ACT 4000, MIDTERM #2 ADVANCED ACTUARIAL TOPICS MARCH 22, 2007 HAL W. PEDERSEN
You have 70 minutes to complete this exam. When the invigilator instructs you to stop writing you must do so immediately. If you do not abide by this instruction you will be penalised. All invigilators have full authority to disqualify your paper if, in their judgement, you are found to have violated the code of academic honesty.
Some questions are easier than others but each counts for 10 points. Provide sufficient reasoning to back up your answer but do not write more than necessary.
This exam consists of 8 questions. Answer each question on a separate page of the exam book. Write your name and student number on each exam book that you use to answer the questions. Good luck!
Question 1. You friend T-Bone is an options trader. T-Bone is interested in trading call options on the S&P 500 Index which is currently sitting at a level of 1400. T-Bone has estimated the long term volatility on the S&P 500 at 20% (i. e. (J = 0.20 in the notation of the textbook).
T-Bone is able to obtain market price quotes for call expiring in April. The April 1460-call has an implied volatility of 22%, the April 1480-call has an implied volatility of 21% and the April 1520-call has an implied volatility of 18%.
(1) [6 points] Explain the concept of implied volatility and discuss how one would compute these implied volatility values. (2) [4 points] Do T-Bone's implied volatility calculations provide any guidance on how to trade these call options and, if so, indicate what T-Bone's trade should be. I L-_ (, J
Suppose XYZ is a nondividend-paying stock. Suppose S = $100. a = 40%. o = 0. and r = 0.06.
What is the price of a lOS-strike call option with I year to expiration? r 1 e-h) [; ~'h) What is the I-year forward price for the stock?
---_.------~.-..---.--- Consider a perpetual put option with S = $50, K = $60, r = 0.06. a = 0.40, and 0 = 0.03.
What is the price of the option and at what stock price should it be exer• cised?
Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two 40-strike puts. If you delta-hedge this position. what investment is required? What is your overnight profit if the stock tomorrow is $39? .
r = 8%. and 0 = O. S = $40, a = 30%. ',.. -rhe.. f ..t~
You have sold one 45-strike put with 180 days to expiration. Compute __ the I-day holding period profit if you delta- and gamma-hedge this position using the stock and a 40-strike call with 180 days to expiration.
r = 8%. and 8 = 0. S = $40. (J = 30%. [i-I.../\ IS. >J
Calculate the delta or an at-the-money 6-month European call option on a non• dividend-paying stock when the risk-free interest rate is 10(10 per annum and the stock price volatility is 25°/rJ per annum.
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What does it mean to assert that the thcta of an option position is -0.1 when time is measured in years? --- A fund manager has a well-diversified portfolio that mirrors the performance of the
S&P 500 and is worth S360 million. The value of the S&P 500 is J .100. and the portfolio manager would like to buy insurance against a reduction of more than 5% in the value of the portfolio over the next 6 months. The risk-free interest rate is 6% per annum. The dividend yield on both the portfolio and the S&P 500 is 3'10. and the volatility of the index is 30% per annum.
If the fund manager buys traded European put options. how much would the insurance cost'?
If the fund manager decides to provide insurance by keeping part of the portfolio in risk-free securities. what should the initial position be? 0.84130.86430.95540.96410.65540.88490.90320.91920.93320.94520.97130.99380.99990.99950.99900.81590.78810.61790.75800.72570.57930.69150.53980.50000.98210.98610.97720.99180.98930.99530.99650.99740.99970.99980.99930.99810.99871.00000.000.82640.79950.80780.63310.77040.87290.70540.55570.56750.030.51600.90990.93820.95250.95910.98380.97930.98750.99690.99450.99960.99970.99770.99980.99840.82380.82890.83400.79100.64060.64430.84850.85080.85310.85770.63680.62550.76730.77340.77940.77640.72910.73570.73890.74220.74860.58710.59480.59870.60640.59100.87490.87080.87900.70190.71230.67000.66280.67360.67720.54780.040.050.070.51990.52390.52790.89250.89070.89440.89800.90820.91470.91310.92510.92650.92920.92360.93940.94180.93570.94950.95050.94840.95820.95990.96160.96710.96640.96780.96930.98420.98500.98300.97980.98030.97380.97500.97320.97440.97560.99250.99290.99270.98840.99110.98780.99010.99040.99060.99710.99570.99590.99600.99610.99490.99991.00000.99940.99950.99920.99780.99790.99760.99880.99890.99850.99820.81860.82120.83150.83890.79390.80230.80510.81330.79670.85540.64800.84610.85990.84380.62930.76110.62170.76420.78520.74540.73240.75490.60260.61030.58320.86860.87700.88100.86650.69500.69850.70880.71570.71900.72240.65910.66640.68080.68440.55170.55960.56360.57530.54380.060.080.010.020.50400.50800.53590.51200.88880.89620.89970.88690.90490.90660.91620.91150.93190.92220.92790.93060.92070.94060.93450.93700.94630.95150.94740.95640.95730.96080.96250.96490.96560.96860.96990.98460.98570.98340.98120.98080.98170.97830.97780.97880.97190.97260.97610.99200.99220.99310.99320.99340.98810.98870.98960.98980.99090.98710.98680.99620.99670.99740.99550.99560.99630.99640.99660.99680.99700.99720.99400.99410.99430.99460.99480.99510.99960.99970.99980.99930.99910.99770.99810.99870.99900.99830.83650.81060.65170.86210.78230.75170.61410.88300.68790.57140.090.53190.90150.91770.94290.94410.95350.95450.96330.97060.98540.98260.97670.99360.98900.99130.99160.98640.99730.99520.99991.00000.99950.99800.99750.99860.99820.9999 Values of z for selectedNORMALvaluesDISTRIBUTIONof Pr(Z As noted on page 400 of the text, computing an implied volatility requires that we: • observe a market price for an option • have an option pricing model with which to infer volatility. One cannot normally solve the equation Market Option Price = V(S, K, a, r, T, 8) directly and thus a numerical procedure will need to be used. A standard approach to this numerical procedure is to use the solver in Excel as was demonstrated in class. (2) If the option pricing model used is the Black-Scholes model, then if that model "fit" the observed market prices the implied volatilities would be constast across all derivatives/options contracts. When the implied volatility is greater than the theo• retical volatility parameter it suggests that that option is relatively expensive. When the implied volatility is lower than the theoretical volatility parameter it suggests that that option is relatively inexpensive. Therefore, in the case of T-Bone, the implied volatilities suggest that T-Bone should consider short positions in the April 1460-calls (0.22 > 0.20) and the April 1480-calls (0.21 > 0.20) and consider long positions in the April 1520-call (0.18 < 0.20). Black-Scholes Option Pricing Model Interest Rate 5_0 VolatilityTimeStockStrikeDividendtoPriceExpirationYield 105.00000100.000000.000000.060000.400001.00000 I Summary I deltaTr sigmaK Call Price 16.32706 Call Delta 0.59019 Call Gamma 0.00972 Put Price 15.21233 Put Delta -0.40981 Put Gamma 0.00972 v Call Price 16.32706 Put Price 15.21233 Weak Lower Bound on Call 1.11472 Call Delta 0.59019 Call Gamma 0.00972 Put Delta -0.40981 Put Gamma 0.00972 A 1\ .5 IN t..-r). -. Coil .oMI] 100 e. I,oc:'. J8 Q>-, ~ 0" h1.. :;.. -. G6 0 8" I 8 ~riC~ 0+ 23.07 ~Lrrf~.1 f"t H~ ~ 7- 2--(,7 .5 . 3.5~ '15 ~t- - . "05hJ QLj -- S :;;. If~ ~ Lf () ..- r~~_ ~ Ol.l.t(\ 1-.5,'t c.; ~ri c..L- _.~8Lfr = -~ bo5 J - L(-, 38't~:: .lb~7. D db~ "+ «.. t- - r t'.L', 0 5~r C iI J cOt) .to dLIb-(I ~(.AJl.- I:.~~J ~.);..ho. ·lJL T'<~ ",..5{"'''~ 'i,5 -/;0 .> ~ 0'- + . j" '11 Id-. b.> IIJ l~\- v ••J <.r l t '< '•• --; • ( \ d. ~ 5 k .•"f' -t . ; b 7 't $k 1\('£.5 ) :>- -.lhLf1(J1-Lfo) + (>.9%.s'1-.s.~5"tf')-1(L177F;-l_S7'1't) - . 0 I .$ ~ ( 't. (. ~ (• 5> 0 t If. ~I .58) ------ , -_,.~ 050 b . ~J2~~ _ ------_. ~LfS3b [ ____ -!.-' ~DL± >: ~ -0 i __ GLCc., - =r~ ------ ___ j,.,.)_h_e-;-t- 1'\5--__ = :J£._2lCl ru -s-I:: .. ~ ~ _,_,, __ b-,::-~_~ _ ____ 4::: h c;:: .=. $ Lf_D_-= ~JU __~~_ Jjl-j_~~'_ .0 o...Q h =- 100"1 2.6 c... -.---~------ ______Y\ S .;:. _~-L- }LiG.~ __ ~e:Qf:i : -=-7' + I~~ol ~~- C~_- /.-LlY63 S' _ ------ 't ~_-tk~..r io~~ (I If y_ J,-~_L _->_ {oe-/c._ j ('-1 -'.- _rj c.)~ -f1r. ~d _11 f _>-trrr I u.f ,f"l . f ro+,-j; If e.. ,08/;/J,_ ~] l' o '" [ G e.'qd\ j.J f ,0 {: i /'") r;i,,-L,~ C D,,>-t It- EV\J.') Vt\I\(~ (t:» t.+ 1t52 L -t ~ ",i~ ::I ••-i ~re.>+- C c>4:- ,..{- Po s ~ -I- j..,", -L"'- J"+"r f-) + - J. LLY(;S - - p~ ( + l.c0726 c... / 51 61G-• ~ In this case So = K, r = 0.1, (J = 0.25, and T = 0.5. Also, (J d( = ~+(O.I +0.25:!j2)0.5 =0.3712 0.25 J(f5 The delta of the ~ion is N(d)) or 0.64. 1/.--5 &.- -- A theta of -IT means that if !!.t years pass with no change in either the stock price or its volatility, the value of the option declines by O.I!!.t. \I ChM)L ~,I\. 0rL ..r, to Ca/lj,-dt •• ./;i •....L ,4 J v- II ••.( r'J II ~o+--t Jv tl.q-{-{~L-(I.-.L.- if <,.5(.c...Q d bv'0 •..•.c £.-))"i'- c. ~ -rat; c. ktt"JJ~~ . tJ C{L\)4> i,,-J"\-..trl..- {\- /J /,r.-., J '- {/ \ •.lJ ( e f.{ •.• ~/-;" .. J " f~f The fund is worth $300,000 times the value of the index. When the value of the portfolio falls by 5% (to $342 million), the value of the S&P 500 also falls by 5% to 1140. The fund manager therefore requires European put options on 300,000 times the S&P 500 with exercise price 1140. (a) So = 1200, K = 1140, r = 0.06, (J = 0.30, T = 0.50 and q = 0.03. Hence: In (1200/] 140) + (0.06 - 0.03 + 0.32/2) x 0.5 dJ = 0.3 J030.5 = 0.4186 d2 = dl - 0.3 V03 = 0.2064 N(dJ) = 0.6622; N(d2) = 0.5818 N(-d,) = 0.3378; N( -d2) = 0.4182 The value of one put option is 1140e-rT N( -d2) - ] 200e-qT N( -dd = ] 140e-0.06x0.5 x 0.4182 - 1200e-0.03x0.5 x 0.3378 = 63.40 The total cost of the insurance is therefore 300,000 x 63.40 = $19,020,000 (b) The delta of one put option is [ r-i,,{u e-qT[N(dd - I] - ~(~);;. f'J (- y.) . = e-0.03xO.5(0.6622 - I) '. _L. f-J(fj -I::. - tff-"f.) J -0.3327 This indicates that 33.27% of the portfolio (i.e., $119.77 miJlion) should be initially sold and invested in risk-free securities. .r;.. 0,...~~ '1> r I~5Q'j~e ("L~LL. JC1"111j\.aJ~("" d dl-o Gf -8-£ Q.~c; V'r'- ~:,..+- 1e~c!v •.L « ....I. skGr-l a . "". ,...b" .... ct sLrl!.J ~lv(,(f•.",b ~~L £-1 < TI<- j,{t-o h'J<- cr,r'" uJ t~ +.II i I-j -'> -r:. 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