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Pricing Swing Options Alex, Devin, Erik, & Laura

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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

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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

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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.

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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)

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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

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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) )

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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)

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

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

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

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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)

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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

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

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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

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Algorithm: Swing Option – The Bad

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Algorithm: Swing Option – The Ugly

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Algorithm: Option Price vs. Refraction Time and Time Steps

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Results…

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Price of Various Put Options

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Swing Option Price vs. Stock Price

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Swing Option Price vs. Strike Price

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Swing Option Price vs. Maturity

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Swing Option Price vs. Refraction Time

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Greeks: Delta

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Greeks: Gamma

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Option Price vs. Maturity and Volatility

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Option Price vs. Exercise Rights and Refraction Time

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Option Price vs. Stock Price and Maturity

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