Lecture 4-5.pptx
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The document covers numerical methods for solving mathematical problems, including equation solving, integration techniques, and stochastic problem-solving methods. It explains various algorithms like the Newton-Raphson method, trapezium and Simpson's rules for integration, and details on option pricing using binomial and trinomial models. Additionally, it introduces Monte Carlo simulation as a method for approximating solutions by repeated random sampling.
![The option
• The option gives the buyer “the right but not the
obligation to purchase an asset at a previously
agreed price, on or before a particular date”
• A put option gives “the right but not the
obligation to sell an asset at a previously agreed
price, on or before a particular date”
• The option has a fixed maximum life and fixed
price (“exercise price or strike price”)
• C=max[0;S-X], P=max [0;X-S], C – value to call, P
value of a put, X - exercise price, S – asset price.
24](https://image.slidesharecdn.com/lecture4-5-221009102745-474bbeff/85/Lecture-4-5-pptx-24-320.jpg)
![S=35; u=1,25; d=0,75; X=35; R=1,1;
r=0,1
30
S=35
Su=43,75
Sd=26,25
c
cu=max[0;43,75-35]=8,75
cd=max[0;26,25-35] =0](https://image.slidesharecdn.com/lecture4-5-221009102745-474bbeff/85/Lecture-4-5-pptx-30-320.jpg)
![36
S =35
4,37
Su =38,68
6,48
Sd =31,67
1,52
Suu =42,75
9,38
Sud =35
2,62
Sdd=28,65
0
Suuu =47,25
13,07
Suud =38,68
4,51
Sudd =31,67
0
Sddd =25,93
0
Sdddd =26,46
c= max[0;23,46-35]=0
Suddd =28,66
c= max[0;28,65-35]=0
Suudd =35
c= max[0;35-35]=0
Suuuu =52,21
c= max[0;52,21-35]=17,21
Suuud =42,75
c= max[0;42,75-35]=7,75
Binomial tree](https://image.slidesharecdn.com/lecture4-5-221009102745-474bbeff/85/Lecture-4-5-pptx-36-320.jpg)
![• C=max[0;S-X], P=max [0;X-S], C – value to call,
P value of a put, X - exercise price, S – asset
price.
39](https://image.slidesharecdn.com/lecture4-5-221009102745-474bbeff/85/Lecture-4-5-pptx-39-320.jpg)
![40
S=35
4,3716
42, 7491
9,3739
35
2,6184
28,6556
0
52,2139
cu2=max[Su2-X,0]=52,2139-35=17,2139
35
cq2=max[Sq2-X,0]=0
42,7491
cuq=max[Suq-X,0]=42, 7491-35=7,7491
28,6556
cdq=max[Sdq-X,0]=0
23,4612
cd2=max[Sd2-X,0]=0
S=35; X=35;
u=1,2214
pu=0,3544
d=0,8187
pd=0,1638
pq=0,4818
δ=0,2](https://image.slidesharecdn.com/lecture4-5-221009102745-474bbeff/85/Lecture-4-5-pptx-40-320.jpg)
Lecture 4-5.pptx
- 1. Topic 3. Numerical methods Kashcha Mariya 1
- 2. Contents 1. Introduction 2. Equation solving 3. Numerical methods for integration 4. Numerical methods to solve stochastic problems 5. Monte Carlo simulation 2
- 3. Introduction 1. Solving mathematical problems by repetitive application of a mathematical procedure 2. To search for a solution or to aggregate many approximations into a final solutoin 3. Monte Carlo simulation, this process solves problems by imitation the random process, i.e. running the mathematical model many, many times and taking the average outcome. 3
- 4. Equation solving • Problem to solving: 1. Non-linear equation 2. If necessary to obtain an approximate solution: x2-2=0, x≈1,4142135624 4
- 5. 5
- 6. Formulas (or algorithms) to equations five and higher degree no such exist 6
- 7. 7
- 8. Bond to maturity 2,5 years paying half- yearly coupons 8
- 9. Bond to maturity 2,5 years paying half-yearly coupons of 5 and is currently priced 98 9 How to we go about our iterative procedure to search for the x?
- 10. Bisection -3 -2 -1 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 x2-2=0 10
- 11. 11
- 12. 12
- 13. The Newton-Raphson method • More rapidly than the bisection method 1. Begins with a guess of the value of x0 2. x1=x0-f(x0)/f’(x0) … 13
- 14. 14
- 15. Numerical methods for integration • Many functions for which no primitive exists – though this does not mean that there is no integral – consequently a numerical method is required. • Two methods: the trapezium rule and Simpson’s rule – dividing the area under the curve into small vertical strips, each with a height equal to the distance between a particular point on the horizontal axis and the curve. The sum of this areas is approximately the area under the curve. 15
- 16. The trapezium rule 16 f(x0) h The area under the curve which is to be evaluated is split into a number of vertical strips of equal width.
- 17. 17 Curve is convex → The area of each trapezium is slightly greater that the true area under the curve. Curve is concave → The area of each trapezium Is understate that the true area under the curve.
- 18. • The area of each trapezium is given by its mean height times ins width, so the total area is 18 -5 0 5 0 1 2 3 4 5 f(x0) f(x1) x0 x1 h/2(‘ends’+2*’middles’)
- 19. Is the probability of a variable with a standard normal distribution taking a value between one and two 19 X 1 1,2 1,4 1,6 1,8 2 F(x) 0,242 0,194 0,150 0,111 0,079 0,054 h = 0.2 “ends” = 0,242+0,054=0,296 “middles” = 0,194+0,150+0,111+0,079=0,534 S≈0,2*(0,296+2*0,534)/2= 0,1364
- 20. We can be improved by increasing the number of strips 20 X 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 f(x) 0,242 0,218 0,194 0,171 0,15 0,13 0,111 0,094 0,079 0,66 0,054 h = 0.1 “ends” = 0,242+0,054=0,296 “middles” = 0,218+0,194+0,171 +0,15+0,13+0,111+0,094+0,079+0,66=1,212 S≈0,1*(0,296+2*1,212)/2= 0,136
- 21. Simpson’s rule • Improvement upon the trapezium rule 1. Divides area under curve into even number of intervals of equal width. 2. On each intervals we giving three points: two ends and middle. 3. Gives this points with weights: 1, 4, 1. 4. S≈h/3*(“ends”+4*”evens”+2*”odds”) 21
- 22. Simpson’s rule 22 1 2 3 4 5 6 7 8 9 10 11 X 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 f(x) 0,242 0,218 0,194 0,171 0,15 0,13 0,111 0,094 0,079 0,66 0,054 h = 0.1 “ends” = 0,242+0,054=0,296 “evens” = 0,218+0,171+0,13+0,094+0,066=0,679 “odds” = 0,194+0,15+0,111+0,079=0,534 S≈0,1*(0,296+4*0,679+2*0,534)/3= 0,1359
- 23. Numerical methods to solve stochastic problems • When the models assume a fixed relationship between the variables, the model are said to be deterministic. • When the models assume some uncertain component such as a random variable, the model are said to be stochastic. • Now we will look at 3 widely used numerical methods for dealing with stochastic problems: 1. The binomial process; 2. The trinomial process; 3. Monte Carlo simulation. 23
- 24. The option • The option gives the buyer “the right but not the obligation to purchase an asset at a previously agreed price, on or before a particular date” • A put option gives “the right but not the obligation to sell an asset at a previously agreed price, on or before a particular date” • The option has a fixed maximum life and fixed price (“exercise price or strike price”) • C=max[0;S-X], P=max [0;X-S], C – value to call, P value of a put, X - exercise price, S – asset price. 24
- 25. The Binomial distribution 1. Only two possible values or outcomes can be taken on by the variable – binomial trial: “success” or “failure”. 2. For each of a succession of trials the probability of each of the two outcomes is constant. 3. Each binomial trial is independent and identical. 25
- 27. The Binomial distribution 27 S, 50 Su, 55 Sd, 45 Suu, 60,50 Sud, 50 Sdd, 40,50 60,5*0,25+50*0,5+40,5*0,25=50,25
- 28. The binomial models • The application of binomial models to option pricing entails modeling the price of the underlying asset as a binomial process. • The binomial models to assumes to possibility of creating a risk free portfolio by hedging a long position in the underlying asset with the short position in a number of fairly priced call options on that asset 28
- 29. The binomial lattice approach • The cost of establishing the risk portfolio is the cost of buying the underlying asset minus the premiums from the written options. • S =35 the underlying asset • X=35 the exercise price • r=10% the risk-free rate • R=1+r=1,1 • 1 year - time to expiry of the option • Price will have either risen by 25%, or fallen by 25%. 29
- 30. S=35; u=1,25; d=0,75; X=35; R=1,1; r=0,1 30 S=35 Su=43,75 Sd=26,25 c cu=max[0;43,75-35]=8,75 cd=max[0;26,25-35] =0
- 31. Find out how many options to sell that makes the portfolio riskless? 1. S(u-d)=35(1,25-0,75)=17,5 -the range of asset prices 2. cu-cd=8,75-0=8,75 - the range of option values 3. H=S(u-d)/(cu-cd)=17,5/8,75=2 – thus as the asset prices have a range twice that of the option prices, a short position in two options is required to achieve the profit that fully offsets the loss on one unit of the long position in the assets 4. Su-Hcu=1,25*35-2*8,75=26,25 – asset rises Sd-Hcd=0,75*35-2*0=26,25 – asset falls 31
- 32. Determine the fair price at which these options should be sold 1. End of year pay-off: Su-Hcu=26,25 2. Present value must be: (Su-Hcu)/R=26,25/1,1=23,86; 3. A single option must be fairly priced at: (S-23,86)/2=(35-23,86)/2=5,57; 32
- 33. The multi-period binomial model • As the time interval between trials becomes infinitesimally small, so that trading is in effect continuous, the binomial model converges to the Black-Scholes model. • Irrespective of the number of binomial trials, the same principles are used to solve the value of the option at each node of the tree, working back from the expiry of the option to the present time period and, therefore, the current price of the option. 33
- 34. • Assumes an asset price S=35 • Exercise price X=35 • Life of the option in years T-t=1 • Number of binomial trials n=4 • Annual risk-free rate r=10% • Annual volatility σ=20% • One-year, divided into four quarterly sub- periods or binomial trials 34
- 35. 35
- 36. 36 S =35 4,37 Su =38,68 6,48 Sd =31,67 1,52 Suu =42,75 9,38 Sud =35 2,62 Sdd=28,65 0 Suuu =47,25 13,07 Suud =38,68 4,51 Sudd =31,67 0 Sddd =25,93 0 Sdddd =26,46 c= max[0;23,46-35]=0 Suddd =28,66 c= max[0;28,65-35]=0 Suudd =35 c= max[0;35-35]=0 Suuuu =52,21 c= max[0;52,21-35]=17,21 Suuud =42,75 c= max[0;42,75-35]=7,75 Binomial tree
- 37. General binomial formula for option valuation: 37
- 38. The trinomial equivalent of the binomial option pricing model • Asset price can take one of the tree possible values at the end of each trinomial trial: rise – u, remain at the current level – q and fall – d. 38 The one-period trinomial option pricing model is given as:
- 39. • C=max[0;S-X], P=max [0;X-S], C – value to call, P value of a put, X - exercise price, S – asset price. 39
- 40. 40 S=35 4,3716 42, 7491 9,3739 35 2,6184 28,6556 0 52,2139 cu2=max[Su2-X,0]=52,2139-35=17,2139 35 cq2=max[Sq2-X,0]=0 42,7491 cuq=max[Suq-X,0]=42, 7491-35=7,7491 28,6556 cdq=max[Sdq-X,0]=0 23,4612 cd2=max[Sd2-X,0]=0 S=35; X=35; u=1,2214 pu=0,3544 d=0,8187 pd=0,1638 pq=0,4818 δ=0,2
- 41. The one-period trinomial option pricing model is given as: 41
- 42. • Monte Carlo Simulation I • Monte Carlo Simulation in Excel: Financial Planning Example 42