1. Nothing below this point Guide @ 2.68 Guide @ 1.57 Guide @ 1.97 Subtitle Guide @ 2.64 Guide @ 2.99 Nothing below this point Guide @ 0.22 Guide @ 4.77 American Options and the Longstaff-Schwartz methodology Gabor Molnar-Saska Morgan Stanley Hungary Analytics Ltd. This material has been prepared for information purposes to support the promotion or marketing of the transaction or matters addressed herein. It is not a solicitation of any offer to buy or sell any security, commodity or other financial instrument or to participate in any trading strategy. This is not a research report and was not prepared by the Morgan Stanley research department. It was prepared by Morgan Stanley sales, trading, banking or other non- research personnel. This material was not intended or written to be used, and it cannot be used by any taxpayer, for the purpose of avoiding penalties that may be imposed on the taxpayer under U.S. federal tax laws. Each taxpayer should seek advice based on the taxpayers particular circumstances from an independent tax advisor. Past performance is not necessarily a guide to future performance. Please see additional important information and qualifications at the end of this material.

2. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 2 Company profile Morgan Stanley is one of the worlds leading investment banks Managing over $600 bn of assets With 600 offices in 30 countries (2 in Hungary) Employing 54,000 people worldwide.

3. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 3 Company profile Morgan Stanley is one of the worlds leading investment banks Managing over $600 bn of assets With 600 offices in 30 countries (2 in Hungary) Employing 54,000 people worldwide. Millennium City CenterDeak Palota

4. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 4 Company profile Morgan Stanley is one of the worlds leading investment banks Managing over $600 bn of assets With 600 offices in 30 countries (2 in Hungary) Employing 54,000 people worldwide. Millennium City Center 500 people ITFinance Securities Operations Deak Palota Analytical Modelling 30 people

5. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 5 A toy example Game: We can throw a regular die at most three times. After each throw we can decide whether to stop playing and win as many thousands of HUF as shown on the die or to continue the game. After the third throw we will win as many thousands of HUF as shown on the die in the last throw. What is the fair price of this game?

6. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 6 A toy example Answer: think backward Throw 3: expected gain 3500 HUF Throw 2: strategy: continue if 1,2, or 3 stop if 4,5, or 6 expected gain 0.5*3500 + 15000/6 = 4250 HUF Throw 1: strategy:continue if 1,2,3, or 4 stop if 5 or 6 expected gain 4250*4/6 + 11000/6 = 4666.67 HUF

7. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 7 European option Stock price today S 0 Strike K Maturity T Interest rate r The dynamics for S t is known Question: E( K - S T ) + ( E V(S T ) = ? ) Note: S t should be martingale (?) Example: dS t =rdt+vdW

8. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 8 American option Stock price today S 0 Strike K Maturity T Interest rate r The dynamics for S t is known Question: E[ max( K - S θ ) + ] = ?, ( E[ V(S θ ) ] = ? ) where 0 < θ < T is a stopping time

9. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 9 American option The simplest case: dS t =vdW, v constant We can approximate with binomial tree log S 0 log S 0 +vT/n log S 0 -vT/n log S 0 +2vT/n log S 0 -2vT/n log S 0

10. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 10 American option Problem: Dynamics are more complex ! General basic idea: Monte Carlo

11. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 11 American option Approach 1: Find a stopping rule from a given class Example (Put option: E[ max( K - S θ ) + ] ): stopping rule: stop if S t < C to find C for the given set of Monte Carlo paths is an optimization problem Problems: 1. How to find the appropriate class? 2. Any decision rule is suboptimal, i.e. we underestimate the option value 3. Optimization can be difficult

12. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 12 American option Approach 2: Try to use the backward idea brute force Monte Carlo simulation: new Monte Carlo generation is need from every point in the future to get the value of the option in a future time (think on toy example) It is too expensive ! NEW IDEA (Longstaff-Schwartz 2001): Combine backward idea with regression

13. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 13 American option Some formulas: Let T 1,T 2,…,T n be a set of exercise dates Let the payoff at time i be P i (if exercise the option) Value of the American option: V n (T n ) = max(0,P n ) V i+1 (T i ) = E[ V i+1 (T i+1 ) | F Ti ], 0 < i < n V i (T i ) = max( V i+1 (T i ), P i ), 0 < i < n V 1 (T 0 ) = E[ V 1 (T 1 ) | F T0 ] = E[ V 1 (T 1 ) ]

14. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 14 American option Algorithm 1: V n+1 = 0 V i (T i ) = E[ V i+1 (T i+1 ) | F Ti ] if P i < E[ V i+1 (T i+1 ) | F Ti ] P i otherwise V 0 = E[ V 1 (T 1 ) ] Algorithm 2: U n+1 =0 U i =U i+1 if P i < E[ U i+1 | F Ti ] P i otherwise U 0 =E[ U 1 ] Observe: V 0 = U 0 is the option price What is the difference between the algorithms?

15. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 15 American option Algorithm 2 requires the conditional expectation operator only to calculate the exercise criterion Longstaff-Schwartz methodology (2001): 1. Generate the appropriate price process (P i ) with MC 2. Use Algorithm 2 to get the option value For the estimation of the conditional expectation use Least-Square Approximation (linear regression)

16. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 16 American put option Example: American (Bermudan) option: Maturity 3 years Exercise dates: 1,2, or 3 years Strike: 1.1 Initial stock price 1.0 Discount factor: 0.94

17. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 17 American put option #Patht_0t_1t_2t_3 11.01.091.081.34 21.01.161.261.54 31.01.221.071.03 41.00.930.970.92 51.01.111.561.52 61.00.760.770.90 71.00.920.841.01 81.00.881.221.34 Strike = 1.1 Stock price paths

18. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 18 American put option #Patht_0t_1t_2t_3 11.081.34 21.261.54 31.071.03 40.970.92 51.561.52 60.770.90 70.841.01 81.221.34 Strike = 1.1

19. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 19 American put option #Patht_0t_1t_2t_3 11.08 1.34 0.00 2 31.07 1.03 0.07 40.97 0.92 0.18 5 60.77 0.90 0.20 70.84 1.01 0.09 8 Strike = 1.1

20. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 20 American put option #Patht_0t_1t_2t_3 11.08 1.34 0.00 2 31.07 1.03 0.07 40.97 0.92 0.18 5 60.77 0.90 0.20 70.84 1.01 0.09 8 XY 1.080.00*0.94 -- 1.070.07*0.94 0.970.18*0.94 -- 0.770.20*0.94 0.840.09*0.94

21. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 21 American put option XY 1.080.00*0.94 = 0.000 1.070.07*0.94 = 0.0658 0.970.18*0.94 = 0.1692 0.770.20*0.94 = 0.1880 0.840.09*0.94 = 0.0846 f 2 (x) = -1.070 + 2.983 x – 1.813 x 2 Basis functions: 1, X, X 2

22. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 22 American put option #Patht_0t_1t_2t_3 11.08 0.02 1.34 0.00 2 31.07 0.03 1.03 0.07 40.97 0.13 0.92 0.18 5 60.77 0.33 0.90 0.20 70.84 0.26 1.01 0.09 8 f 2 (x) Decision 0.0369 Continue 0.0461 Continue 0.1176 Exercise 0.1520 Exercise 0.1565 Exercise

23. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 23 American put option #Patht_0t_1t_2Cash flow if we follow decision at time t 2 11.01.091.08(continue) 0.00 21.01.16 31.01.22 41.00.930.97(exercise) 0.13*0.94 51.01.11 61.00.760.77(exercise) 0.33*0.94 71.00.920.84(exercise) 0.26*0.94 81.00.881.22(continue) 0.00

24. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 24 American put option XY 1.090.00 0.930.13*0.94 = 0.1222 0.760.33*0.94 = 0.3102 0.920.26*0.94 = 0.2068 0.880.00*0.94 = 0.0000 f 1 (x) = 2.038 – 3.335 x + 1.356 x 2 Basis functions: 1, X, X 2

25. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 25 American put option #Patht_0t_1t_2 11.09 0.01 2 3 40.93 0.17 5 60.76 0.34 70.92 0.18 80.88 0.22 f 1 (x) Decision 0.014 Continue 0.109 Exercise 0.286 Exercise 0.117 Exercise 0.153 Exercise

26. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 26 American put option t_1t_2t_3 1.091.081.34 1.161.261.54 1.221.071.03 0.930.970.92 1.111.561.52 0.760.770.90 0.920.841.01 0.881.221.34 t_1t_2t_3 000 000 001 100 000 100 100 100 t_1t_2t_3 0 0 0.7*0.94 3 0.17*0.94 0 0.34*0.94 0.18*0.94 0.22*0.94 Stopping rule Cashflows Stock price paths

27. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 27 American option Advantages: fast computation (linear regression) price dynamics is arbitrary Problem: Option price is overestimated (foresight bias) Reason:regression coefficients are not adapted to the filtration of the price process (we use too much information from the future!)

28. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 28 American option Further improvements: 1. Use two independent sets of MC paths in the Longstaff-Schwartz algorithm: one to estimate the exercise criterion one to apply the criterion in pricing 2. Use one set of MC paths in the Longstaff-Schwartz algorithm and adjust the option value with the estimation of the foresight bias (Christian Fries 2006) 3. Mix Longstaff-Schwartz idea with stopping time optimization

29. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 29 We are currently hiring Analytic Modellers Ph.D. or near-Ph.D. in a quantitative area Academic research experience preferred Finance background not needed but helpful Must love analytic thinking Morgan Stanley Mathematical Modeling Center Deak Ferenc u. 15 1052 Budapest www.morganstanley.hu

30. Only Source / Footnotes below this line Guide @ 2.68 Guide @ 1.64 Guide @ 1.95 Subtitle Guide @ 2.64 Guide @ 2.80 Only Source / Footnotes below this line Guide @ 0.22 Guide @ 4.69 prototype template (5428278)\print library_new_final.ppt 6/1/2014 30 This material was prepared by sales, trading, banking or other non-research personnel of one of the following: Morgan Stanley & Co. Incorporated, Morgan Stanley & Co. International Limited, Morgan Stanley Japan Limited, Morgan Stanley Capital Group Inc. and/or Morgan Stanley Dean Witter Asia Limited (together with their affiliates, hereinafter Morgan Stanley). Unless otherwise indicated, these views (if any) are the authors and may differ from those of the Morgan Stanley fixed income or equity research department or others in the firm. 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