Course MFE/3F Practice Exam 1 Solutions
|
|
- Ira Lindsey
- 5 years ago
- Views:
Transcription
1 Course MFE/3F Practice Exam 1 Solutions he chapter references below refer to the chapters of the ActuraialBrew.com Study Manual. Solution 1 C Chapter 16, Sharpe Ratio If we (incorrectly) assume that the cost of the shares needed to delta-hedge the put option is $18.5, then we have: S18.5 is positive But the delta of a put option must be negative, so the equation above cannot be correct. Since (iv) tells us that the put option is delta-hedged by selling shares of stock, the original position must be a short position in the put option. he cost of the shares required to delta-hedge the put option is the number of shares required,, times the cost of each share, S. Since shares are sold, the cost is negative. herefore, based on statement (iv) in the question, we have: S 18.5 he elasticity of the put option is: S 18.5 V 7 he risk premium of the put option is equal to the elasticity of the put option times the risk premium of the underlying bond: r ( r) ( ) Solution D Chapter 17, Black Formula he option expires in 1 year, so 1. he underlying bond matures 1 year after the option expires, so s 1. he bond forward price is: P(0, s) 0.84 F P0 (, s) P(0, ) 0.96 he volatility of the forward price is: 0.08 ActuarialBrew.com 016 Page 1
2 We have: d 1 F ln 0.5 ln 0.5(0.08) (1) K d d N( d ) N(0.4154) N( d ) N(0.3154) he Black formula for the put price is: P P(0, ) K N( d ) F N( d ) Solution 3 C Chapter 3, wo-period Binomial Model he factors u and d are constant in the model since: u d he risk-neutral probability is: ( r ) h ( )1 e d e 0.8 p* ud If the final stock price is $144, then the payoff of the call option is $50. If the final stock price is $96, then the payoff of the call option is $. If the final stock price is $64, then the payoff of the call option is $0. he value of the call option is: n rhn ( ) n j nj j nj V( S0, K,0) e ( p*) (1 p*) V( S0u d, K, hn) j j () e [(0.6818) (50) (0.6818)( ) 0] ActuarialBrew.com 016 Page
3 Solution 4 C Chapter 8, Elasticity All three statements are discussed in the last two paragraphs preceding Example 1.8 on page 363 of the third edition of the Derivatives Markets textbook. Statement I is false. he elasticity of a call option decreases as the option becomes more in the money. As the strike price decreases, the option becomes more in the money, and therefore the elasticity decreases. Statement II is false. he upper bound for a put option is 0, not 1. he correct statement is 0 Put. Statement III is true. he elasticity is equal to the percentage of the replicating portfolio invested in the stock. Since a call option is replicated by a leveraged investment in the stock, Call 1. Solution 5 E Chapter 11, All-or-Nothing Options For the square of the final stock price to be greater than 64, the final stock price must be greater than 8: S S (1) 64 (1) 8 herefore, the option described in the question is 51 cash-or-nothing call options that have a strike price of 8. he current value of the option is: r CashCall( SK,, ) 51 e Nd ( ) 51 e Nd ( ) o find the delta of the option, we must find the derivative of the price with respect to the stock price: ( ) ( ) e N d 0.03 Nd 0.03 d 51e 51 e N'( d ) S S S he derivative of d with respect to the stock price is: Se ln r Ke r Ke e d d1 r Se Ke 1 S S S S ActuarialBrew.com 016 Page 3
4 he current value of d is: d 0.0 Se 8e 0.30 ln ln r 0.03 Ke 8e he density function for the standard normal random variable is: 1 N'( x) e 0.5x We can now calculate the delta of the option: 0.03 d d 51 e N'( d ) 51e e S ( ) 51e e herefore, 8.03 shares must be purchased to delta-hedge the option. Solution 6 C Chapter 15, Forward Price of S a he stock price follows geometric Brownian motion, and the claim pays S ( ) a, where a 5. he forward price on the claim is therefore: ( r) 0.5 ( 1) F a 0, S( ) a a a a S(0) e a ( r) 0.5 ( 1) S e aa e (0) a 0.5 ( 1) F0, ( S) aa e (1.5) (5)(51)(0.5) () e Solution 7 C Chapter 9, Delta-Gamma Hedging he gamma of the position to be hedged is: 1, ActuarialBrew.com 016 Page 4
5 We can solve for the quantity, Q, of the other put option that must be purchased to bring the hedged portfolio s gamma to zero: Q 0.00 Q 1,496.5 he delta of the position becomes: 1,000 ( ) 1,496.5 ( 0.339) 79.9 he quantity of underlying stock that must be purchased, of the position being hedged: QS 79.9 Q S, is the opposite of the delta herefore, in order to delta-hedge and gamma-hedge the position, we must sell 79.9 units of stock and purchase 1,496.5 units of Put-II. Solution 8 B Chapter 15, Geometric Brownian Motion Equivalencies A lognormal stock price implies that changes in the stock price follow geometric Brownian motion: ( 0.5 ) tz( t) St S e dst Stdt StdZt () (0) () ( ) () () () Substituting 0.4 for, we have: ( ) t0.4 Z( t) St () S(0) e dst () ( ) Stdt () 0.4 StdZt () () ( 0.08) t0.4 Z( t) St () S(0) e dst () ( ) Stdt () 0.4 StdZt () () he expression for ds( t ) can be rewritten to match Choice B: dst () ( ) Stdt () 0.4 StdZt () () ( 0.08) S( t) dt 0.4 S( t) dz( t) 0.08 S( t) dt Solution 9 E Chapter 7, Black-Scholes Call Price he first step is to calculate d 1 and d : 1 ln( S/ K) ( r 0.5 ) ln(30 / 35) ( ) 0.5 d d d ActuarialBrew.com 016 Page 5
6 We have: Nd ( ) N ( ) Nd ( ) N ( ) he value of one European call option is: r C Se N( d ) Ke N( d ) Eur 1 0.0(0.5) 0.06(0.5) 30e e he value of 100 of the European call options is: Solution 10 D Chapter 5, Stock Price Probability he stock price follows geometric Brownian motion with: o answer this question, we use: Prob S ( ˆ K N d ) When calculating ˆd, we use the absolute value of the volatility coefficient,, so below we have 0.3 : dˆ St St t K 1.10S t ln ( 0.5 )( ) 0.75 t ln (0.3) he probability that the stock price does not increase by more than 10% over the next 6 months is: Prob S 1.10 S N( d ˆ ) N(0.75) t0.5 t ActuarialBrew.com 016 Page 6
7 Solution 11 E Chapter 13, Historical Volatility he Derivatives Markets textbook provides two formulas for estimating the volatility of the stock: 1. Chapter 11 Formula: ( ri r) 1 1 ˆ i h k 1. Chapter 4 Formula: k k ( ri ) 1 ˆ i 1 h h k 1 If we were to use the Chapter 11 formula, then we would need to obtain r : r k 14 r i i i1 i1 k 14 r Since we are not given the first 9 prices, we are not able to obtain r. herefore we use the Chapter 4 formula instead. he original volatility estimate is based on 10 prices. here are 9 returns, so k 9 : ˆ h h k ( ri ) i1 k 1 1 1/ 5 9 ( ri ) i1 9 ( ri ) i1 91 ActuarialBrew.com 016 Page 7
8 After a week passes, the analyst can add 5 more returns into the sum of squared returns: ( ri) ( ri) ( ri) ( ri) i1 i1 i10 i ln ln ln ln ln Now we can calculate the volatility using all 15 prices. here are 14 returns, so k 14 : k 14 ( ri) ( ri) i 1 i 1 h ˆ 34.4% h k 1 1/ / We can use the I-30XS Multiview calculator to obtain the sum of the 5 additional squared returns above: [data] [data] 4 (to clear the data table) (enter the data below) L1 L L (place cursor in the L3 column) [data] (to highlight FORMULA) 1 [ln] [data] / [data] 1 ) [enter] [ nd ] [quit] [ nd ] [stat] 1 DAA: (highlight L3) FRQ: (highlight one) (select CALC) [enter] Statistic number 6 is Solution 1 D Chapter 18, Black-Derman-oy Model In each column of rates, each rate is greater than the rate below it by a factor of: i h e ActuarialBrew.com 016 Page 8
9 herefore, the missing rate in the third column is: e i he missing rate in the fourth column is: e i he tree of short-term rates is then: 41.79% 7.85% 17.47% 9.5% 14.00% 0.86% 14.59% 0.85% 15.6% 14.73% he caplet pays off only if the interest rate at the end of the fourth year is greater than 4.00%. he payoff table is: he payments have been converted to their equivalents payable at the end of 3 years. he calculations are shown below: 100 ( ) ( ) he expected present value of these payments is the value of the 3-year caplet: 1 * 1 V0 E V (1 r ) i0 i (1.14)(1.1747)(1.785) (1.14)(1.1747)(1.785) (1.14)(1.1747)(1.086) (1.14)(1.1459)(1.086) 1.89 ActuarialBrew.com 016 Page 9
10 Alternatively, the value of the caplet can be found recursively, as shown in the tree below: Solution 13 D Chapter 1, Currency Options he first option gives its owner the right to: Give up $1.00 Get 0.4 pounds he value of this option is: pounds dollars $ he second option gives its owner the right to: Give up $.50 Get 1.0 pound he payoff is.5 times the payoff of the first option, so the value of the option is.5 times the value of the first option:.5 $ $ Solution 14 E Chapter 4, Options on Futures Contracts he values of u F and d F are: h uf e e h df e e he futures price tree and the call option tree are below. here is no need to find the current value of the call option: F 0 F 1 European Call 1, , ActuarialBrew.com 016 Page 10
11 he number of futures contracts that the investor must be long is: Vu Vd Fu ( F df ) 1, Solution 15 B Chapter 3, Realistic Probability In the Cox-Ross-Rubinstein model, the values of u and d are: h u e e h d e e We can solve for p, the true probability of the stock price going up, using the following formula: ( ) h ( )0.5 e d e p ud Solution 16 A Chapter 7, Options on Futures he 6-month futures price is: ( r )( 0) F0, F S F te ( )(0.5) F0,0.5 50e he values of d 1 and d are: We have: F0, K ln / 0.5 ln / (0.30) (0.5) d F d d Nd ( ) N ( ) Nd ( ) N ( ) he value of the call option is: r r Eur 0, F 0, F (0.5) 0.09(0.5) C ( F, K,, r,, r) F e N( d ) Ke N( d ) e e ActuarialBrew.com 016 Page 11
12 Solution 17 A Chapter 9, Frequency of Re-Hedging he gamma the 100 calls is: he return on the delta-hedged position is: 1 Rh S (1 z ) h We can solve for the random values that produce a profit greater than $1.40: (1 z ) (1 z ) z z z and z z and z Since z is a standard normal random variable, the probability that both of the inequalities above are satisfied is: N( ) N( ) N( ) 1 N ( ) We have: N ( ) he probability of the profit exceeding $1.40 is therefore: N( ) 1 N ( ) ( ) Solution 18 A Chapter 11, Exchange Options Since we need to find the value of the option in yen, we use yen as the base currency. Let s define the underlying asset to be 1 euro and the strike asset to be 1.45 Canadian dollars. his makes the option an exchange put option with: S 165 K he volatility of ln( S/ K ) is: S K S K (0.59)(0.18)(0.3) ActuarialBrew.com 016 Page 1
13 he values of d 1 and d are: d 1 S 0.05(1) Se 165e (1) ln ln 0.07(1) K Ke e d d We have: N( d ) N( ) N d N ( ) ( ) he value of the exchange put is: ExchangePutPrice K ( ) S Ke N d Se N( d1) 0.07(1) 0.05(1) e ( ) 165 e (0.4438) Solution 19 E Chapter 1, Put-Call Parity he dividends paid before the expiration of the options occur at time 1 month, 4 months, and 7 months. We can use put-call parity to find the value of the put option: r Eur(, ) 0 0, ( ) Eur (, ) 0.1(0.75) 0.1(1/1) 0.1(4 /1) 0.1(7 /1) C K Ke S PV Div P K e 73 e e e P (75,0.75) P Eur (75,0.75) Eur Solution 0 D Chapter 19, Risk-Neutral Cox-Ingersoll-Ross Model Ann uses the CIR model for the short rate. We can use the Sharpe ratio in Ann s model to solve for the parameter : (0.16,0) 1.0 r ActuarialBrew.com 016 Page 13
14 he risk-neutral version of Ann s model is: dr a( r) ( r) ( r, t) dt ( r) dz r a( r) r dt ( r) dz a ( r) r dt ( r) dz 0.4( r) 0.15rdt 0.05 rdz 0.5(0.15 rdt ) 0.05 rdz Mike s model will produce the same prices (and therefore the same yields) as Ann s model if the risk-neutral version of his model is the same as the risk-neutral version of Ann s model. We can rule out Choices A and B, because they are based on the Vasicek model. Ann s model is a CIR model. We can use the Sharpe ratio in Choice C to solve for the parameter : he risk-neutral version of Choice C is: dr a( r) r dt ( r) dz 0.5(0.15 r) rdt 0.05 rdz ( rdt ) 0.05 rdz he risk-neutral version of Choice C is not the same as the risk-neutral version of Ann s model. We can use the Sharpe ratio in Choice D to solve for the parameter : he risk-neutral version of Choice D is: dr a( r) r dt ( r) dz 0.3(0.15 r) 0.05rdt0.05 rdz 0.5(0.15 rdt ) 0.05 rdz Choice D has the same risk-neutral process as Ann s model, and therefore it will produce the same prices and yields as Ann s model. herefore, Choice D is the correct answer. ActuarialBrew.com 016 Page 14
15 For the sake of thoroughness, let s consider Choice E as well. he Sharpe ratio for Choice E is the same as the Sharpe ratio we found for Choice C: he risk-neutral version of Choice E is: dr a( r) r dt ( r) dz 0.3(0.15 r) rdt 0.05 rdz ( rdt ) 0.05 rdz he risk-neutral version of Choice E is not the same as the risk-neutral version of Ann s model. Solution 1 C Chapter 5, Estimating Parameters from Observed Data he quickest way to do this problem is to input the data into a calculator. Using the I-30X IIS, the procedure is: [nd][sa] (Select 1-VAR) [DAA] [ENER] X1= ln 95 /100 (Hit the down arrow twice) X= ln 105 / 95 X3= ln 111/105 X4= ln 103 /111 [ENER] [SAVAR] (Arrow over to Sx) 1 [ENER] (he result is ) [SAVAR] (Arrow over to x ) 1 [ENER] (he result is ) o exit the statistics mode: [nd] [EXISA] [ENER] Alternatively, using the BA II Plus calculator, the procedure is: [nd][daa] [nd][clr WORK] X01= 95/100 = LN [ENER] (Hit the down arrow twice) X0= 105/95 = LN [ENER] ActuarialBrew.com 016 Page 15
16 X03= 111/105 = LN [ENER] X04= 103/111 = LN [ENER] [nd][sa] 1 = (he result is ) [nd][sa] 1 = (he result is ) o exit the statistics mode: [nd][qui] Alternatively, using the I-30XS MultiView, the procedure is: [data] [data] 4 (enter the data below) (to clear the data table) L1 L L (place cursor in the L3 column) [data] (to highlight FORMULA) 1 [ln] [data] / [data] 1 ) [enter] [ nd ] [quit] [ nd ] [stat] 1 DAA: (highlight L3) FRQ: (highlight one) (select CALC) [enter] (to obtain x ) x 1 [enter] he result is [ nd ] [stat] 3 3 (to obtain Sx) Sx 1 [enter] he result is herefore, the annualized estimates for the mean and standard deviation of the normal distribution are: ˆ ˆ ˆ he estimate for the annualized expected return is: r ˆ ˆ (0.9060) h ActuarialBrew.com 016 Page 16
17 We know that the stock price at the end of month 4 is $103. he expected value of the stock at the end of month 1 is: ( )( t) ES [ ] Se t ( )(1 4 /1) / 3 ES [ 1 ] S 4 e 103e Solution A Chapter 9, Market-Maker Profit Question 47 of the Sample Exam Questions uses a constant risk-free interest rate, but in the solution provided by the Society of Actuaries a remark describes how the question could still be answered if the risk-free interest rate is deterministic but not necessarily constant. Let s assume that several months ago was time 0. Further, let s assume that the options expire at time and that the current time is time t. he interest rate is not necessarily constant, so it can vary over time. herefore, we replace the usual discount factors as described below: e e r r ( t) is replaced by e 0 is replaced by e rsds ( ) t rsds ( ) We can use put-call parity to obtain a system of equations: rsds ( ) 17.6 Ke rsds ( ) 8.41 Ke t his can be solved to find t rsds ( ) e : t r( s) ds r( s) ds r( s) ds t Ke Ke e t rsds ( ) 9.08 e 0 rsds ( ) rsds ( ) Ke t 9.08 Ke t 9.08 he market-maker sold 100 of the put options. From the market-maker s perspective, the value of this position was equal to the quantity owned times the price: he delta of the call option can be used to determine the delta of the put option: 0 Put Call e e he delta of the position, from the perspective of the market-maker, was: Delta of a short position in 100 puts 100 ( 0.303) 30.3 ActuarialBrew.com 016 Page 17
18 o delta-hedge the position, the market-maker sold 30.3 shares of stock. he value of this position was the quantity owned times the price: , he proceeds from selling the put options and the stock were available to be lent at the risk-free rate. he value of this cash position at time 0 was: , , he initial position, from the perspective of the market-maker, was: Component Value Options Shares 3, Risk-Free Asset 3, Net 0.00 After t years elapsed, the value of the options changed by: 100 ( ) After t years elapsed, the value of the shares of stock changed by: 30.3 ( ) After t years elapsed, the value of the funds that were lent at the risk-free rate changed by: t 0 rsds 3, e ( ) 1 3, he sum of these changes is the profit. Component Change Gain on Options Gain on Stock Interest Overnight Profit he change in the value of the position is $41.65, and this is the profit. Solution 3 E Chapter 1, Forward Price with Monte Carlo Valuation o find the value of the option at time 0, we use the risk-neutral distribution. he Monte Carlo estimate for the current price of the contingent claim is: r ( t) n e V Vi ( ) n i e 1,000 1,000 1,000 1,000 1,000 1,000 1, ActuarialBrew.com 016 Page 18
19 he forward price for an asset that does not pay dividends is its current price accumulated forward at the risk-free rate of return. herefore, the forward price of the contingent claim is the expression above, accumulated at 8%: e 1,000 1,000 1,000 1,000 1,000 1,000 1, e 1 1,000 1,000 1,000 1,000 1,000 1,000 1, he Monte Carlo estimate for the forward price of the contingent claim is he quickest way to answer this question is to use the I-30XS MultiView Calculator: [data] [data] 4 (to clear the data table) (enter the data below) L1 L L (place cursor in the L column) [data] (to highlight FORMULA) 1 1,000 / [data] 1 ) [enter] [ nd ] [quit] [ nd ] [stat] 1 DAA: (highlight L) FRQ: (highlight one) (select CALC) [enter] x is Solution 4 D Chapter 8, Elasticity he elasticity of the contingent claim is: S 43 V.33 o find the delta of this contingent claim, we break it down into its component parts. he contingent claim's payoff is the same as a straddle's payoff, except that the contingent claim's payoff is shifted down by $10 compared to the straddle's payoff. A straddle consists of a long position in a call and a long position in a put. he payoff of the straddle can be shifted down by $10 by borrowing the present value of $10 at time 0. ActuarialBrew.com 016 Page 19
20 he lowest payoff for a straddle occurs at its strike price, so the straddle has strike price of $40. he contingent claim can be replicated with a portfolio consisting of a long call with a strike price of $40, a long put with a strike price of $40, and borrowing the present value of $10. he current value of the contingent claim is: 0.09 C(40) P(40) 10e We need to find the call option's delta and the put option's delta. he first step is to calculate d 1 : d1 We have: ln( S/ K) ( r 0.5 ) ln(43 / 40) ( ) Nd ( 1) N(0.5449) he delta of the call option is: Call e N( d1 ) e he delta of the put option is: Put Call e e he delta of the contingent claim is the delta of the call plus the delta of the put minus the delta of the loan (i.e., zero): ( 0.881) he elasticity of the contingent claim is: S 43 ( ) V.33 Solution 5 E Chapter, Strike Price Grows Over ime he prices of the options decrease as time to maturity increases. herefore, if the strike price increases at a rate that is greater than the risk-free rate, then arbitrage is available. Option C expires 0.5 years after Option B, so let s accumulate Option B s strike price for 0.5 years at the risk-free rate: e ActuarialBrew.com 016 Page 0
21 Since the strike price of Option C is $57, the strike price grows from time 1.5 to time.0 at a rate that is greater than the risk-free rate of return. Consequently, arbitrage can be earned by purchasing Option C and selling Option B. he arbitrageur buys the -year option for $3.00 and sells the 1.5-year option for $3.50 he difference of $0.50 is lent at the risk-free rate of return. he 1.5-year option After 1.5 years, the stock price is $5.00. herefore, the 1.5-year option is exercised against the arbitrageur. he arbitrageur borrows $53 and uses it to buy a share of stock. As a result, at the end of years the arbitrageur owns the share of stock and owes the accumulated value of the $53. his position results in the following cash flow at the end of years: e he -year option he stock price of $58.00 at the end of years is greater than the strike price of the -year option, which is $57. herefore, the -year put option expires worthless, and the resulting cash flow is zero. he net cash flow he net cash flow at the end of years is the sum of the accumulated value of the $0.50 that was obtained by establishing the position, the $ resulting from the 1.5-year option, and the $0.00 resulting from the -year option: e Solution 6 B Chapter 3, Expected Return he up and down factors are constant throughout the tree, and therefore the risk-neutral probability of an upward movement is also constant and can be calculated using any node. Below we use the first node: ( )(1) 56 ( r ) h e e d p* u d ActuarialBrew.com 016 Page 1
22 Working from right to left, we create the tree of prices for the European put option: Stock European Put he realistic probability is also constant throughout the tree, so as with the risk-neutral probability, we can use the values at the first node to calculate it: ( )(1) 56 ( ) h e e d p u d When the stock price is $80.00, we have: h ( pv ) u (1 pv ) d e V (0.7697)(7.9918) ( )( ) e (0.7697)(7.9918) ( )( ) ln When the stock price is $104.00, we have: h ( pv ) u (1 pv ) d e V (0.7697)(0.00) ( )(7.0) e (0.7697)(0.00) ( )(7.0) ln When the stock price is $56.00, we have: h ( pv ) u (1 pv ) d e V (0.7697)(7.0) ( )(60.80) e (0.7697)(7.0) ( )(60.80) ln ActuarialBrew.com 016 Page
23 he tree of values for is filled in below: N/A N/A N/A he highest of the three values above is 1.31%. Solution 7 E Chapter 19, Rendleman-Bartter Model In the Rendleman-Bartter Model, the short-term rate follows geometric Brownian motion. When a stock's price follows geometric Brownian motion, we can find its expected value as shown below at right: ( ) t dst () ( ) Stdt () StdZt () () ESt () S(0) e In this case, the short rate follows geometric Brownian motion, so we have: 0.1 drt () 0.1() rtdt0.0() rtdzt () Ert () r(0) e he expected value of r (1) is: Er(1) r(0) e 0.08e t Solution 8 C Chapter 7, Black-Scholes Formula Using Prepaid Forward Prices he values of d 1 and d for Stock Y are: P F0, ( S) 40 ln e P ln F , ( K) Ke d ln Ke 0.48 d d d 0.4 d ActuarialBrew.com 016 Page 3
24 he values of d 1 and d for Stock Z are: P F0, ( S) 016 ln e P ln F , ( K) 0.5Ke e d ln ln Ke Ke d d 16 d d Note that the call on Stock Y and the call on Stock Z have the same values for d 1 and d. he Black-Scholes formula for the call option on Stock Y is: r C ( S, K,, r,, ) Se N( d ) Ke N( d ) Eur e N( d ) Ke N( d ) Nd ( ) Ke Nd ( ) We can use the Black-Scholes formula for the call option on Stock Z to see that the call option on stock Z has a value that is half the value of the call option on Stock Y: r e N d1 Ke e N d 0.8 Nd1 Ke Nd 0.8 Nd1 Ke Nd C ( S, K,, r,, ) Se N( d ) Ke N( d ) Eur 40 ( ) 0.5 ( ) 40 ( ) 0.5 ( ) ( ) ( ) Solution 9 C Chapter 10, Compound Options Let S be the ex-dividend price of the stock. Since it doesn't makes sense to exercise the 6-strike American call, the exercise value must be less than the value of a 6-strike 1- year European call option. Below we use put-call parity to substitute for the value of the European call option: 0 S 6 CEur(6,1) 0 S 6 6 S PEur(6,1) PEur(6,1) Since the value of the European put option is greater than 7.60, it makes sense to exercise Option A and not exercise Option B. herefore Option A is worth more than zero. ActuarialBrew.com 016 Page 4
25 Since it makes sense to exercise the 44-strike American call, the exercise value must be less than the value of a 44-strike -year European call option. Below we use put-call parity to substitute for the value of the European call option: 0 S44 CEur(44,) 44 0 S44 S PEur(44,) 4.16 P (44,) Eur 1.5 Since the value of the European put option is less than 4.16, it makes sense to exercise Option D and not exercise Option C. herefore Option D is worth more than zero. Options A and D are worth more than zero, and options B and C expire worthless. Solution 30 B Chapter 15, Drift We can use the theta of the claim to determine : Vt () 0 t 0 t 3 St () t 0 t he value of the claim is therefore: We have: Vt () St () 3 VS 3S VSS 6S V t 0 From Itô s Lemma: S 0.5 SS( ) t dv V ds V ds V dt 3S 0.08Sdt 0.0SdZ 0.5 6S 0.08Sdt 0.0SdZ S dt dz S S dt S dt 0.60S dz 0.36Vdt 0.60VdZ he drift is the expected change per unit of time in the price of the claim. From the stochastic differential equation above, we observe that the drift is 0.36 Vt. ( ) ActuarialBrew.com 016 Page 5
Course MFE/3F Practice Exam 1 Solutions
Course MFE/3F Practice Exam Solutions he chapter references below refer to the chapters of the ActuraialBrew.com Study Manual. Solution C Chapter 6, Sharpe Ratio If we (incorrectly) assume that the cost
More informationCourse MFE/3F Practice Exam 2 Solutions
Course MFE/3F Practice Exam Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 A Chapter 16, Black-Scholes Equation The expressions for the value
More informationCourse MFE/3F Practice Exam 4 Solutions
Course MFE/3F Practice Exam 4 Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 D Chapter 1, Prepaid Forward Price of $1 We don t need the information
More informationSOA Exam MFE Solutions: May 2007
Exam MFE May 007 SOA Exam MFE Solutions: May 007 Solution 1 B Chapter 1, Put-Call Parity Let each dividend amount be D. The first dividend occurs at the end of months, and the second dividend occurs at
More informationCourse MFE/3F Practice Exam 4 Solutions
Course MFE/3F Practice Exam 4 Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution D Chapter, Prepaid Forward Price of $ We don t need the information
More informationCourse MFE/3F Practice Exam 3 Solutions
Course MFE/3F ractice Exam 3 Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 C Chapter 15, repaid Forward rice of S a The payoff consists
More informationCourse MFE/3F Practice Exam 3 Solutions
Course MFE/3F ractice Exam 3 Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution C Chapter 5, repaid Forward rice of S a The payoff consists of
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationActuarialBrew.com. Exam MFE / 3F. Actuarial Models Financial Economics Segment. Solutions 2014, 2nd edition
ActuarialBrew.com Exam MFE / 3F Actuarial Models Financial Economics Segment Solutions 04, nd edition www.actuarialbrew.com Brewing Better Actuarial Exam Preparation Materials ActuarialBrew.com 04 Please
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationActuarialBrew.com. Exam MFE / 3F. Actuarial Models Financial Economics Segment. Solutions 2014, 1 st edition
ActuarialBrew.com Exam MFE / 3F Actuarial Models Financial Economics Segment Solutions 04, st edition www.actuarialbrew.com Brewing Better Actuarial Exam Preparation Materials ActuarialBrew.com 04 Please
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationErrata and updates for ASM Exam MFE/3F (Ninth Edition) sorted by page.
Errata for ASM Exam MFE/3F Study Manual (Ninth Edition) Sorted by Page 1 Errata and updates for ASM Exam MFE/3F (Ninth Edition) sorted by page. Note the corrections to Practice Exam 6:9 (page 613) and
More information(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given:
(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationA&J Flashcards for Exam MFE/3F Spring Alvin Soh
A&J Flashcards for Exam MFE/3F Spring 2010 Alvin Soh Outline DM chapter 9 DM chapter 10&11 DM chapter 12 DM chapter 13 DM chapter 14&22 DM chapter 18 DM chapter 19 DM chapter 20&21 DM chapter 24 Parity
More informationActuarial Models : Financial Economics
` Actuarial Models : Financial Economics An Introductory Guide for Actuaries and other Business Professionals First Edition BPP Professional Education Phoenix, AZ Copyright 2010 by BPP Professional Education,
More informationLecture 18. More on option pricing. Lecture 18 1 / 21
Lecture 18 More on option pricing Lecture 18 1 / 21 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21 Greeks (1) The price f of a derivative depends
More informationIntroduction. Financial Economics Slides
Introduction. Financial Economics Slides Howard C. Mahler, FCAS, MAAA These are slides that I have presented at a seminar or weekly class. The whole syllabus of Exam MFE is covered. At the end is my section
More informationChapter 2 Questions Sample Comparing Options
Chapter 2 Questions Sample Comparing Options Questions 2.16 through 2.21 from Chapter 2 are provided below as a Sample of our Questions, followed by the corresponding full Solutions. At the beginning of
More informationB.4 Solutions to Exam MFE/3F, Spring 2009
SOLUTIONS TO EXAM MFE/3F, SPRING 29, QUESTIONS 1 3 775 B.4 Solutions to Exam MFE/3F, Spring 29 The questions for this exam may be downloaded from http://www.soa.org/files/pdf/edu-29-5-mfe-exam.pdf 1. [Section
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationInterest-Sensitive Financial Instruments
Interest-Sensitive Financial Instruments Valuing fixed cash flows Two basic rules: - Value additivity: Find the portfolio of zero-coupon bonds which replicates the cash flows of the security, the price
More informationOptions. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Options
Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Definitions and Terminology Definition An option is the right, but not the obligation, to buy or sell a security such
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationPricing Options with Binomial Trees
Pricing Options with Binomial Trees MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will learn: a simple discrete framework for pricing options, how to calculate risk-neutral
More informationMFE/3F Study Manual Sample from Chapter 10
MFE/3F Study Manual Sample from Chapter 10 Introduction Exotic Options Online Excerpt of Section 10.4 his document provides an excerpt of Section 10.4 of the ActuarialBrew.com Study Manual. Our Study Manual
More informationBinomial Option Pricing
Binomial Option Pricing The wonderful Cox Ross Rubinstein model Nico van der Wijst 1 D. van der Wijst Finance for science and technology students 1 Introduction 2 3 4 2 D. van der Wijst Finance for science
More information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More information(atm) Option (time) value by discounted risk-neutral expected value
(atm) Option (time) value by discounted risk-neutral expected value Model-based option Optional - risk-adjusted inputs P-risk neutral S-future C-Call value value S*Q-true underlying (not Current Spot (S0)
More informationArbitrage, Martingales, and Pricing Kernels
Arbitrage, Martingales, and Pricing Kernels Arbitrage, Martingales, and Pricing Kernels 1/ 36 Introduction A contingent claim s price process can be transformed into a martingale process by 1 Adjusting
More informationDynamic Hedging and PDE Valuation
Dynamic Hedging and PDE Valuation Dynamic Hedging and PDE Valuation 1/ 36 Introduction Asset prices are modeled as following di usion processes, permitting the possibility of continuous trading. This environment
More informationFinancial Stochastic Calculus E-Book Draft 2 Posted On Actuarial Outpost 10/25/08
Financial Stochastic Calculus E-Book Draft Posted On Actuarial Outpost 10/5/08 Written by Colby Schaeffer Dedicated to the students who are sitting for SOA Exam MFE in Nov. 008 SOA Exam MFE Fall 008 ebook
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 18 & 5 th, 13 he Black-Scholes-Merton Model for Options plus Applications 11.1 Where we are Last Week: Modeling the Stochastic Process for
More informationDeeper Understanding, Faster Calc: SOA MFE and CAS Exam 3F. Yufeng Guo
Deeper Understanding, Faster Calc: SOA MFE and CAS Exam 3F Yufeng Guo Contents Introduction ix 9 Parity and other option relationships 1 9.1 Put-callparity... 1 9.1.1 Optiononstocks... 1 9.1. Optionsoncurrencies...
More informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 19 & 6 th, 1 he Black-Scholes-Merton Model for Options plus Applications Where we are Previously: Modeling the Stochastic Process for Derivative
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives November 5, 212 Option Analysis and Modeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis and Modeling:
More informationErrata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1
Errata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1 1B, p. 72: (60%)(0.39) + (40%)(0.75) = 0.534. 1D, page 131, solution to the first Exercise: 2.5 2.5 λ(t) dt = 3t 2 dt 2 2 = t 3 ]
More information2 f. f t S 2. Delta measures the sensitivityof the portfolio value to changes in the price of the underlying
Sensitivity analysis Simulating the Greeks Meet the Greeks he value of a derivative on a single underlying asset depends upon the current asset price S and its volatility Σ, the risk-free interest rate
More informationOptions Markets: Introduction
17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationThe Binomial Model. Chapter 3
Chapter 3 The Binomial Model In Chapter 1 the linear derivatives were considered. They were priced with static replication and payo tables. For the non-linear derivatives in Chapter 2 this will not work
More information1. 2 marks each True/False: briefly explain (no formal proofs/derivations are required for full mark).
The University of Toronto ACT460/STA2502 Stochastic Methods for Actuarial Science Fall 2016 Midterm Test You must show your steps or no marks will be awarded 1 Name Student # 1. 2 marks each True/False:
More informationIntroduction to Binomial Trees. Chapter 12
Introduction to Binomial Trees Chapter 12 Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. Hull 2013 1 A Simple Binomial Model A stock price is currently $20. In three months
More informationRisk Neutral Valuation, the Black-
Risk Neutral Valuation, the Black- Scholes Model and Monte Carlo Stephen M Schaefer London Business School Credit Risk Elective Summer 01 C = SN( d )-PV( X ) N( ) N he Black-Scholes formula 1 d (.) : cumulative
More informationReview of Derivatives I. Matti Suominen, Aalto
Review of Derivatives I Matti Suominen, Aalto 25 SOME STATISTICS: World Financial Markets (trillion USD) 2 15 1 5 Securitized loans Corporate bonds Financial institutions' bonds Public debt Equity market
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationB. Combinations. 1. Synthetic Call (Put-Call Parity). 2. Writing a Covered Call. 3. Straddle, Strangle. 4. Spreads (Bull, Bear, Butterfly).
1 EG, Ch. 22; Options I. Overview. A. Definitions. 1. Option - contract in entitling holder to buy/sell a certain asset at or before a certain time at a specified price. Gives holder the right, but not
More informationSubject CT8 Financial Economics Core Technical Syllabus
Subject CT8 Financial Economics Core Technical Syllabus for the 2018 exams 1 June 2017 Aim The aim of the Financial Economics subject is to develop the necessary skills to construct asset liability models
More informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationP-1. Preface. Thank you for choosing ACTEX.
Preface P- Preface Thank you for choosing ACTEX ince Exam MFE was introduced in May 007, there have been quite a few changes to its syllabus and its learning objectives To cope with these changes, ACTEX
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationChapter 24 Interest Rate Models
Chapter 4 Interest Rate Models Question 4.1. a F = P (0, /P (0, 1 =.8495/.959 =.91749. b Using Black s Formula, BSCall (.8495,.9009.959,.1, 0, 1, 0 = $0.0418. (1 c Using put call parity for futures options,
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationACTSC 445 Final Exam Summary Asset and Liability Management
CTSC 445 Final Exam Summary sset and Liability Management Unit 5 - Interest Rate Risk (References Only) Dollar Value of a Basis Point (DV0): Given by the absolute change in the price of a bond for a basis
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 23 rd March 2017 Subject CT8 Financial Economics Time allowed: Three Hours (10.30 13.30 Hours) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please read
More informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M375T/M396C Introduction to Financial Mathematics for Actuarial Applications Spring 2013 University of Texas at Austin Sample In-Term Exam II - Solutions This problem set is aimed at making up the lost
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
More informationContinuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a
Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history
More informationCHAPTER 10 OPTION PRICING - II. Derivatives and Risk Management By Rajiv Srivastava. Copyright Oxford University Press
CHAPTER 10 OPTION PRICING - II Options Pricing II Intrinsic Value and Time Value Boundary Conditions for Option Pricing Arbitrage Based Relationship for Option Pricing Put Call Parity 2 Binomial Option
More informationMahlerʼs Guide to. Financial Economics. Joint Exam MFE/3F. prepared by Howard C. Mahler, FCAS Copyright 2013 by Howard C. Mahler.
Mahlerʼs Guide to Financial Economics Joint Exam MFE/3F prepared by Howard C. Mahler, FCAS Copyright 2013 by Howard C. Mahler. Study Aid 2013-MFE/3F Howard Mahler hmahler@mac.com www.howardmahler.com/teaching
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationReplication strategies of derivatives under proportional transaction costs - An extension to the Boyle and Vorst model.
Replication strategies of derivatives under proportional transaction costs - An extension to the Boyle and Vorst model Henrik Brunlid September 16, 2005 Abstract When we introduce transaction costs
More informationOption Pricing. Simple Arbitrage Relations. Payoffs to Call and Put Options. Black-Scholes Model. Put-Call Parity. Implied Volatility
Simple Arbitrage Relations Payoffs to Call and Put Options Black-Scholes Model Put-Call Parity Implied Volatility Option Pricing Options: Definitions A call option gives the buyer the right, but not the
More informationDerivatives Options on Bonds and Interest Rates. Professor André Farber Solvay Business School Université Libre de Bruxelles
Derivatives Options on Bonds and Interest Rates Professor André Farber Solvay Business School Université Libre de Bruxelles Caps Floors Swaption Options on IR futures Options on Government bond futures
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More informationMORNING SESSION. Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quantitative Finance and Investment Core Exam QFICORE MORNING SESSION Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Instructions 1.
More informationAspects of Financial Mathematics:
Aspects of Financial Mathematics: Options, Derivatives, Arbitrage, and the Black-Scholes Pricing Formula J. Robert Buchanan Millersville University of Pennsylvania email: Bob.Buchanan@millersville.edu
More informationIntroduction to Energy Derivatives and Fundamentals of Modelling and Pricing
1 Introduction to Energy Derivatives and Fundamentals of Modelling and Pricing 1.1 Introduction to Energy Derivatives Energy markets around the world are under going rapid deregulation, leading to more
More informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
More informationLecture 16: Delta Hedging
Lecture 16: Delta Hedging We are now going to look at the construction of binomial trees as a first technique for pricing options in an approximative way. These techniques were first proposed in: J.C.
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 12. Binomial Option Pricing Binomial option pricing enables us to determine the price of an option, given the characteristics of the stock other underlying asset
More informationTHE USE OF NUMERAIRES IN MULTI-DIMENSIONAL BLACK- SCHOLES PARTIAL DIFFERENTIAL EQUATIONS. Hyong-chol O *, Yong-hwa Ro **, Ning Wan*** 1.
THE USE OF NUMERAIRES IN MULTI-DIMENSIONAL BLACK- SCHOLES PARTIAL DIFFERENTIAL EQUATIONS Hyong-chol O *, Yong-hwa Ro **, Ning Wan*** Abstract The change of numeraire gives very important computational
More informationThe Multistep Binomial Model
Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationM339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina
M339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. Time: 50 minutes
More informationForwards and Futures. Chapter Basics of forwards and futures Forwards
Chapter 7 Forwards and Futures Copyright c 2008 2011 Hyeong In Choi, All rights reserved. 7.1 Basics of forwards and futures The financial assets typically stocks we have been dealing with so far are the
More informationINTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS. Jakša Cvitanić and Fernando Zapatero
INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Jakša Cvitanić and Fernando Zapatero INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Table of Contents PREFACE...1
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationFinancial derivatives exam Winter term 2014/2015
Financial derivatives exam Winter term 2014/2015 Problem 1: [max. 13 points] Determine whether the following assertions are true or false. Write your answers, without explanations. Grading: correct answer
More informationRisk Neutral Pricing Black-Scholes Formula Lecture 19. Dr. Vasily Strela (Morgan Stanley and MIT)
Risk Neutral Pricing Black-Scholes Formula Lecture 19 Dr. Vasily Strela (Morgan Stanley and MIT) Risk Neutral Valuation: Two-Horse Race Example One horse has 20% chance to win another has 80% chance $10000
More informationFIXED INCOME SECURITIES
FIXED INCOME SECURITIES Valuation, Risk, and Risk Management Pietro Veronesi University of Chicago WILEY JOHN WILEY & SONS, INC. CONTENTS Preface Acknowledgments PART I BASICS xix xxxiii AN INTRODUCTION
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationReading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,
More informationwww.coachingactuaries.com Raise Your Odds The Problem What is Adapt? How does Adapt work? Adapt statistics What are people saying about Adapt? So how will these flashcards help you? The Problem Your confidence
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationCourse FM/2 Practice Exam 2 Solutions
Course FM/ Practice Exam Solutions Solution 1 E Nominal discount rate The equation of value is: 410 45 (4) (4) d d 5,000 1 30,000 1 146,84.60 4 4 We let 0 (4) d x 1 4, and we can determine x using the
More informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems
Math489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems Steve Dunbar No Due Date: Practice Only. Find the mode (the value of the independent variable with the
More informationLecture 11: Ito Calculus. Tuesday, October 23, 12
Lecture 11: Ito Calculus Continuous time models We start with the model from Chapter 3 log S j log S j 1 = µ t + p tz j Sum it over j: log S N log S 0 = NX µ t + NX p tzj j=1 j=1 Can we take the limit
More informationChapter 14. Exotic Options: I. Question Question Question Question The geometric averages for stocks will always be lower.
Chapter 14 Exotic Options: I Question 14.1 The geometric averages for stocks will always be lower. Question 14.2 The arithmetic average is 5 (three 5s, one 4, and one 6) and the geometric average is (5
More informationDr. Maddah ENMG 625 Financial Eng g II 10/16/06
Dr. Maddah ENMG 65 Financial Eng g II 10/16/06 Chapter 11 Models of Asset Dynamics () Random Walk A random process, z, is an additive process defined over times t 0, t 1,, t k, t k+1,, such that z( t )
More informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M375T/M396C Introduction to Financial Mathematics for Actuarial Applications Spring 2013 University of Texas at Austin Sample In-Term Exam II Post-test Instructor: Milica Čudina Notes: This is a closed
More informationDeriving and Solving the Black-Scholes Equation
Introduction Deriving and Solving the Black-Scholes Equation Shane Moore April 27, 2014 The Black-Scholes equation, named after Fischer Black and Myron Scholes, is a partial differential equation, which
More informationFIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M339D/M389D Introduction to Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam II - Solutions Instructor: Milica Čudina Notes: This is a closed book and
More informationMixing Di usion and Jump Processes
Mixing Di usion and Jump Processes Mixing Di usion and Jump Processes 1/ 27 Introduction Using a mixture of jump and di usion processes can model asset prices that are subject to large, discontinuous changes,
More information