Pricing Swing Options

Alex, Devin, Erik, & Laura

Intro: Swing Options Holder has right to exercise N times during period [T0, T]

When N = 1, identical to American Option Separated by minimum refraction time τR

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 τR

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

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

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.

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)

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 Ti = T – (i – 1) τR, i = 1, ... , N, where τR is the recovery period

Theory: Swing Put Options Let PN(St) = the price of a swing option with N

rights where St = the price of the stock at time t

Let g(St) = (K – St)+ denote the payoff function of the swing put where K is the strike price

Let{ θi }, i = 1, ... , N, t ≤ θi ≤ T, θi+1+ τ ≤ θi 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) )

Theory: Swing Put/Call Options Previous formula works for Call Options but

the set of optimal exercise times will be θi = T-(N-i)τR, i = 1, ... , N

For a dual-style swing option g(St) = abs(St-K)

Algorithm: Naïve Pricing of American Call

F(0,0) is the option price Can be implemented directly, no real thinking

involved

Algorithm: Naïve Pricing of American Call

F(0,0) is the option price Can be implemented directly, no real thinking

involved TOO SLOW

Algorithm: Naïve Pricing of American Call

We compute things more than once Complexity is O(2^N)

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)

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

Algorithm: Swing Option

Much messier! Fundamental principles of pricing the

American Call still apply Naïve approach is NOT computationally

feasible

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

Algorithm: Swing Option – The Bad

Algorithm: Swing Option – The Ugly

Algorithm: Option Price vs. Refraction Time and Time Steps

Results…

Price of Various Put Options

Swing Option Price vs. Stock Price

Swing Option Price vs. Strike Price

Swing Option Price vs. Maturity

Swing Option Price vs. Refraction Time

Greeks: Delta

Greeks: Gamma

Option Price vs. Maturity and Volatility

Option Price vs. Exercise Rights and Refraction Time

Option Price vs. Stock Price and Maturity

FIN

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