put call parity 说明
期权期货与其他衍生产品第九版课后习题与答案Chapter
CHAPTER 31Interest Rate Derivatives: Models of the Short Rate Practice QuestionsProblem 31.1.What is the difference between an equilibrium model and a no-arbitrage model?Equilibrium models usually start with assumptions about economic variables and derive the behavior of interest rates. The initial term structure is an output from the model. In ano-arbitrage model the initial term structure is an input. The behavior of interest rates in ano-arbitrage model is designed to be consistent with the initial term structure.Problem 31.2.Suppose that the short rate is currently 4% and its standard deviation is 1% per annum. What happens to the standard deviation when the short rate increases to 8% in (a) Vasicek’s model;(b) Rendleman and Bartter’s mod el; and (c) the Cox, Ingersoll, and Ross model?In Vasicek’s model the standard deviation stays at 1%. In the Rendleman and Bartter model the standard deviation is proportional to the level of the short rate. When the short rate increases from 4% to 8% the standard deviation increases from 1% to 2%. In the Cox, Ingersoll, and Ross model the standard deviation of the short rate is proportional to the square root of the short rate. When the short rate increases from 4% to 8% the standard deviation increases from 1% to 1.414%.Problem 31.3.If a stock price were mean reverting or followed a path-dependent process there would be market inefficiency. Why is there not market inefficiency when the short-term interest rate does so?If the price of a traded security followed a mean-reverting or path-dependent process there would be market inefficiency. The short-term interest rate is not the price of a traded security. In other words we cannot trade something whose price is always the short-term interest rate. There is therefore no market inefficiency when the short-term interest rate follows amean-reverting or path-dependent process. We can trade bonds and other instruments whose prices do depend on the short rate. The prices of these instruments do not followmean-reverting or path-dependent processes.Problem 31.4.Explain the difference between a one-factor and a two-factor interest rate model.In a one-factor model there is one source of uncertainty driving all rates. This usually means that in any short period of time all rates move in the same direction (but not necessarily by the same amount). In a two-factor model, there are two sources of uncertainty driving all rates. The first source of uncertainty usually gives rise to a roughly parallel shift in rates. The second gives rise to a twist where long and short rates moves in opposite directions.Problem 31.5.Can the approach described in Section 31.4 for decomposing an option on a coupon-bearing bond into a portfolio of options on zero-coupon bonds be used in conjunction with a two-factor model? Explain your answer.No. The approach in Section 31.4 relies on the argument that, at any given time, all bond prices are moving in the same direction. This is not true when there is more than one factor.Problem 31.6.Suppose that 01a =. and 01b =. in both the Vasicek and the Cox, Ingersoll, Ross model. In both models, the initial short rate is 10% and the initial standard deviation of the short rate change in a short time t ∆is 0.zero-coupon bond that matures in year 10.In Vasicek’s model, 01a =., 01b =., and 002σ=. so that01101(10)(1)63212101B t t e -.⨯,+=-=..22(63212110)(010100002)00004632121(10)exp 00104A t t ⎡⎤.-.⨯.-..⨯.,+=-⎢⎥..⎣⎦071587=. The bond price is therefore 63212101071587038046e -.⨯..=.In the Cox, Ingersoll, and Ross model, 01a =., 01b =.and 00200632σ=.=.. Also013416γ==. Define10()(1)2092992a e γβγγ=+-+=.102(1)(10)607650e B t t γβ-,+==.225()2(10)069746ab a e A t t σγγβ/+⎛⎫,+==. ⎪⎝⎭The bond price is therefore 60765001069746037986e -.⨯..=.Problem 31.7.Suppose that 01a =., 008b =., and 0015σ=. in Vasicek’s model with the initial value of the short rate being 5%. Calculate the price of a one-year European call option on azero-coupon bond with a principal of $100 that matures in three years when the strike price is $87.Using the notation in the text, 3s =, 1T =, 100L =, 87K =, and2010015(1002588601P e σ-⨯..=-=.. From equation (31.6), (01)094988P ,=., (03)085092P ,=., and 114277h =. so thatequation (31.20) gives the call price as call price is 100085092(114277)87094988(111688)259N N ⨯.⨯.-⨯.⨯.=. or $2.59.Problem 31.8.Repeat Problem 31.7 valuing a European put option with a strike of $87. What is the put –call parity relationship between the prices of European call and put options? Show that the put and call option prices satisfy put –call parity in this case.As mentioned in the text, equation (31.20) for a call option is essentially the same as Bl ack’s model. By analogy with Black’s formulas corresponding expression for a put option is (0)()(0)()P KP T N h LP s N h σ,-+-,- In this case the put price is 87094988(111688)100085092(114277)014N N ⨯.⨯-.-⨯.⨯-.=.Since the underlying bond pays no coupon, put –call parity states that the put price plus the bond price should equal the call price plus the present value of the strike price. The bond price is 85.09 and the present value of the strike price is 870949888264⨯.=.. Put –call parity is therefore satisfied:82642598509014.+.=.+.Problem 31.9.Suppose that 005a =., 008b =., and 0015σ=. in Vasicek’s model with the initialshort-term interest rate being 6%. Calculate the price of a 2.1-year European call option on a bond that will mature in three years. Suppose that the bond pays a coupon of 5% semiannually. The principal of the bond is 100 and the strike price of the option is 99. The strike price is the cash price (not the quoted price) that will be paid for the bond.As explained in Section 31.4, the first stage is to calculate the value of r at time 2.1 years which is such that the value of the bond at that time is 99. Denoting this value of r by r *, we must solve(2125)(2130)25(2125)1025(2130)99B r B r A e A e **-.,.-.,...,.+..,.=where the A and B functions are given by equations (31.7) and (31.8). In this case A (2.1, 2.5)=0.999685, A (2.1,3.0)=0.998432, B(2.1,2.5)=0.396027, and B (2.1, 3.0)= 0.88005. and Solver shows that 065989.0*=r . Since434745.2)5.2,1.2(5.2*)5.2,1.2(=⨯-r B e Aand56535.96)0.3,1.2(5.102*)0.3,1.2(=⨯-r B e Athe call option on the coupon-bearing bond can be decomposed into a call option with a strike price of 2.434745 on a bond that pays off 2.5 at time 2.5 years and a call option with a strike price of 96.56535 on a bond that pays off 102.5 at time 3.0 years. The options are valued using equation (31.20).For the first option L =2.5, K = 2.434745, T = 2.1, and s =2.5. Also, A (0,T )=0.991836, B (0,T ) = 1.99351, P (0,T )=0.880022 while A (0,s )=0.988604, B (0,s )=2.350062, andP (0,s )=0.858589. Furthermore σP = 0.008176 and h = 0.223351. so that the option price is 0.009084.For the second option L =102.5, K = 96.56535, T = 2.1, and s =3.0. Also, A (0,T )=0.991836, B (0,T ) = 1.99351, P (0,T )=0.880022 while A (0,s )=0.983904, B (0,s )=2.78584, andP (0,s )=0.832454. Furthermore σP = 0.018168 and h = 0.233343. so that the option price is0.806105.The total value of the option is therefore 0.0090084+0.806105=0.815189.Problem 31.10.Use the answer to Problem 31.9 and put –call parity arguments to calculate the price of a put option that has the same terms as the call option in Problem 31.9.Put-call parity shows that: 0()c I PV K p B ++=+ or 0()()p c PV K B I =+--where c is the call price, K is the strike price, I is the present value of the coupons, and 0B is the bond price. In this case 08152c =., ()99(021)871222PV K P =⨯,.=., 025(025)1025(03)874730B I P P -=.⨯,.+.⨯,=. so that the put price is0815287122287473004644.+.-.=.Problem 31.11.In the Hull –White model, 008a =. and 001σ=.. Calculate the price of a one-year European call option on a zero-coupon bond that will mature in five years when the term structure is flat at 10%, the principal of the bond is $100, and the strike price is $68.Using the notation in the text 011(0)09048P T e -.⨯,==. and 015(0)06065P s e -.⨯,==.. Also4008001(100329008P e σ-⨯..=-=.. and 04192h =-. so that the call price is10006065()6809048()0439P N h N h σ⨯.-⨯.-=.Problem 31.12.Suppose that 005a =. and 0015σ=. in the Hull –White model with the initial term structure being flat at 6% with semiannual compounding. Calculate the price of a 2.1-year European call option on a bond that will mature in three years. Suppose that the bond pays a coupon of 5% per annum semiannually. The principal of the bond is 100 and the strike price of the option is 99. The strike price is the cash price (not the quoted price) that will be paid for the bond.This problem is similar to Problem 31.9. The difference is that the Hull –White model, which fits an initial term structure, is used instead of Vasicek’s model where the initial term structure is determined by the model.The yield curve is flat with a continuously compounded rate of 5.9118%.As explained in Section 31.4, the first stage is to calculate the value of r at time 2.1 years which is such that the value of the bond at that time is 99. Denoting this value of r by r *, we must solve(2125)(2130)25(2125)1025(2130)99B r B r A e A e **-.,.-.,...,.+..,.=where the A and B functions are given by equations (31.16) and (31.17). In this case A (2.1, 2.5)=0.999732, A (2.1,3.0)=0.998656, B(2.1,2.5)=0.396027, and B (2.1, 3.0)= 0.88005. and Solver shows that 066244.0*=r . Since434614.2)5.2,1.2(5.2*)5.2,1.2(=⨯-r B e Aand56539.96)0.3,1.2(5.102*)0.3,1.2(=⨯-r B e Athe call option on the coupon-bearing bond can be decomposed into a call option with a strike price of 2.434614 on a bond that pays off 2.5 at time 2.5 years and a call option with a strike price of 96.56539 on a bond that pays off 102.5 at time 3.0 years. The options are valued using equation (31.20).For the first option L =2.5, K = 2.434614, T = 2.1, and s =2.5. Also, P (0,T )=exp(-0.059118×2.1)=0.88325 and P (0,s )= exp(-0.059118×2.5)=0.862609. Furthermore σP = 0.008176 and h = 0.353374. so that the option price is 0.010523. For the second option L =102.5, K = 96.56539, T = 2.1, and s =3.0. Also, P (0,T )=exp(-0.059118×2.1)=0.88325 and P (0,s )= exp(-0.059118×3.0)=0.837484. Furthermore σP = 0.018168 and h = 0.363366. so that the option price is 0.934074.The total value of the option is therefore 0.010523+0.934074=0.944596.Problem 31.13.Observations spaced at intervals ∆t are taken on the short rate. The ith observation is r i (1 ≤ i ≤ m). Show that the maximum likelihood estimates of a, b, and σ in Vasicek’s model are given by maximizing[]∑=--⎪⎪⎭⎫ ⎝⎛∆σ∆----∆σ-m i i i i t t r b a r r t 122112)()ln(What is the corresponding result for the CIR model?The change r i –r i -1 is normally distributed with mean a (b − r i -1) and variance σ2∆t. The probability density of the observation is⎪⎭⎫⎝⎛∆σ---∆πσ--t r b a r r ti i i 21122)(exp 21We wish to maximize∏=--⎪⎭⎫⎝⎛∆σ---∆πσmi i i i t r b a r r t121122)(exp 21Taking logarithms, this is the same as maximizing[]∑=--⎪⎪⎭⎫ ⎝⎛∆σ∆----∆σ-m i i i i t t r b a r r t 122112)()ln(In the case of the CIR model, the change r i –r i -1 is normally distributed with mean a (b − r i -1) and variance t r i ∆σ-12and the maximum likelihood function becomes[]∑=----⎪⎪⎭⎫ ⎝⎛∆σ∆----∆σ-mi i i i i i t r t r b a r r t r 11221112)()ln(Problem 31.14.Suppose 005a =., 0015σ=., and the term structure is flat at 10%. Construct a trinomial tree for the Hull –White model where there are two time steps, each one year in length.The time step, t ∆, is 1 so that 0002598r ∆=.=.. Also max 4j = showing that the branching method should change four steps from the center of the tree. With only three steps we never reach the point where the branching changes. The tree is shown in Figure S31.1.Figure S31.1: Tree for Problem 31.14Problem 31.15.Calculate the price of a two-year zero-coupon bond from the tree in Figure 31.6.A two-year zero-coupon bond pays off $100 at the ends of the final branches. At nodeB it is worth 01211008869e -.⨯=.. At nodeC it is worth 010********e -.⨯=.. At nodeD it is worth 00811009231e -.⨯=.. It follows that at node A the bond is worth 011(88690259048059231025)8188e -.⨯.⨯.+.⨯.+.⨯.=. or $81.88Problem 31.16.Calculate the price of a two-year zero-coupon bond from the tree in Figure 31.9 and verify that it agrees with the initial term structure.A two-year zero-coupon bond pays off $100 at time two years. At nodeB it is worth 0069311009330e -.⨯=.. At nodeC it is worth 0052011009493e -.⨯=.. At nodeD it is worth 0034711009659e -.⨯=.. It follows that at node A the bond is worth 003821(933001679493066696590167)9137e -.⨯.⨯.+.⨯.+.⨯.=.or $91.37. Because 00451229137100e -.⨯.=, the price of the two-year bond agrees with the initial term structure.Problem 31.17.Calculate the price of an 18-month zero-coupon bond from the tree in Figure 31.10 and verify that it agrees with the initial term structure.An 18-month zero-coupon bond pays off $100 at the final nodes of the tree. At node E it is worth 0088051009570e -.⨯.=.. At node F it is worth 00648051009681e -.⨯.=.. At node G it is worth 00477051009764e -.⨯.=.. At node H it is worth 00351051009826e -.⨯.=.. At node I it is worth 00259051009871e .⨯.=.. At node B it is worth 0056405(011895700654968102289764)9417e -.⨯..⨯.+.⨯.+.⨯.=.Similarly at nodes C and D it is worth 95.60 and 96.68. The value at node A is therefore 0034305(016794170666956001679668)9392e -.⨯..⨯.+.⨯.+.⨯.=.The 18-month zero rate is 0181500800500418e -.⨯..-.=.. This gives the price of the 18-month zero-coupon bond as 00418151009392e -.⨯.=. showing that the tree agrees with the initial term structure.Problem 31.18.What does the calibration of a one-factor term structure model involve?The calibration of a one-factor interest rate model involves determining its volatility parameters so that it matches the market prices of actively traded interest rate options as closely as possible.Problem 31.19.Use the DerivaGem software to value 14⨯, 23⨯, 32⨯, and 41⨯ European swap options to receive fixed and pay floating. Assume that the one, two, three, four, and five year interest rates are 6%, 5.5%, 6%, 6.5%, and 7%, respectively. The payment frequency on the swap is semiannual and the fixed rate is 6% per annum with semiannual compounding. Use the Hull –White model with 3a %= and 1%σ=. Calcula te the volatility implied by Black’s model for each option.The option prices are 0.1302, 0.0814, 0.0580, and 0.0274. The implied Black volatilities are 14.28%, 13.64%, 13.24%, and 12.81%Problem 31.20.Prove equations (31.25), (31.26), and (31.27).From equation (31.15) ()()()()r t B t t t P t t t A t t t e -,+∆,+∆=,+∆ Also ()()R t t P t t t e -∆,+∆= so that()()()()R t t r t B t t t e A t t t e -∆-,+∆=,+∆or()()()()()()()()R t B t T t B t t t r t B t T B t T B t t t e eA t t t -,∆/,+∆-,,/,+∆=,+∆ Hence equation (31.25) is true with()ˆ()(B t T t Bt T B t t t ,∆,=,+∆ and()()()ˆ()()B t T B t t t A t T At T A t t t ,/,+∆,,=,+∆ or()ˆln ()ln ()ln ()()B t T At T A t T A t t t B t t t ,,=,-,+∆,+∆Problem 31.21.(a) What is the second partial derivative of P(t,T) with respect to r in the Vasicek and CIR models?(b) In Section 31.2, ˆDis presented as an alternative to the standard duration measure D. What is a similar alternative ˆCto the convexity measure in Section 4.9? (c) What is ˆCfor P(t,T)? How would you calculate ˆC for a coupon-bearing bond? (d) Give a Taylor Series expansion for ∆P(t,T) in terms of ∆r and (∆r)2 for Vasicek and CIR.(a) ),(),(),(),(),(2),(222T t P T t B e T t A T t B rT t P r T t B ==∂∂- (b) A corresponding definition for Cˆis 221r QQ ∂∂(c) When Q =P (t ,T ), 2),(ˆT t B C=For a coupon-bearing bond C ˆis a weighted average of the Cˆ’s for the constituent zero -coupon bonds where weights are proportional to bond prices. (d)+∆+∆-=+∆∂∂+∆∂∂=∆22222),(),(21),(),(),(21),(),(r T t P T t B r T t P T t B r rT t P r r T t P T t PProblem 31.22.Suppose that the short rate r is 4% and its real-world process is0.1[0.05]0.01dr r dt dz =-+while the risk-neutral process is0.1[0.11]0.01dr r dt dz =-+(a) What is the market price of interest rate risk?(b) What is the expected return and volatility for a 5-year zero-coupon bond in the risk-neutral world?(c) What is the expected return and volatility for a 5-year zero-coupon bond in the real world?(a) The risk neutral process for r has a drift rate which is 0.006/r higher than the real world process. The volatility is 0.01/r . This means that the market price of interest rate risk is −0.006/0.01 or −0.6.(b) The expected return on the bond in the risk-neutral world is the risk free rate of 4%. The volatility is 0.01×B (0,5) where935.31.01)5,0(51.0=-=⨯-e Bi.e., the volatility is 3.935%.(c) The process followed by the bond price in a risk-neutral world isPdz Pdt dP 03935.004.0-=Note that the coefficient of dz is negative because bond prices are negatively correlated with interest rates. When we move to the real world the return increases by the product of the market price of dz risk and −0.03935. The bond price process becomes:Pdz Pdt dP 03935.0)]03935.06.0(04.0[--⨯-+=orPdz Pdt dP 03935.006361.0-=The expected return on the bond increases from 4% to 6.361% as we move from the risk-neutral world to the real world.Further QuestionsProblem 31.23.Construct a trinomial tree for the Ho and Lee model where 002σ=.. Suppose that the initial zero-coupon interest rate for a maturities of 0.5, 1.0, and 1.5 years are 7.5%, 8%, and 8.5%. Use two time steps, each six months long. Calculate the value of a zero-coupon bond with a face value of $100 and a remaining life of six months at the ends of the final nodes of the tree. Use the tree to value a one-year European put option with a strike price of 95 on the bond. Compare the price given by your tree with the analytic price given by DerivaGem.The tree is shown in Figure S31.2. The probability on each upper branch is 1/6; the probability on each middle branch is 2/3; the probability on each lower branch is 1/6. The six month bond prices nodes E, F, G, H, I are 0144205100e -.⨯., 0119705100e -.⨯., 0095205100e -.⨯., 0070705100e -.⨯., and 0046205100e -.⨯., respectively. These are 93.04, 94.19, 95.35, 96.53, and 97.72. The payoffs from the option at nodes E, F, G, H, and I are therefore 1.96, 0.81, 0, 0, and 0. The value at node B is 0109505(0166719606667081)08192e -.⨯..⨯.+.⨯.=.. The value at node C is00851050166708101292e -.⨯..⨯.⨯=.. The value at node D is zero. The value at node A is 0075005(01667081920666701292)0215e -.⨯..⨯.+.⨯.=. The answer given by DerivaGem using the analytic approach is 0.209.Figure S31.2: Tree for Problem 31.23Problem 31.24.A trader wishes to compute the price of a one-year American call option on a five-year bond with a face value of 100. The bond pays a coupon of 6% semiannually and the (quoted) strike price of the option is $100. The continuously compounded zero rates for maturities of sixmonths, one year, two years, three years, four years, and five years are 4.5%, 5%, 5.5%, 5.8%, 6.1%, and 6.3%. The best fit reversion rate for either the normal or the lognormal model has been estimated as 5%.A one year European call option with a (quoted) strike price of 100 on the bond is actively traded. Its market price is $0.50. The trader decides to use this option for calibration. Use the DerivaGem software with ten time steps to answer the following questions.(a) Assuming a normal model, imply the σ parameter from the price of the European option.(b) Use the σ parameter to calculate the price of the option when it is American. (c) Repeat (a) and (b) for the lognormal model. Show that the model used does notsignificantly affect the price obtained providing it is calibrated to the known European price.(d) Display the tree for the normal model and calculate the probability of a negative interest rate occurring.(e) Display the tree for the lognormal model and verify that the option price is correctly calculated at the node where, with the notation of Section 31.7, 9i = and 1j =-.Using 10 time steps:(a) The implied value of σ is 1.12%.(b) The value of the American option is 0.595(c) The implied value of σ is 18.45% and the value of the American option is 0.595. Thetwo models give the same answer providing they are both calibrated to the same European price.(d) We get a negative interest rate if there are 10 down moves. The probability of this is0.16667×0.16418×0.16172×0.15928×0.15687×0.15448×0.15212×0.14978×0.14747 ×0.14518=8.3×10-9 (e) The calculation is0052880101641791707502789e -.⨯..⨯.⨯=.Problem 31.25.Use the DerivaGem software to value 14⨯, 23⨯, 32⨯, and 41⨯ European swap options to receive floating and pay fixed. Assume that the one, two, three, four, and five year interest rates are 3%, 3.5%, 3.8%, 4.0%, and 4.1%, respectively. The payment frequency on the swap is semiannual and the fixed rate is 4% per annum with semiannual compounding. Use thelognormal model with 5a %=, 15%σ=, and 50 time steps. Calculate the volatility implied by Black’s model for each option.The values of the four European swap options are 1.72, 1.73, 1.30, and 0.65, respectively. The implied Black volatilities are 13.37%, 13.41%, 13.43%, and 13.42%, respectively.Problem 31.26.Verify that the DerivaGem software gives Figure 31.11 for the example considered. Use the software to calculate the price of the American bond option for the lognormal and normalmodels when the strike price is 95, 100, and 105. In the case of the normal model, assume that a = 5% and σ = 1%. Discuss the results in the context of the heavy-tails arguments of Chapter 20.With 100 time steps the lognormal model gives prices of 5.585, 2.443, and 0.703 for strike prices of 95, 100, and 105. With 100 time steps the normal model gives prices of 5.508, 2.522, and 0.895 for the three strike prices respectively. The normal model gives a heavier left tail and a less heavy right tail than the lognormal model for interest rates. This translates into a less heavy left tail and a heavier right tail for bond prices. The arguments in Chapter 20 show that we expect the normal model to give higher option prices for high strike prices and lower option prices for low strike. This is indeed what we find.Problem 31.27.Modify Sample Application G in the DerivaGem Application Builder software to test theconvergence of the price of the trinomial tree when it is used to price a two-year call option on a five-year bond with a face value of 100. Suppose that the strike price (quoted) is 100, the coupon rate is 7% with coupons being paid twice a year. Assume that the zero curve is as in Table 31.2. Compare results for the following cases:(a) Option is European; normal model with 001σ=. and 005a =..(b) Option is European; lognormal model with 015σ=. and 005a =..(c) Option is American; normal model with 001σ=. and 005a =..(d) Option is American; lognormal model with 015σ=. and 005a =..The results are shown in Figure S31.3.Figure S31.3: Tree for Problem 31.27Problem 31.28.Suppose that the (CIR) process for short-rate movements in the (traditional) risk-neutral world is()dr a b r dt =-+and the market price of interest rate risk is λ(a) What is the real world process for r?(b) What is the expected return and volatility for a 10-year bond in the risk-neutral world? (c) What is the expected return and volatility for a 10-year bond in the real world?(a) The volatility of r (i.e., the coefficient of rdz in the process for r ) is real world process for r is therefore increased by r r λσ⨯ so that the process isdz r dt r r b a dr σ+λσ+-=])([(b) The expected return is r and the volatility is (,B t T σin the risk-neutral world.(c) The expected return is r T t B r ),(λσ+ and the volatility is as in (b) in the real world.。
期权基础知识
5 履约日 是指期权权利可以执行的日期。可以分为欧式期权与 美式期权。 6 到期日 是期权合约双方当事人预先设立的期权买方可以行使 期权的最终有效期限。 7 停板额 又称每日交易限价，即涨跌停板额，是期权合约每日 价格的波动幅度高于或低于上一交易日结算价的幅度。
8 合约月份 即买卖双方交付、接受实货以履行职权合约的 月份。 9 交易时间 即各相关期权市场规定的每个交易日及其进场 交易的具体时间。
一份美国芝加哥期货交易所的玉米期 权合约格式。
交易单位 最小变动价位 每日价格最大波 动限制 执行价格 合约月份 交易时间 最后交易日 合约到期日
一个CBOT期货合约交易单位（5000蒲式 耳）
每蒲式耳1/8美分 每蒲式耳不高于或低于上一交 易日结算权利金各10美分 每蒲式耳10美分的整数倍 1、2、3、5、7、9 同期货
一、期权市场概述
(一) 金融期权合约的定义与种类
(二) 金融期权的交易
(三) 期权交易与期货交易的区别
(一) 金融期权合约的定义与种类
金融期权 (Option)，是指它的持有者有权在规 定期限内按双方约定的价格购买或出售一定数 量某种金融资产(称为标的金融资产Underlying Financial Assets) 的合约。
(六) 障碍期权
障碍期权 (Barrier Option) 是指其收益依赖于标 的资产价格在一段特定时期内是否达到了一个 特定水平的期权。常见的障碍期权有两种，一 是封顶期权 (Caps)，一是失效期权 (Knockout Option)。
(七) 两值期权
两值期权 (Binary Option) 是具有不连续收益的期 权，当到期日标的资产价格低于协议价格时， 该期权作废，而当到期日标的资产价格高于协 议价格时，期权持有者将得到一个固定的金额。
看涨看跌平价定理
看涨看跌平价定理（一）欧式看涨期权与看跌期权之间的平价关系1．无收益资产的欧式期权在标的资产没有收益的情况下，为了推导c和p之间的关系，我们考虑如下两个组合：组合A：一份欧式看涨期权加上金额为Xe-r(T-t) 的现金组合B：一份有效期和协议价格与看涨期权相同的欧式看跌期权加上一单位标的资产在期权到期时，两个组合的价值均为max(ST,X)。
由于欧式期权不能提前执行，因此两组合在时刻t必须具有相等的价值，即：c+Xe-r(T-t)=p+S（1.1）这就是无收益资产欧式看涨期权与看跌期权之间的平价关系（Parity）。
它表明欧式看涨期权的价值可根据相同协议价格和到期日的欧式看跌期权的价值推导出来，反之亦然。
如果式（1.1）不成立，则存在无风险套利机会。
套利活动将最终促使式（1.1）成立。
2．有收益资产欧式期权在标的资产有收益的情况下，我们只要把前面的组合A中的现金改为D+Xe-r(T-t) ，我们就可推导有收益资产欧式看涨期权和看跌期权的平价关系：c+D+Xe-r(T-t)=p+S（1.2）（二）美式看涨期权和看跌期权之间的关系1．无收益资产美式期权由于P>p,从式（1.1）中我们可得：P>c+Xe-r(T-t)-S对于无收益资产看涨期权来说，由于c=C，因此：P>C+Xe-r(T-t)-SC-P<S-Xe-r(T-t)（1.3）为了推导出C和P的更严密的关系，我们考虑以下两个组合：组合A：一份欧式看涨期权加上金额为X的现金组合B：一份美式看跌期权加上一单位标的资产如果美式期权没有提前执行，则在T时刻组合B的价值为max(ST,X)，而此时组合A 的价值为max(ST,X)+ Xe-r(T-t)-X。
因此组合A的价值大于组合B。
如果美式期权在T-t 时刻提前执行，则在T-t 时刻，组合B的价值为X，而此时组合A的价值大于等于Xe-r(T-t)。
因此组合A的价值也大于组合B。
北美精算师 fm教材 《衍生品市场》课后答案m48-ch03 shrunk v02
Chapter3Insurance,Collars,and Other StrategiesQuestion3.1.This question is a direct application of the Put-Call-Parity(equation(3.1))of the textbook.Mim-icking Table3.1.,we have:S&R Index S&R Put Loan Payoff−(Cost+Interest)Profit900.00100.00−1000.000.00−95.68−95.68950.0050.00−1000.000.00−95.68−95.681000.000.00−1000.000.00−95.68−95.681050.000.00−1000.0050.00−95.68−45.681100.000.00−1000.00100.00−95.68 4.321150.000.00−1000.00150.00−95.6854.321200.000.00−1000.00200.00−95.68104.32The payoff diagram looks as follows:We can see from the table and from the payoff diagram that we have in fact reproduced a call with the instruments given in the exercise.The profit diagram on the next page confirms this hypothesis.Part1Insurance,Hedging,and Simple StrategiesQuestion3.2.This question constructs a position that is the opposite to the position of Table3.1.Therefore,we should get the exact opposite results from Table3.1.and the associatedfigures.Mimicking Table 3.1.,we indeed have:S&R Index S&R Put Payoff−(Cost+Interest)Profit−900.00−100.00−1000.001095.6895.68−950.00−50.00−1000.001095.6895.68−1000.000.00−1000.001095.6895.68−1050.000.00−1050.001095.6845.68−1100.000.00−1100.001095.68−4.32−1150.000.00−1150.001095.68−54.32−1200.000.00−1200.001095.68−104.32On the next page,we see the corresponding payoff and profit diagrams.Please note that they match the combined payoff and profit diagrams of Figure3.5.Only the axes have different scales.Chapter3Insurance,Collars,and Other Strategies Payoff-diagram:Profit diagram:Part1Insurance,Hedging,and Simple StrategiesQuestion3.3.In order to be able to draw profit diagrams,we need tofind the future value of the put premium,the call premium and the investment in zero-coupon bonds.We have for:the put premium:$51.777×(1+0.02)=$52.81,the call premium:$120.405×(1+0.02)=$122.81andthe zero-coupon bond:$931.37×(1+0.02)=$950.00Now,we can construct the payoff and profit diagrams of the aggregate position:Payoff diagram:From thisfigure,we can already see that the combination of a long put and the long index looks exactly like a certain payoff of$950,plus a call with a strike price of950.But this is the alternative given to us in the question.We have thus confirmed the equivalence of the two combined positions for the payoff diagrams.The profit diagrams on the next page confirm the equivalence of the two positions(which is again an application of the Put-Call-Parity).Chapter 3Insurance,Collars,and Other StrategiesProfit Diagram for a long 950-strike put and a long index combined:Question 3.4.This question is another application of Put-Call-Parity.Initially,we have the following cost to enter into the combined position:We receive $1,000from the short sale of the index,and we have to pay the call premium.Therefore,the future value of our cost is: $120.405−$1,000 ×(1+0.02)=−$897.19.Note that a negative cost means that we initially have an inflow of money.Now,we can directly proceed to draw the payoff diagram:Part 1Insurance,Hedging,and Simple StrategiesWe can clearly see from the figure that the payoff graph of the short index and the long call looks like a fixed obligation of $950,which is alleviated by a long put position with a strike price of 950.The following profit diagram,including the cost for the combined position,confirms this:Question 3.5.This question is another application of Put-Call-Parity.Initially,we have the following cost to enter into the combined position:We receive $1,000from the short sale of the index,and we have to pay the call premium.Therefore,the future value of our cost is: $71.802−$1,000 ×(1+0.02)=−$946.76.Note that a negative cost means that we initially have an inflow of money.Chapter3Insurance,Collars,and Other Strategies Now,we can directly proceed to draw the payoff diagram:In order to be able to compare this position to the other suggested position of the exercise,we need tofind the future value of the borrowed$1,029.41.We have:$1,029.41×(1+0.02)=$1,050. We can now see from thefigure that the payoff graph of the short index and the long call looks like afixed obligation of$1,050,which is exactly the future value of the borrowed amount,and a long put position with a strike price of1,050.The following profit diagram,including the cost for the combined position we calculated above,confirms this.The profit diagram is the same as the profit diagram for a loan and a long1,050-strike put with an initial premium of$101.214.Part 1Insurance,Hedging,and Simple StrategiesProfit Diagram of going short the index and buying a 1,050-strike call:Question 3.6.We now move from a graphical representation and verification of the Put-Call-Parity to a mathe-matical representation.Let us first consider the payoff of (a).If we buy the index (let us name it S),we receive at the time of expiration T of the options simply S T .The payoffs of part (b)are a little bit more complicated.If we deal with options and the maximum function,it is convenient to split the future values of the index into two regions:one where S T <K and another one where S T ≥K .We then look at each region separately,and hope to be able to draw a conclusion when we look at the aggregate position.We have for the payoffs in (b):InstrumentS T <K =950S T ≥K =950Get repayment of loan $931.37×1.02=$950$931.37×1.02=$950Long Call Option max (S T−950,0)=0S T −950Short Put Option −max $950−S T ,0 0=S T −$950TotalS TS TChapter 3Insurance,Collars,and Other StrategiesWe now see that the total aggregate position only gives us S T ,no matter what the final index value is—but this is the same payoff as in part (a).Our proof for the payoff equivalence is complete.Now let us turn to the profits.If we buy the index today,we need to finance it.Therefore,we borrow $1,000,and have to repay $1,020after one year.The profit for part (a)is thus:S T −$1,020.The profits of the aggregate position in part (b)are the payoffs,less the future value of the call premium plus the future value of the put premium (because we have sold the put),and less the future value of the loan we gave initially.Note that we already know that a risk-less bond is canceling out of the profit calculations.We can again tabulate:InstrumentS T <KS T ≥KFuture value of given loan −$950−$950Long Call Optionmax (S T −950,0)=0S T −950Future value call premium −$120.405×1.02=−$122.81−$120.405×1.02=−$122.81Short Put Option −max$950−S T ,0 0=S T −$950Future value put premium $51.777×1.02=$52.81$51.777×1.02=$52.81TotalS T −1020S T −1020Indeed,we see that the profits for part (a)and part (b)are identical as well.Question 3.7.Let us first consider the payoff of (a).If we short the index (let us name it S),we have to pay at the time of expiration T of the options:−S T .The payoffs of part (b)are more complicated.Let us look again at each region separately,and hope to be able to draw a conclusion when we look at the aggregate position.We have for the payoffs in (b):InstrumentS T <K S T ≥KMake repayment of loan −$1029.41×1.02=−$1050−$1029.41×1.02=−$1050Short Call Option −max (S T −1050,0)=0−max (S T −1050,0)=1050−S TLong Put Option max$1050−S T ,00=$1050−S TTotal−S T−S TPart 1Insurance,Hedging,and Simple StrategiesWe see that the total aggregate position gives us −S T ,no matter what the final index value is—but this is the same payoff as part (a).Our proof for the payoff equivalence is complete.Now let us turn to the profits.If we sell the index today,we receive money that we can lend out.Therefore,we can lend $1,000,and be repaid $1,020after one year.The profit for part (a)is thus:$1,020−S T .The profits of the aggregate position in part (b)are the payoffs,less the future value of the put premium plus the future value of the call premium (because we sold the call),and less the future value of the loan we gave initially.Note that we know already that a risk-less bond is canceling out of the profit calculations.We can again tabulate:InstrumentS T <KS T ≥KMake repayment of loan −$1029.41×1.02=−$1050−$1029.41×1.02=−$1050Future value of borrowed $1050$1050moneyShort Call Option −max (S T −1050,0)=0−max (S T −1050,0)=1050−S TFuture value of premium $71.802×1.02=$73.24$71.802×1.02=$73.24Long Put Option max $1050−S T ,0 =0$1050−S TFuture value of premium −$101.214×1.02=−$103.24−$101.214×1.02=−$103.24Total$1,020−S T $1,020−S TIndeed,we see that the profits for part (a)and part (b)are identical as well.Question 3.8.This question is a direct application of the Put-Call-Parity.We will use equation (3.1)in the follow-ing,and input the given variables:Call (K,t)−P ut (K,t)=P V F 0,t −K⇔Call (K,t)−P ut (K,t)−P V F 0,t=−P V (K)⇔Call (K,t)−P ut (K,t)−S 0=−P V (K)⇔$109.20−$60.18−$1,000=−K 1.02⇔K =$970.00Question 3.9.The strategy of buying a call (or put)and selling a call (or put)at a higher strike is called call (put)bull spread.In order to draw the profit diagrams,we need to find the future value of the cost of entering in the bull spread positions.We have:Cost of call bull spread:$120.405−$93.809×1.02=$27.13Cost of put bull spread:$51.777−$74.201×1.02=−$22.87The payoff diagram shows that the payoffs to the put bull spread are uniformly less than the payoffs to the call bull spread.There is a difference,because the put bull spread has a negative initial cost, i.e.,we are receiving money if we enter into it.The difference is exactly$50,the value of the difference between the two strike prices.It is equivalent as well to the value of the difference of the future values of the initial premia.Yet,the higher initial cost for the call bull spread is exactly offset by the higher payoff so that the profits of both strategies are the same.It is easy to show this with equation(3.1),the put-call-parity. Payoff-Diagram:Profit diagram:Question 3.10.The strategy of selling a call (or put)and buying a call (or put)at a higher strike is called call (put)bear spread.In order to draw the profit diagrams,we need to find the future value of the cost of entering in the bull spread positions.We have:Cost of call bear spread: $71.802−$120.405 ×1.02=−$49.575Cost of put bear spread: $101.214−$51.777 ×1.02=$50.426The payoff diagram shows that the payoff to the call bear spread is uniformly less than the payoffs to the put bear spread.The difference is exactly $100,equal to the difference in strikes and as well equal to the difference in the future value of the costs of the spreads.There is a difference,because the call bear spread has a negative initial cost,i.e.,we are receiving money if we enter into it.The higher initial cost for the put bear spread is exactly offset by the higher payoff so that the profits of both strategies turn out to be the same.It is easy to show this with equation (3.1),the put-call-parity.Payoff-Diagram:Profit Diagram:Question 3.11.In order to be able to draw the profit diagram,we need to find the future value of the costs of establishing the suggested position.We need to finance the index purchase,buy the 950-strike put and we receive the premium of the sold call.Therefore,the future value of our cost is: $1,000−$71.802+$51.777 ×1.02=$999.57.Now we can draw the profit diagram:The net option premium cost today is:−$71.802+$51.777=−$20.025.We receive about $20if we enter into this collar.If we want to construct a zero-cost collar and keep the 950-strike put,we would need to increase the strike price of the call.By increasing the strike price of the call,the buyer of the call must wait for larger increases in the underlying index before the option pays off.This makes the call option less attractive,and the buyer of the option is only willing to pay a smaller premium.We receive less money,thus pushing the net option premium towards zero.Question 3.12.Our initial cash required to put on the collar,i.e.the net option premium,is as follows:−$51.873+$51.777=−$0.096.Therefore,we receive only 10cents if we enter into this collar.The position is very close to a zero-cost collar.The profit diagram looks as follows:If we compare this profit diagram with the profit diagram of the previous exercise(3.11.),we see that we traded in the additional call premium(that limited our losses after index decreases)against more participation on the upside.Please note that both maximum loss and gain are higher than in question3.11.Question3.13.The followingfigure depicts the requested profit diagrams.We can see that the aggregation of the bought and sold straddle resembles a bear spread.It is bearish,because we sold the straddle with the smaller strike price.a)b)c)Question3.14.a)This question deals with the option trading strategy known as Box spread.We saw earlier that if we deal with options and the maximum function,it is convenient to split the future values of the index into different regions.Let us name thefinal value of the S&R index S T.We have two strike prices,therefore we will use three regions:One in which S T<950,one in which950≤S T<1,000 and another one in which S T≥1,000.We then look at each region separately,and hope to be ableto see that indeed when we aggregate,there is no S&R risk when we look at the aggregate position.Instrument S T<950950≤S T<1,000S T≥1,000long950call0S T−$950S T−$950short1000call00$1,000−S Tshort950put S T−$95000long1000put$1,000−S T$1,000−S T0Total$50$50$50We see that there is no occurrence of thefinal index value in the row labeled total.The option position does not contain S&R price risk.b)The initial cost is the sum of the long option premia less the premia we receive for the sold options.We have:Cost $120.405−$93.809−$51.77+$74.201=$49.027c)The payoff of the position after 6months is $50,as we can see from the above table.d)The implicit interest rate of the cash flows is:$50.00÷$49.027=1.019∼=1.02.The im-plicit interest rate is indeed 2percent.Question 3.15.a)Profit diagram of the 1:2950-,1050-strike ratio call spread (the future value of the initial cost of which is calculated as: $120.405−2×$71.802 ×1.02=−$23.66):b)Profit diagram of the 2:3950-,1050-strike ratio call spread (the future value of the initial cost of which is calculated as: 2×$120.405−3×$71.802 ×1.02=$25.91.c)We saw that in part a),we were receiving money from our position,and in part b),we had to pay a net option premium to establish the position.This suggests that the true ratio n/m lies between 1:2and2:3.Indeed,we can calculate the ratio n/m as:n×$120.405−m×$71.802=0⇔n×$120.405=m×$71.802⇔n/m=$71.802/$120.405⇔n/m=0.596which is approximately3:5.Question3.16.A bull spread or a bear spread can never have an initial premium of zero,because we are buying the same number of calls(or puts)that we are selling and the two legs of the bull and bear spreads have different strikes.A zero initial premium would mean that two calls(or puts)with different strikes have the same price—and we know by now that two instruments that have different payoff structures and the same underlying risk cannot have the same price without creating an arbitrage opportunity.A symmetric butterfly spread cannot have a premium of zero because it would violate the convexitycondition of options.Question3.17.From the textbook we learn how to calculate the right ratioλ.It is equal to:λ=K3−K2K3−K1=1050−10201050−950=0.3In order to construct the asymmetric butterfly,for every1020-strike call we write,we buyλ950-strike calls and1−λ1050-strike calls.Since we can only buy whole units of calls,we will in this example buy three950-strike and seven1050-strike calls,and sell ten1020-strike calls.The profit diagram looks as follows:Question3.18.The following threefigures show the individual legs of each of the three suggested strategies.The last subplot shows the aggregate position.It is evident from thefigures that you can indeed use all the suggested strategies to construct the same butterfly spread.Another method to show the claim of3.18.mathematically would be to establish the equivalence by using the Put-Call-Parity on b) and c)and showing that you can write it in terms of the instruments of a).profit diagram part a)profit diagram part b)Chapter3Insurance,Collars,and Other Strategies profit diagram part c)Question3.19.a)We know from the Put-Call-Parity that if we buy a call and sell a put that are at the money (i.e.,S(0)=K),then the call option is slightly more expensive than the put option,the difference being the value of the stock minus the present value of the strike.Therefore,we can tell that the strike price must be a bit higher than the current stock price,and more precisely,it should be equal to the forward price.b)We sold a collar with no difference in strike prices.The profit diagram will be a straight line,which means that we effectively created a long forward contract.c)Remember that you are buying at the ask and selling at the bid,and that the bid price is always smaller than the ask.Suppose we had established a zero-cost synthetic at the forward price, and now we introduce the bid-ask spread.This means that we have to pay a little more for the call, and receive a little less for the put.We are paying money for the position,and in order to correct it,we must make the put a bit more attractive,and the call less attractive.We do so by shifting the strike price to the right of the forward:Now the buyer of the call must wait a little bit longer before his call pays off,and he is only willing to buy it for less.As the opposite is true for the put,we have established that the strike must be to the right of the forward.Part1Insurance,Hedging,and Simple Strategiesd)If we are creating a synthetic short stock,we buy the put option and sell the call option. We are buying at the ask and selling at the bid,and the bid price is always smaller than the ask. Suppose we had established a zero-cost synthetic short at the forward price,and now we introduce the bid-ask spread.This means that we have to pay a little more for the put,and receive a little less for the call.We are paying money for the position,and in order to correct it,we must make the call a bit more attractive.The call gets more attractive if the strike price decreases,because thefinal payoff is max(S−K,0).Therefore,we have to shift the strike price to the left of the forward price.e)No,transaction fees are not a wash,because we are paying implicitly the bid-ask spread:If we bought a stock today and held it until the expiration of the options,we would get the future stock price less the forward price(which is equivalent to the loan we got tofinance the stock purchase). Now,we established in c)that the strike price is to the right of the forward price.Therefore,we will receive from the collar of part c)the stock price less something that is larger than the forward price: We make a loss compared to the self-financed outright purchase of the stock.These considerations do not yet take into account that we incur transaction costs on two instruments,compared to only one time brokerage fees if we buy the stock directly.It is thus very important to be aware of transaction costs when comparing different investment strategies.Question3.20.Use separate cells for the strike price and the quantities you buy and sell for each strike(i.e.,make use of the plus or minus sign).Then,use the maximum function to calculate payoffs and profits. The best way to solve this problem is probably to have the calculations necessary for the payoff and profit diagrams run in the background,e.g.,in another auxiliary table that you are referencing to.Define the boundaries for the calculations dynamically and symmetrically around the current stock price.Then use the diagram function with the line style to draw the diagrams.。
2011-2012FRM备考详尽学习笔记
2007年note Topic 35波动率微笑1. 理解为什么在计算看涨期权和看跌期权价格时，所使用的隐含波动率（implied volatility）是相同的：这是根据put-call parity得出的，并不受关于股票价格分布的限制。
2. 实证研究表明：外汇期权（foreign currency option）隐含波动率呈现出微笑的形态（当strike较大或较小的时候，波动率会变大）——因为参与外汇期权的人们预期汇率发生大幅波动的可能性要大于由对数正态分布模型中所预测出的概率。
3. 股票期权（equity option）隐含波动率呈现出半笑的形态（向左偏：当strike较小时，波动率会变大）——人们认为股票价格下跌的可能性要大于其上涨的可能性。
原因：leverage效应：标的资产的价值与其波动性成反比；crashophobia效应：人们恐惧股灾的出现，因此对股票价格下跌提供保护的期权定以较高的价格(只是表明在实际中，标的资产价格下降的概率要大于BS 公式中所假设的那样，但并不意味着随着股票价格降低波动率会提高)。
————这是探究波动率与施权价的关系。
4. 波动率期限结构（volatility term structure）：平价期权的波动率与期权到期期限之间的关系——当短期波动率非常低时，波动率函数是距离到期期限时间的增函数；当短期波动率非常高时，波动率函数是距离到期期限时间的减函数（与波动率的均值回归性质有关）。
————这是探究波动率与到期期限的关系。
5. Volatility surface：同时考虑了期限结构和微笑效应，即同时考虑了strike和maturity term。
因此交易者可以使用volatility term structure和surfaces来判断一个期权定价模型得出结论的准确性和一致性。
6. 在计算期权的Greek的时候所遵守的两大原则：i. sticky strike rule(期权的隐含波动率在短期内应当是保持不变的)；ii. sticky delta rule（期权价格和标的价格与施权价格的比率之间的关系应当是保持不变的，根据这个假设得出的delta要大于由BS公式得到的delta）。
Option
Option1. Put-call parity(1) Fiduciary Call = C(X) + B(X)B: zero-coupon bond that pays X in T yearsC(X) : X is the exercise priceS : buying a stockPayoff to it0f00P0 = C0 - S0 + [X* exp (- R c f * T)](4) Arbitrage strategy2. Binomial Option Pricing Model(1) Basic Model(2) Arbitrage OpportunityOption – fractional share of the stockFractional share of stock in the arbitrage trade for each option traded (hedge ratio)= Delta = (C+1– C-1) / (S+1– S -1)(3) Two-period Binomial Model3. Options on a fixed-income instrument using a binomial tree(1) Basic SituationStep 1: Price the bond at each node using the projected interest ratesStep 2: Calculate the intrinsic value of the option at each node at the maturity of the option Step 3: Bring the terminal option values determined in Step 2 back to todayRisk-neutral probability of an up- and down- move here is always 50%.If we need to value an American option, the value at any node will be equal to the greater of the PV of the future payoffs or the current intrinsic value (S T– X for call or X - S T for put).(2) Caps and FloorsCaps and Floors are just bundles of European-style options on interest rates, called caplets and floorlets. The value of a cap or floor is the sum of the values of the individual caplets or floorlets.e.g. value of 2-year cap = value of a 1-year caplet + value of a 2-year caplet- expiration value of caplet = max {0, [(one-year rate – cap rate)* notional principal]} / (1+ one-year rate)- expiration value of floorlet = max {0, [floor rate –one-year rate]*notional principal}/(1+ one-year rate)4. Black-Scholes-Merton (BSM) Model(1) continuous time and no-arbitrage assumption. To derive the BSM model, an“instantaneously” riskless portfolio (one that is riskless over the next instant) is used to solve for the option price.(2) BSM assumptions and limitations - P 265Exception: the put value may increase as the option approaches maturity if the option is deep in the money and close to maturity.(4) DeltaDelta call = (C1– C0) / (S1– S0)0 = < Delta call < = 1- 1 = < Delta put < = 0C - Price of the callS - Price of the underlying stockFrom BSM:N(d1) *[N(d1) - 1] *The approximation gets worse when the△S gets larger.Delta is the slope of the prior-to-expiration curve, which evaluates the time value.- intrinsic value0, when call option is out of moneyS T– X, when call option is at the money- time valueThe prior-to-expiration curve lies above the at-expiration diagram by the amount of the time value.5. Dynamic Hedging(1) goal of delta-neutral portfolio (delta-neutral hedge) is to combine a long position in astock with a short position in a call option so that the value of the portfolio doesn’t change when the value of the stock changes.(2) Number of options needed to delta hedge = number of shares hedged / delta of calloption(3) The delta-neutral portfolio must be continually rebalanced to maintain the hedge –dynamic hedge.6. Gamma(1) It is used to measure the rate of change in delta as the underlying stock pricechanges.(2) Long positions in calls and puts have positive gammas. And, call and put on the sameunderlying assets with the same exercise price and time to maturity will have equal gammas.(3) Gamma is the largest when a call or put is at the money and close to expiration. If theoption is either deep in or deep out of the money, gamma approaches zero.(4) Gamma can be used to measure how poorly a dynamic hedge will perform when it isnot rebalanced in response to a change in the asset price.7. Effect of underlying asset cash flowExistence of cash flows on the underlying asset:Decrease the value of a call and increase the value of a putC0(X) + [X/(1 + R f)T] = P0 + (S0– PVCF)8. Options on ForwardsPut-call parity for options on forwardsPortfolio IC0 + (X - F T )/(1+R f)TA call on the forward contract with an exercise price of X that matures at time T on a forward contract at F TPortfolio IIP0A put on the forward contract with an exercise price of XA long position in the forward contractC0 + (X - F T )/(1+R f)T = P09. Options on futures & Options on forwardsAmerican options on futures are more valuable than comparable European options because there is mark to market on futures, early exercise can accelerate the payment of any gains.Since there is no mark to market on forwards, early exercise doesn’t accelerate the payment of any gains. So the value of American and European options on forwards are the same.。
期权定价的有限差分法
Liaoning Normal University教师指导本科生科研训练项目研究论文题目：期权定价的有限差分法学院：数学学院专业：数学与应用数学（金融数学）班级序号：6班号学号：**************学生姓名：***指导教师：***2011年12月期权定价的有限差分法学生：刘小芹指导教师：包振华数学学院数学与应用数学(金融数学)专业2008级摘要：期权定价是所有金融应用领域数学上最复杂的问题之一。
第一个完整的期权定价模型由Fisher Black和Myron Scholes创立并于1973年公之于世。
B—S期权定价模型发表的时间和芝加哥期权交易所正式挂牌交易标准化期权合约几乎是同时。
不久，德克萨斯仪器公司就推出了装有根据这一模型计算期权价值程序的计算器。
现在，几乎所有从事期权交易的经纪人都持有各家公司出品的此类计算机，利用按照这一模型开发的程序对交易估价。
这项工作对金融创新和各种新兴金融产品的面世起到了重大的推动作用。
有限差分方法(FDM)是计算机数值模拟最早采用的方法，至今仍被广泛运用，该方法是一种直接将微分问题变为代数问题的近似数值解法，数学概念直观，表达简单，是发展较早且比较成熟的数值方法。
关键词：期权定价；有限差分方法；Abstract: Hedging is one of the important reasons for the development of the financial derivatives. When we use derivatives to hedge the risk for other asset or derivatives, it is the nature method to compute some common insensitive index for hedging tool and the underlying asset，then we can construct the portfolio. In this paper we take into account the options, which is called the dynamic hedging. Most investors use some complex hedging strategies in order to reduce the risk they faced. They try to make their portfolio immune to the small changes in the underlying asset in the short future, which is the so called Delta hedging.Keywords: Hedging; Delta; risk neutral hedging1、引言期权，也即期货合约的选择权，指的是其购买者在交付一定数量的权利金之后，所拥有的在未来一定时间内以一定价格买进或卖出一定数量相关商品合约（不论是实物商品，金融证券或期货）的权利，但不负有必须买进或卖出的义务。
公司理财精要版原书第12版教师手册RWJ_Fund_12e_IM_Chapter24
CHAPTER 24OPTION VALUATIONSLIDES24.1 Chapter 2424.2 Key Concepts and Skills24.3 Chapter Outline24.4 Protective Put24.5 An Alternative strategy24.6 Comparing the Strategies24.7 Put-Call Parity24.8 Example: Finding the Call Price24.9 Continuous Compounding24.10 Example: Continuous Compounding24.11 Example: PCP with Continuous Compounding24.12 Black-Scholes Option Pricing Model24.13 Example: Black-Scholes OPM24.14 Example: Black-Scholes OPM in a Spreadsheet24.15 Put Values24.16 European vs. American Options24.17 Table 24.424.18 Varying Stock Price and Delta24.19 Work the Web Example24.20 Figure 24.1 Option Prices vs. Stock Price24.21 Example: Delta24.22 Varying Time to Expiration and Theta24.23 Figure 24.2 Option Prices vs. Time to Expiration24.24 Example: Time Premiums24.25 Varying Standard Deviation and Vega24.26 Figure 24.3 Option Prices vs. Return Standard Deviation 24.27 Varying the Risk-free Rate and Rho24.28 Figure 24.4 Option Prices vs. Risk-free Interest Rate 24.29 Implied Standard Deviations24.30 Work the Web Example – 224.31 Equity as a Call Option24.32 Valuing Equity and Changes in Assets24.33 Put-Call Parity and the Balance Sheet Identity24.34 Mergers and Diversification24.35 Extended Example – Part I24.36 Extended Example – Part II24.37 Extended Example – Part III24.38 M&A Conclusions24.39 Extended Example: Low NPV – Part I24.40 Extended Example: Low NPV – Part II24.41 Extended Example: Low NPV – Part III24.42 Extended Example: Negative NPV – Part I24.43 Extended Example: Negative NPV – Part II24.44 Extended Example: Negative NPV – Part III24.45 Extended Example: Negative NPV – Part IV24.46 Conclusions24.47 Quick Quiz24.48 Ethics Issues24.49 Comprehensive Problem24.50 End of ChapterCHAPTER WEB SITESCHAPTER ORGANIZATION24.1 Put-Call ParityProtective PutsAn Alternative StrategyThe ResultContinuous Compounding: A Refresher Course 24.2 The Black-Scholes Option Pricing ModelThe Call Option Pricing FormulaPut Option ValuationA Cautionary Note24.3 More about Black-ScholesVarying the Stock PriceVarying the Time to ExpirationVarying the Standard DeviationVarying the Risk-Free RateImplied Standard Deviations24.4 Valuation of Equity and Debt in a Leveraged FirmValuing the Equity in a Leveraged FirmOptions and the Valuation of Risky Bonds 24.5 Options and Corporate Decisions: Some ApplicationsMergers and DiversificationOptions and Capital Budgeting24.6 Summary and ConclusionsANNOTATED CHAPTER OULTINESlide 1: Chapter 24Slide 2: Key Concepts and SkillsSlide 3: Chapter Outline25.1 Put-Call ParityTerminology Review:Call – right, but not the obligation, to buy the underlying asset at thespecified price on or before a specified datePut – right, but not the obligation, to sell the underlying asset at the specifiedprice on or before a specified dateExercise or strike price – price specified in the option contractA.Protective PutsSlide 4: Protective PutThe strategy:Buy one share of stock at price, S.Buy one put option with strike price, E, and put premium, P.Example: Suppose you buy Citigroup stock for $45, and at thesame time, you purchase a put option with a strike price of $40.You pay $1.80 for the option, and it expires in one year. You planto sell the stock in one year.Consider the following possible payoffs:Stock Price Put Value Combined Value Total Gain or Loss25 15 40 −5.8030 10 40 −5.8035 5 40 −5.8040 0 40 −5.8045 0 45 −1.80Stock Price Put Value Combined Value Total Gain or Loss50 0 50 +3.2055 0 55 +8.2060 0 60 +13.2065 0 65 +18.20The maximum loss has been limited to $5.80.B.An Alternative StrategySlide 5: An Alternative StrategySuppose, instead, you buy a call option with a strike price of E anda call price of C. You invest the remainder in a Treasury Bill.Example: A $40 call option on Citigroup stock is selling for $7.78,and the T-bill has an interest rate of 2.5%. We want to look at thesame investment as in part A, so you invest a total of $46.80. So,you invest 46.80 − 7.78 = 39.02 in T-bills. Consider the payoffs.Combined Value Total Gain or Loss Stock Price Call Value T-bill39.02(1.025)25 0 40 40 −5.8030 0 40 40 −5.8035 0 40 40 −5.8040 0 40 40 −5.8045 5 40 45 −1.8050 10 40 50 +3.2055 15 40 55 +8.2060 20 40 60 +13.2065 25 40 65 +18.20The payoffs are the same with both strategies.C.The ResultSlide 6: Comparing the StrategiesIf the combined value is the same at the end, under all situations,then the cost today must be the same.Slide 7: Put-Call ParityThis leads to the famous put-call parity (PCP) condition:S + P = C + PV(E)where the present value is computed using the risk-free rate. Slide 8: Example: Finding the Call PriceThe PCP condition can be rearranged to solve for any of thecomponents.D.Continuous Compounding: A Refresher CourseSlide 9: Continuous CompoundingEffective annual rate with continuous compounding:EAR = e q– 1where q is the quoted rate.Suppose you have a quoted rate of 5% per year with continuouscompounding:EAR = e.05− 1 = .05127 or 5.127%Time value of money calculations with continuous compounding:FV = PVe RtPV = FVe-Rtwhere R = continuously compounded rate, and t = number ofperiods in terms of yearsSlide 10: Example: Continuous CompoundingExample: What is the present value of $1000 to be received inthree months if the annual continuously compounded rate is 8%?PV = 1000e-.08(3/12) = 980.20Slide 11: Example: PCP with Continuous CompoundingPCP with continuous compoundingS + P = C + Ee−RtExample: Given the following, what does the call have to sell forto prevent arbitrage?S = 80; P = 6; E = 85; R = 10% with continuous compounding; t =9 months (9/12)80 + 6 = C + 85e−.1(9/12)C = 86 − 78.86 = 7.1424.2The Black-Scholes Option Pricing ModelA. The Call Option Pricing Formula Slide 12:Black-Scholes Option Pricing ModelThe Formula: C = SN(d 1) − Ee −Rt N(d 2) where N(d 1) and N(d 2) are probabilities that we compute using the following formulas and then look the numbers up in the standard normal tables. where σ is the standard deviation (or volatility) of the underlying asset returns. Slide 13:Example: Black-Scholes OPMExample: Consider a stock that is currently selling for $35. You are looking at a call option that has an exercise price of $30 and expires in 6 months. The risk-free rate is 4%, compounded continuously. The volatility of stock returns is .25. What is the call price? ()89675.12625.07353.1d 07353.112625.126225.04.3035ln d 221=-==⎪⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛= From Table 24.3 N(d 1) = N(1.07) = (.8554 + .8599)/2 = .8577 N(d 2) = N(.90) = .8159 C = 35(.8577) – 30e -.04(6/12)(.8159) = $6.03t σd d t σt 2σR E S ln d 1221-=⎪⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛=Slide 14: Example: Black-Scholes OPM in a SpreadsheetB.Put Option ValuationSlide 15: Put ValuesLecture Tip: The Black-Scholes model can also be adjusted tosolve for the value of a put option directly.P = Ee−Rt N(−d2) − SN(−d1)where d1 and d2 are computed as before. You just change the signbefore looking it up in the table.As an example, consider the previous information but find thevalue of a put instead of a call.N(−d1) = N(−1.07) = (.1401 + .1446) ⁄ 2 = .1424N(−d2) = N(−.9) = .1841P = 30e−.04(6/12)(.1841) − 35(.1424) = $0.43A slightly different answer will be found below using the PCP. Thedifference is due to rounding.Example: Consider the previous call option example and use put-call parity to find the value of the put.S + P = C + PV(E)35 + P = 6.03 + 30e−.04(6/12)P = $0.44C. A Cautionary NoteSlide 16: European vs. American OptionsBoth PCP and the Black-Scholes model are strictly for Europeanoptions that can be exercised only at expiration. There are timeswhen it would be optimal to exercise a put option early, but thesemodels will not capture that additional value, called the “earlyexercise premium.”24.3 More about Black-ScholesSlide 17: Table 24.4Table 24.4 illustrates the relationship between option values andthe five major inputs.A.Varying the Stock PriceSlide 18: Varying Stock Price and DeltaSlide 19: Work the Web ExampleSlide 20: Figure 24.1 Option Prices vs. Stock PriceSlide 21: Example: DeltaCall prices have a direct relationship with the stock price, while putprices have an inverse relationship. This relationship is calleddelta.For European options:Call delta = N(d1)Put delta = N(d1) − 1You can use delta to estimate the new option value given a smallchange in the stock price.Lecture Tip: Delta is the first derivative of the OPM with respectto S. Gamma is the second derivative with respect to S.An option’s delta changes as S changes, and gamma measures therate of change. Delta is often used to determine how many optionsare needed to hedge a portfolio. As S changes, the number ofoptions needed will change because delta depends on S.The larger the gamma, the smaller the change in S required tocause a significant change in delta. The larger the change in delta,the greater the need to rebalance the portfolio and the higher thetrading costs. Therefore, portfolio managers will often look at boththe gamma and the delta when deciding which options to use forhedging.B.Varying the Time to ExpirationSlide 22: Varying Time to Expiration and ThetaSlide 23: Figure 24.2 Option Prices vs. Time to ExpirationFor American calls and puts, the value of the option increases asthe time to expiration increases.It is never optimal to exercise call options on non-dividend payingstocks early. Therefore, the value of a European call will alsoincrease as time increases.However, it may be optimal to exercise a put option early, and aEuropean put prevents early exercise. Therefore, there aresituations in which a shorter time to expiration would actually bemore valuable, and the relationship between European put valueand time is ambiguous.The relationship between option value and time to expiration iscalled theta.Intrinsic valuecall: max[S − E, 0]put: max[E − S, 0]Option value = Intrinsic value + Time premiumTime premium – option value associated with the time left toexpiration, decreases as expiration approachesSlide 24: Example: Time PremiumsExample: Consider the previous option valuation examples. Whatis the intrinsic value and the time premium for each option?Call: C = 6.03intrinsic value = max[35 − 30, 0] = 5time premium = 6.03 − 5 = 1.03Put: P = .44intrinsic value = max[30 − 35, 0] = 0time premium = .44 − 0 = .44C.Varying the Standard DeviationSlide 25: Varying Standard Deviation and VegaSlide 26: Figure 24.3 Option Prices vs. Return Standard DeviationThe relationship between volatility and option value is called vega.As volatility increases, the value of the option increases.The potential loss is limited to your premium. However, the greaterthe volatility, the larger the potential gain.D.Varying the Risk-Free RateSlide 27: Varying the Risk-free Rate and RhoSlide 28: Figure 24.4 Option Prices vs. Risk-free Interest RateThe value of a call increases as the risk-free rate increases. Theopposite is true for puts. However, the impact is very small,especially for “realistic” rates.The relationship between the risk-free rate and option value iscalled rho.E.Implied Standard DeviationsSlide 29: Implied Standard DeviationsSlide 30: Work the Web ExampleWe can observe option values, underlying asset values, exerciseprices, and risk-free rates in the market. The one variable that isnot observable is the standard deviation, or volatility.The OPM can be used to estimate the expected standard deviationof returns – called the implied standard deviation or impliedvolatility.There is not a closed-form solution—the easiest way to find theimplied volatility is to use an options calculator.24.4 Valuation of Equity and Debt in a Leveraged FirmSlide 31: Equity as a Call OptionEquity can be viewed as a call option on the assets of a business.When a debt payment is due, stockholders can choose not toexercise the option and the assets pass to the bondholders.Paying off the debt is the same as exercising the option.A.Valuing the Equity in a Leveraged FirmSlide 32: Valuing Equity and Changes in AssetsExample: For simplicity, assume a firm has a 5-year, zero coupon bond with a face value of $20 million. The firm’s assets have a market value of $30 million. The volatility of asset returns is .3, and the continuously compounded risk-free rate is 5%. What is the market value of equity? Of debt? S = 30; E = 20; t = 5; σ = .3; R = .05 N(1.31) = (.9032 + .9066) ⁄ 2 = .9049 N(.64) = .7389 Equity = 30(.9049) − 20e −.05(5)(.7389) = 15.63788 million Debt = 30 − 15.63788 = 14.36212 millionWhat is the firm’s cost of debt? 14.36212 = 20e −R(5) R = .06623 = 6.623%B. Options and the Valuation of Risky Bonds A protective put can be used to reduce the risk of bonds. Buy a put with an exercise price of $20 million Value of risky bond + put = value of risk-free bond 14.36212 + P = 20e −.05(5) P = 1.2139 million Increasing the value of the put decreases the value of the risky bond. Slide 33:Put-Call Parity and the Balance Sheet IdentityValue of debt = Value of risk-free debt − Put Debt = Ee −Rt − PPCP: S + P = C + Ee −RtS = C + (Ee −Rt − P)()()64.53.31.1d 31.153.523.05.2030ln d 221=-==⎪⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛=Assets = Equity + DebtPCP and the balance sheet identity are the same.24.4 Options and Corporate Decisions: Some ApplicationsC.Mergers and DiversificationSlide 34: Mergers and DiversificationSlide 35: Extended Example – Part ISlide 36: Extended Example – Part IISlide 37: Extended Example – Part IIIUse option valuation to investigate whether diversification is agood reason for a merger from a stockholder’s viewpoint.If synergies do not exist, then a merger will reduce volatilitywithout increasing cash flow.Decreasing volatility decreases the value of the call option (equity)and the put option. Decreasing the value of the put increases thevalue of the debt.Slide 38: M&A ConclusionsSo, a merger for diversification reasons, transfers value from thestockholders to the bondholders.D.Options and Capital BudgetingSlide 39: Extended Example: Low NPV – Part ISlide 40: Extended Example: Low NPV – Part IISlide 41: Extended Example: Low NPV – Part IIIIf a firm has a substantial amount of debt, stockholders may preferriskier projects, even if they have a lower NPV.The riskier project increases the volatility of the asset returns. Theincreased volatility increases the value of the call (equity) and theput. The increased put value decreases the value of the debt. Thistransfers wealth from the bondholders to the stockholders.Slide 42: Extended Example: Negative NPV – Part ISlide 43: Extended Example: Negative NPV – Part IISlide 44: Extended Example: Negative NPV – Part IIISlide 45: Extended Example: Negative NPV – Part IVStockholders may even prefer a negative NPV project if itincreases volatility enough.The wealth transfer from bondholders to stockholders mayoutweigh the negative NPV.Slide 46: ConclusionsLecture Tip: Bondholders recognize the desire of stockholders totake on riskier projects. Consequently, provisions are put into thebond indentures to try to prevent this wealth transfer. Theseprovisions add to the firm’s cost either directly through a higherinterest rate or through additional monitoring costs. These costsare all considered agency costs.24.5 Summary and ConclusionsSlide 47: Quick QuizSlide 48: Ethics IssuesSlide 49: Comprehensive ProblemSlide 50: End of Chapter。