two-period binomial model: Calculate the stock values at the end of two periods (there are three possible outcomes because an up-down move gets you to the same place as a down-up move) Calculate option payoffs at the end of two periods Calculate expected values at the end of two periods using the up- and down-move probabilities Discount these back one period at the risk-free rate to find the option values at the end of the first period Calculate expected value at the end of period one using the up- and down-move probabilities Discount this back one period to find the option value today To price an option on a bond using a binomial tree, (1) price the bond at each node using projected interest rates, (2) calculate the intrinsic value of the option at each node at maturity of the option, and (3) calculate the value of the option today LOS 38.b The value of a European call option using the binomial option valuation model is the present value of the expected value of the option in the states C0 = (πU ×C+ +πD ×C−) (1+Rf ) The value of an American-style call option on a non-dividend paying stock is the same as the value of an equivalent European-style call option American-style put options may be more valuable than equivalent European-style put options due to the ability to exercise early and earn interest on the intrinsic value LOS 38.c Synthetic call and put options can be created using a replicating portfolio A replication portfolio for a call option consists of a leveraged position in h shares where h is the hedge ratio or delta of the option A replication portfolio for a put option consists of a long position in a risk-free bond and a short position in h shares If the value of the option exceeds the value of the replicating portfolio, an arbitrage profit can be earned by writing the option and purchasing the replicating portfolio LOS 38.d The value of an interest rate option is computed similarly to the value of options on stocks: as the present value of the expected future payoff Unlike binomial stock price trees, binomial interest rate trees have equal (risk-neutral) probabilities of the up and down states occurring LOS 38.e Option values can be calculated as present value of expected payoffs on the option, discounted at the risk-free rate The probabilities used to calculate the expected value are riskneutral probabilities LOS 38.f The assumptions underlying the BSM model are: The price of the underlying asset changes smoothly (i.e., does not jump) and follows a lognormal distribution The (continuous) risk-free rate is constant and known The volatility of the underlying asset is constant and known Markets are “frictionless.” The underlying asset generates no cash flows The options are European LOS 38.g Calls can be thought of as leveraged stock investment where N(d1) units of stock is purchased using e-rTXN(d2) of borrowed funds A portfolio that replicates a put option can be constructed by combining a long position in N(–d2) bonds and a short position in N(–d1) stocks LOS 38.h European options on dividend-paying stock can be valued by adjusting the model to incorporate the yield on the stock: the current stock price is adjusted by subtracting the present value of dividends expected up until option expiration Options on currencies incorporate a yield on the foreign currency based on the interest rate in that currency LOS 38.i c The Black model is simply the BSM model with e−Rf × TF T substituted for S0 LOS 38.j The Black model can be used to value interest rate options by substituting the current forward rate in place of the stock price and the exercise rate in place of exercise price The value is adjusted for accrual period (i.e., the period covered by the underlying rate) A swaption is an option that gives the holder the right to enter into an interest rate swap A payer (receiver) swaption is the right to enter into a swap as the fixed-rate payer (receiver) A payer (receiver) swaption gains value when interest rates increase (decrease) LOS 38.k Direction of BSM option value changes for an increase in the five model inputs: Sensitivity Factor (Greek) Delta Gamma Vega Rho Theta Input Calls Puts Asset price (S) Delta Volatility (σ) Risk-free rate (r) Time to expiration (T) Exercise price (X) Positively related Delta > Positively related Gamma > Positively related Vega > Negatively related Delta < Positively related Gamma > Positively related Vega > Positively related Rho > Negatively related Rho < Time value → $0 as call → maturity Theta < Time value → $0 as put → maturity Theta < Negatively related Positively related Delta is the change in the price of an option for a one-unit change in the price of the underlying security e–δTN(d ) from the BSM model is the delta of a call option, while –e– δTN(–d ) is the put option delta As stock price increases, delta for a call option increases from to e–δT, while delta for a put option increases from –e–δT to LOS 38.l The goal of a delta-neutral portfolio (or delta-neutral hedge) is to combine a long position in a stock with a short position in call options (or a long position in put options) so that the portfolio value does not change when the stock value changes Given that delta changes when stock price changes, a delta hedged portfolio needs to be continuously rebalanced Gamma measures how much delta changes as the asset price changes and, thus, offers a measure of how poorly a fixed hedge will perform as the price of the underlying asset changes LOS 38.m When the price of the underlying stock abruptly jumps, a violation of BSM, the delta of the option would change (captured by the option gamma), leaving a previously delta hedged portfolio unhedged This is the gamma risk of a delta hedged portfolio LOS 38.n Implied volatility is the volatility that, when used in the Black-Scholes formula, produces the current market price of the option If an option is overvalued, implied volatility is too high ANSWER KEY FOR MODULE QUIZZES Module Quiz 38.1, 38.2 B Stock Tree Option Tree U = 1.15 D = 0.87 πU = 1.04−0.87 1.15−0.87 = 0.61 πD = − 0.61 = 0.39 C++ = $43.80 −+ C+− = C = $18.00 C−− = $0 C+ = (0.61×$43.80)+(0.39 ×$18.00) 1.04 C− = (0.61×$18.00)+(0.39 ×$0) 1.04 C0 = (0.61 ×$32.44)+(0.39× $10.56) 1.04 = $32.44 = $10.56 = $22.99 (Module 38.2, LOS 38.b) Module Quiz 38.3 B In the upper node at the end of the first year, the European option is worth $0.29, but the American option can be exercised and a profit of $0.57 realized (the difference between the bond price of $100.57 and the exercise price of $100) In the lower node at the end of the first year, the European option is worth $1.35, but the American option can be exercised and a profit of $3.80 realized (the difference between the bond price of $103.80 and the exercise price of $100) The value of the American option today is therefore: American option price = ($0.57×0.5)+($3.80× 0.5) 1.03 = $2.12 (LOS 38.b) Module Quiz 38.4 C The hedge ratio in a one-period model is equivalent to a delta, the ratio of the call price change to the stock price change We will sell the 1,000 calls because they are overpriced Buying 350 shares of stock will produce a riskless hedge The payoff at expiration will return more than the riskless rate on the net cost of the hedge portfolio Borrowing to finance the hedge portfolio and earning a higher rate than the borrowing rate produces arbitrage profits (LOS 38.c) A A synthetic European put option is formed by: Buying a European call option Short-selling the stock Buying (i.e., investing) the present value of the exercise price worth of a purediscount riskless bond (LOS 38.c) Module Quiz 38.5, 38.6 A The dividend affects option values because if you own the option, you not have access to the dividend Hence, if the firm pays a dividend during the life of the option, this must be considered in the valuation formula Dividends decrease the value of call options, all else equal, and they increase the value of put options (Module 38.6, LOS 38.h) B To derive the BSM model, we need to assume no arbitrage is possible and that: Asset returns (price changes) follow a lognormal distribution The (continuous) risk-free rate is constant The volatility of the underlying asset is constant Markets are “frictionless.” The asset has no cash flows The options are European (i.e., they can only be exercised at maturity) (Module 38.6, LOS 38.f) A According to put/call parity, the put’s value is: P0 = C0 − S0 + (X × e−Rc ×T) = $4.09 − $60.00 + [$60.00 × e−(0.05× 1.0)] = $ f (Module 38.6, LOS 38.h) A ABC and Chefron stock are identical in all respects except Chefron pays a dividend Therefore, the call option on Chefron stock must be worth less than the call on ABC (i.e., less than $4.09) $3.51 is the only possible answer (Module 38.6, LOS 38.h) Module Quiz 38.7 A The value of a call and the risk-free rate are positively related, so as the risk-free rate increases, the value of the call will increase (LOS 38.k) A Volatility increases will increase the values of both puts and calls (LOS 38.k) A Implied volatility is the volatility that produces market option prices from the BSM model Its use for pricing options is limited because it is based on market prices Past returns are used to calculate historical volatility (LOS 38.n) C The put option will decrease in value as the underlying stock price increases: –0.43 × $4 = –$1.72 (LOS 38.k) A The call option is deep in-the-money and must have a delta close to one The put option is deep out-of-the-money and will have a delta close to zero Therefore, the value of the in-the-money call will decrease by close to $1 (e.g., $0.94), and the value of the out-of-the-money put will increase by a much smaller amount (e.g., $0.08) The call price will fall by more than the put price will increase (LOS 38.k) C The put option is currently at-the-money because its exercise price is equal to the stock price of $45 As stock price increases, the put option’s delta (which is less than zero) will increase toward zero, becoming less negative The put option’s gamma, which measures the rate of change in delta as the stock price changes, is at a maximum when the option is at-the-money Therefore, as the option moves out-of-the-money, its gamma will fall (LOS 38.k) A If ∆S = –$1.00, ∆C ≈ 0.42 × (–1.00) = –$0.42, and ∆P ≈ (0.42 – 1) × (–1.00) = $0.58, the call will decrease by less ($0.42) than the increase in the price of the put ($0.58) (LOS 38.k) TOPIC ASSESSMENT: DERIVATIVES You have now finished the Derivatives topic section The following topic assessment will provide immediate feedback on how effective your study of this material has been The test is best taken timed; allow minutes per subquestion (18 minutes per item set) This topic assessment is more exam-like than typical Module Quizzes or QBank questions A score less than 70% suggests that additional review of this topic is needed Use the following information to answer Questions through Derrick Honny, CFA, has operated his own portfolio management business for many years Several of his clients have fixed income positions, and one of Honny’s analysts has advised him that the firm could improve the performance of these portfolios through swaps Honny has begun investigating the properties of swaps His plan is first to establish some minor positions to gain some experience before actively using swaps on behalf of his clients Honny knows that the most basic type of swap is the plain vanilla swap, where one counterparty pays LIBOR as the floating rate and the other counterparty pays a fixed rate determined by the swap market He feels this would be a good place to begin and plans to enter into a 2-year, annual-pay plain vanilla swap where Honny pays LIBOR and receives the fixed swap rate from the other counterparty To get an idea regarding the swap rate he can expect on the 2-year swap, he collects market data on LIBOR Details are shown in Figure Figure 1: Market Data on Term Structure of Interest Rates Year LIBOR Discount Factor 5.00% 4.60% 0.9524 0.9158 Honny knows that as interest rates change, the value of a swap position will change Suppose that one year after inception, the LIBOR term structure is as given in Figure 2: Figure 2: Term Structure of Interest Rates After One Year Year LIBOR Discount Factor 0.5 1.5 4.80% 4.88% 4.90% 5.02% 0.9766 0.9535 0.9315 0.9088 One of Honny’s clients, George Rosen, is aware of Honny’s plans to use swaps and other derivatives in the management of Honny’s clients’ portfolios Rosen has a position for which he thinks a swap strategy might be appropriate Rosen asks Honny to arrange for him a payer swaption that matures in three years Honny is uncertain of the level of Rosen’s familiarity with swaps and swaptions, so Honny wants to make sure that the derivative is appropriate for the client He asks Rosen exactly what he intends to accomplish by entering into the swaption Honny also discusses the possible use of a covered call strategy for Rosen’s portfolio, whereby they would write one call option per unit of long stock held in order to generate an income stream Which of the following would be the least appropriate position to replicate the exposure Honny will get from the 2-year, plain vanilla swap position that he plans to take? A Long a series of interest rate puts and short a series of interest rate calls B Short a series of bond futures C Short a series of forward rate agreements Given the 1- and 2-year rates, the 2-year swap fixed rate would be closest to: A 4.20% B 4.51% C 4.80% Which of the following is most likely to be a conclusion that Honny would reach if the payer swaption has the same exercise rate as the market swap fixed rate for the underlying swap? A The payer swaption would be out of the money B The value of the payer swaption would be same as the value of an otherwise identical receiver swaption C The payer swaption would be worth more than an otherwise identical receiver swaption For this question only, assume that the swap fixed rate is 4.50% and the notional principal is $1 million Based on the information in Figure 2, the value of the swap to Honny is closest to: A –$3,623 B –$3,800 C –$6,790 If the current 30-day, 90-day, and 120-day LIBOR rates are 5%, 6%, and 7% respectively, the price (expressed as an annualized rate) of a × forward rate agreement (FRA) should be closest to: A 4.77% B 6.15% C 7.63% The gamma position of a covered call strategy is most likely to be: A positive B negative C zero TOPIC ASSESSMENT ANSWERS: DERIVATIVES B Since Honny will pay the floating rate in the 2-year swap, he gains when the floating rate goes down and loses when it goes up (relative to expectations at inception) This exposure could be replicated with either a short position in a series of FRAs, or a series of short interest rate calls and long interest rate puts Since short bond futures gain when floating rates increase and lose when floating rates decrease, such a position would give Honny an exposure opposite to the floating rate payer position in a fixed-for-floating interest rate swap (Study Session 14, Module 38.6, LOS 38.j) B Given the discount factors, the swap-fixed rate can be calculated as: swap fixed rate = 1−Z2-year Z1-year +Z 2-year = 1−0.9158 0.9524+0.9158 = 0451 Since the rates are already in annual terms, no further adjustment is necessary (Study Session 14, Module 37.7, LOS 37.d) B If the exercise rate of a receiver option and a payer swaption are equal to the atmarket forward swap rate, then the receiver and payer swaptions will have the same value When the exercise rate is equal to the market SFR, the payer option will be atthe-money (Study Session 14, Module 38.6, LOS 38.j) A Value to the payer = ΣDF ×(SFRNew − SFROld )× days 360 × notional There is only one settlement date remaining (one year away) Hence the sum of discount factors = the discount factor for the last settlement (one year away) = 0.9535 Also, given the single settlement date, the new swap fixed rate has to be the LIBOR rate for that settlement date, or 4.88% (given in Figure 2) value to the payer = 0.9535 × (0.0488 – 0.0450) × 360/360 × $1,000,000 = $3,623 The swap discussed is a receiver swap value to the receiver = –$3,623 (Study Session 14, Module 37.7, LOS 37.d) C 1 × FRA refers to a 90-day loan, beginning 30 days from today The actual rate on a 30-day loan is R30 = 0.05 × 30/360 = 0.004167 The actual rate on a 120-day loan is R120 = 0.07 × 120/360 = 0.02333 FR(30,90) = [(1 + R120) / (1 + R30)] – = (1.023333 / 1.004167) – = 0.0190871 The annualized 90-day rate = 0.0190871 × 360/90 = 0.07634 = 7.63% (Study Session 14, Module 37.4, LOS 37.a) B A covered call strategy entails a long position combined with a short call A long position in stock has zero gamma Calls have positive gamma, and a short position in call would have a negative gamma This negative gamma position from short call combined with zero gamma of long stock results in a net negative gamma position of a covered call portfolio (Study Session 14, Module 38.7, LOS 38.k) FORMULAS Study Sessions 12 and 13: Fixed Income price of a T-period zero-coupon bond PT = (1+ST) T forward price (at t = j) of a zero-coupon bond maturing at (j+k) F(j,k) = [1+ f(j,k)]k forward pricing model P(j+k) = PjF(j,k) Therefore: F(j,k) = P(j+k) Pj forward rate model [1 + S(j+k)](j+k) = (1 + Sj)j [1 + f(j,k)]k or [1 + f(j,k)]k = [1 + S(j+k)](j+k) / (1 + Sj)j swap spread swap spreadt = swap ratet – Treasury yieldt TED Spread TED Spread = (3-month LIBOR rate) – (3-month T-bill rate) LIBOR-OIS spread LIBOR-OIS spread = LIBOR rate – “overnight indexed swap” rate portfolio value change due to level, steepness, and curvature movements ΔP P ≈ −DL ΔxL − D SΔx S − DC ΔxC callable bond Vcall = Vstraight – Vcallable putable bond Vputable = Vstraight + Vput Vput = Vputable – Vstraight effective duration = ED = BV−Δy −BV +Δy 2×BV 0× Δy BV + BV − BV effective convexity = EC = × × BV−Δy + BV+Δy −(2×BV 0) BV 0× Δy2 convertible bond minimum value of convertible bond = greater of conversion value or straight value market conversion price = market price of convertible bond conversion ratio market conversion premium per share = market conversion price – stock’s market price market conversion premium ratio = premium over straight value = ( market conversion premium per share market price of common stock market price of convertible bond ) straight value −1 callable and putable convertible bond value = straight value of bond + value of call option on stock – value of call option on bond + value of put option on bond credit analysis recovery rate = percentage of money received upon default of the issuer loss given default (%) = 100 – recovery rate expected loss = probability of default × loss given default present value of expected loss = (value of risk-free bond) – (value of credit-risky bond) upfront premium % (paid by protection buyer) ≈ (CDS spread – CDS coupon) × duration price of CDS (per $100 notional) ≈ $100 – upfront premium (%) profit for protection buyer ≈ change in spread × duration × notional principal Study Session 14: Derivatives forward contract price (cost-of-carry model) FP = S0 × (1 + Rf) T or S0 = FP (1+Rf ) T no-arbitrage price of an equity forward contract with discrete dividends FP (on an equity security) = (S0 − PVD) × (1 + Rf)T FP (on an equity security) = [S0 × (1 + Rf) T] − FVD value of the long position in a forward contract on a dividend-paying stock Vt (long position) = [St − PVDt ] − [ FP (1+Rf ) (T−t) ] price of an equity index forward contract with continuous dividends FP (on an equity index) = S0 × e(Rf −δ )× T = (S0 × e−δ ×T ) × eRf ×T c c c c where: Rcf = continuously compounded risk-free rate δc = continuously compounded dividend yield forward price on a coupon-paying bond = (S0 − PVC) × (1 + Rf)T FP (on a fixed income security) or = S0 × (1 + Rf )T − FVC value prior to expiration of a forward contract on a coupon-paying bond Vt (long position) = [St − PVC t] − [ FP (1+Rf )(T−t) ] price of a bond futures contract FP = [(full price)(1 + Rf)T − AIT − FVC] quoted bond futures price based on conversion factor (CF) QFP = FP/CF = [(full price)(1 + Rf )T − AIT − FVC] ( CF ) price of a currency forward contract FT (currency forward contract) = S0 × (1+RPC )T (1+RBC ) T value of a currency forward contract Vt (long base currency) = Vt = [FPt −FP] (1+rPC) (T−t) price and value for a currency forward contract (continuous time) (Rc − Rc )×T FT = (currency forward contract) = S0 × e PC BC swap fixed rate SFR (periodic) = 1−final discount factor sum of discount factors swap fixed rate(annual) = SFR(periodic) × number of settlements periods per year value of plain vanilla interest rate swap (to payer) after inception value to the payer = ∑ DF × (SFRNew − SFROld ) × days 360 × notional principal probability of an up-move or down-move in a binomial stock tree πU = probability of an up move = πU = 1+Rf −D U−D πD = probability of a down move = (1 – πU) put-call parity S0 + P0 = C0 + PV(X) put-call parity when the stock pays dividends P0 + S0e–δT = C0 + e–rTX dynamic hedging number of short call options needed to delta hedge = number of shares hedged delta of call option number of long put options needed to delta hedge = − change in option value ∆C ≈ call delta ì S + ẵ gamma ì S2 P put delta ì S + ẵ gamma ì S2 option value using arbitrage-free pricing portfolio C0 = hS0 + (−hS − +C−) (1+Rf ) − P0 = hS0 + − (−hS +P ) (1+Rf ) = hS0 + = hS0 + (−hS+ +C+) (1+Rf ) (−hS+ +P+ ) BSM model C0 = S0e–δTN(d1) – e–rTXN(d2) P0 = e–rTXN(–d2) – S0e–δTN(–d1) (1+Rf ) number of shares delta of the put option All rights reserved under International and Pan-American Copyright Conventions By payment 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