Download Electricity Spot Prices - Banking - Lecture Slides and more Slides Banking and Finance in PDF only on Docsity! De 19 Electricity Spot Prices • Mean reversion. • Seasonality • Heteroscedasticity • Jumps. • GBM is not a good fit. • Ornstein-Uhlenbeck with jump component.(Xt=log(St)) PN JM IS O $ /M W H 20 30 40 50 60 01/01/1998 03/12/1998 05/21/1998 07/30/1998 10/08/1998 12/17/1998 dqdBdtXdX ttt ησθα ++−= )( Docsity.com D 20 Swing Option Example • Three swing rights call option with strike K=$40, maturity in 1 month, daily exercise. • 50,000 paths simulated from GOU with Poisson jumps, Xt = log(St), dXt = α(θ -Xt)dt + σdBt + ηdq - Volatility σ = 200% - Mean reversion level (MRL) θ = $40, is at-the-money. - Speed of mean reversion α = 50 (half life ≈ 5 days ). - Jump intensity parameter λ = 2 and amplitude η = 0.4. Docsity.com D 23 American Call days to expiration C oe ffi ci en t S ta nd ar d Er ro r 0 5 10 15 20 25 30 0 10 20 30 40 50 Standard Errors Docsity.com D 24 Swing Exercise Boundaries days to expiration C rit ic al P ric e 0 5 10 15 20 25 30 40 45 50 55 One Swing Right days to expiration C rit ic al P ric e 0 5 10 15 20 25 30 40 45 50 55 Two Swing Rights days to expiration C rit ic al P ric e 0 5 10 15 20 25 30 40 45 50 55 Three Swing Rights days to expiration C rit ic al P ric e 0 5 10 15 20 25 30 40 45 50 55 1st Swing 2nd Swing 3rd Swing Three Swing Boundaries Docsity.com D 25 Three Swing Rights Call Option Docsity.com D 28 Implicit Exercise Boundary weeks to expiration Bo un da ry 0 10 20 30 40 50 32 34 36 38 40 alpha=5, sig=50%, MRL=40 weeks to expiration Bo un da ry 0 10 20 30 40 50 34 36 38 40 alpha=25, sig=50%, MRL=40 weeks to expiration Bo un da ry 0 10 20 30 40 50 25 30 35 40 alpha=5, sig=100%, MRL=40 weeks to expiration Bo un da ry 0 10 20 30 40 50 25 30 35 40 alpha=25, sig=100%, MRL=40 Docsity.com D 29 Gas Storage • Goal: To value a lease of a gas storage facility for one year if optimally managed. • Possible capacity levels = 0, 1, 2, 3 units. • Each week can buy, sell, or hold 1 unit. • Proportional injection/withdrawal cost = c percent. Docsity.com D 30 week C rit ic al P ric e 0 10 20 30 40 50 2. 0 2. 5 3. 0 3. 5 Sell Levels Buy Levels Alpha=10, Sigma=100%, MRL=3 Docsity.com D 33 Optimal Operating Levels weeks to expiration sp ot p ric e 0 10 20 30 40 50 32 34 36 38 40 42 44 46 off = 0 low = 1 high = 8 weeks to expiration sp ot p ric e 0 10 20 30 40 50 32 34 36 38 40 42 44 46 off = 0 low = 2 high = 8 weeks to expiration sp ot p ric e 0 10 20 30 40 50 32 34 36 38 40 42 44 46 off = 0 low = 4 high = 8 weeks to expiration sp ot p ric e 0 10 20 30 40 50 32 34 36 38 40 42 44 46 off = 0 low = 6 high = 8 Docsity.com D 34 Conclusions • Extended LSM to do pricing and hedging of multiple early-exercise instruments. • Boundaries and prices seem reliable even if regression coefficients and coefficient SEs do not. Docsity.com
