1. Pricing Swing Options Alex, Devin, Erik, & Laura

2. Intro: Swing Options  Holder has right to exercise N times during period [T 0, T]  When N = 1, identical to American Option  Separated by minimum refraction time τ R  Prevents multiple exercising at one time instant  If expected payoff is not optimal, one should not exercise  However, waiting too long prevents use of all exercise rights  At a given node, one may:  a) Exercise, collect payoff with (N – 1 ) times left to exercise after τ R  b) Not exercise, collect no payoff but maintain ability to exercise at any moment  Bounds  Lower: Series of European Options  Upper: Series of American Options

3. Intro: Energy Applications  Also referred to as “Take-or-Pay”, “Variable Volume”, or “Variable Take” Options  Usually a Dual Option  Complex patterns of consumption and limited storability of commodities create need to hedge for pricing and demand spikes  Allow holder to repeatedly choose to receive or deliver a specified amount of commodity  A penalty function may be applied if the exchanged amount is outside the set boundary  When the penalty function is non-zero, the Swing Option can no longer be approximated or bounded by American or European Options  A seasonality factor may be applied to create a mean-reverting process

4. Intro: Finance Applications  Relatively new to Stock Market  Similar to Flexi-Options which hedge against interest rate spikes  Similar to Multi-Callable Options  In contrast to Energy Market, “Bang-Bang” Control  When the market suggests that it is best to exercise, you will exercise as much as possible  Not limited by season, weather, storage capacity, etc.

5. Intro: Pricing Methods in Literature  Dynamic Programming  Binomial Forest/Multi-Layered Tree  Our method  Jaillet, Ronn, & Tompaidis (2003)  Sequence of Multiple Optimal Stopping Problems  Solved by Hamilton-Jacobi-Bellman Variational Inequalities (HJBVI)  Dahlgren & Korn (2003)  Above method reduced to cascade of Stopping Time Problems  Finite Element Analysis  Wilhelm & Winter (2006)

6. Theory: Swing Call Options  Bounded above by strip of N American options  Bounded below by a strip of N European options  For a Swing Call with N exercise rights:  Same price as a strip of N European options with maturities T i = T – (i – 1 ) τ R, i = 1,..., N, where τ R is the recovery period

7. Theory: Swing Put Options  Let P N (S t ) = the price of a swing option with N rights where S t = the price of the stock at time t  Let g(S t ) = (K – S t ) + denote the payoff function of the swing put where K is the strike price  Let{ θ i }, i = 1,..., N, t ≤ θ i ≤ T, θ i+1 + τ ≤ θ i be the set of allowable optimal exercise times  The price of a swing option is given by: (For proof of existence see M. Dahlgren and R. Korn, The Swing Option On The Stock Market, International Journal of Mathematical Finance Vol. 8. No.1 (2005) )

8. Theory: Swing Put/Call Options  Previous formula works for Call Options but the set of optimal exercise times will be θ i = T-(N-i) τ R, i = 1,..., N  For a dual-style swing option g(S t ) = abs(S t -K)

9. Algorithm: Naïve Pricing of American Call  F(0,0) is the option price  Can be implemented directly, no real thinking involved

10. Algorithm: Naïve Pricing of American Call  F(0,0) is the option price  Can be implemented directly, no real thinking involved  TOO SLOW

11. Algorithm: Naïve Pricing of American Call  We compute things more than once  Complexity is O(2^N)

12. Algorithm: Dynamic Programming  Identical subproblems should be solved only once  Work backwards, save intermediate results  This is just how one would price an option by hand  Complexity is O(N^2)

13. Algorithm: Overview of Implementation  Recursive computation converted to iterative computation  Results stored in a giant (n+ 1 ) x (n+ 1 ) array  Work backwards, from the (known) values to our desired price

14. Algorithm: Swing Option  Much messier!  Fundamental principles of pricing the American Call still apply  Naïve approach is NOT computationally feasible

15. Algorithm: Swing Option – The Good  We can directly translate this into an iterative problem, working backwards and saving intermediate results  Complexity is O(N^3 * C * D)  For the most part, this is good enough

16. Algorithm: Swing Option – The Bad

17. Algorithm: Swing Option – The Ugly

18. Algorithm: Option Price vs. Refraction Time and Time Steps

19. Results…

20. Price of Various Put Options

21. Swing Option Price vs. Stock Price

22. Swing Option Price vs. Strike Price

23. Swing Option Price vs. Maturity

24. Swing Option Price vs. Refraction Time

25. Greeks: Delta

26. Greeks: Gamma

27. Option Price vs. Maturity and Volatility

28. Option Price vs. Exercise Rights and Refraction Time

29. Option Price vs. Stock Price and Maturity

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