Chapter 22 of Hull Details in Chapters of Wilmott, Howison, and Dewynne Binomial Tree for a Down and Out Option
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1 EXOTIC OPTIONS 10.1 Overview of Exotic Options Chapter 22 of Hull Details in Chapters of Wilmott, Howison, and Dewynne 10.2 Binomial Tree for a Down and Out Option Clewlow and Strickland, Chapter /13/04
2 10.1 OVERVIEW OF EXOTIC OPTIONS 10.2 Vanilla Options The European and American options that we have treated so far American puts are weakly path dependent (also calls with dividends) Exotic Options All other options Usually traded only over the counter (OTC) Common types: Binaries, Compounds, Choosers, Barriers, Asians, Lookbacks, Packages, Forward Starts Barriers, Asians, and Lookbacks are strongly path dependent.
3 10.3 Binary Options Cash-or-nothing call: Payoff is Q if S T > K, and 0 otherwise Can be valued with Black-Scholes machinery Can check that value is Q I( S K) T Qe rt ( t) Φ( d ) Prob. of being in the money I( u) 1 u Cash-or-nothing put: Payoff is Q I( K S T )
4 10.4 Compound Options Options on options, e.g., a call on a call Simplest case: vanilla options, European options can be valued with Black-Scholes machinery Call on a call, call on a put Put on a call, put on a put
5 Choosers 10.5 Regular chooser Right to purchase at time T 1, for amount K 1, either a call or a put, with exercise price K 2 at time T 2 These are strictly speaking path dependent, but can be valued by solving the Black-Scholes PDE (at least in the European case) Complex choosers K 2, put, T 2, put, K 2, call, T 2, call These prices/times may be different May give rights to sell (rather than buy)
6 Barrier Options 10.6 Weakly path dependent: Can be valued using only current values of S and t Up-and-in (see figure at right) Up-and-out Down-and-in Down-and-out H (cutoff) worth something Can use the Black-Scholes machinery (at least for European case) worthless
7 Asian Options 10.7 Strongly path dependent Payoff depends on history via an averaging Average Strike Call Payoff: max( "avg"( S ) K,0) Variations Period of averaging? Arithmetic or geometric average? Weighted or un-weighted average Discrete or continuous sampling for average? Complex Analysis Chapters 13 and 14 of Wilmott, Howison, & Dewynne t average over some period of time prior to expiry
8 Lookback Options 10.8 Strongly path dependent Payoff: max( "max"( S ) K,0) t some maximum of S t over some period of time prior to expiry Complex analysis (see W.H.D., Chapter 15)
9 10.9 Forward Starts Will start at some time in the future, usually strike defined so they are initially at the money Example: employee stock options with a vesting schedule Packages Portfolio of standard European calls, puts, forward contracts, underlying contract, cash
10 ADDITIVE BINOMIAL TREE PRICING OF AN AMERICAN DOWN-AND-OUT OPTION * H = Knock-out level. Option ceases to exist (value = 0 and pays nothing) if S t falls below H * Clewlow and Strickland (1998) pp
11 Equal Jump Size Additive Model = + = xu σ t ν ( t), ν r 2 σ 2 p = +, x = x 1 1 ν t u 2 2 x d u S = exp x+ j x + ( i j) x ( ) ij u d S = S exp x x ( ) i, j+ 1 ij u d
12 { set coefficients Trigeorgis } dt = T/N nu = r 0.5*sig^2 dxu = sqrt( sig^2*dt + (nu*dt)^2 ) dxd = -dxu pu = ½ + ½*(nu*dt/dxu) pd = 1- pu { precompute constants } disc = exp(-r*dt) dpu = disc*pu dpd = disc*pd edxud = exp( dxu dxd ) edxd = exp( dxd ) { Initialize asset prices at maturity N } St[0] = S*exp( N*dxd ) for j = 1 to N do St[j] = St[j-1]*edxud { Initialize option values at maturity } for j = 0 to N do if ( St[j] > H ) then C[j] = max( 0.0, St[j] K ) else C[j] = 0.0 next j
13 { step back through the tree applying the barrier and early exercise condition } for i = (N-1) downto 0 do for j = 0 to i do { adjust asset price to current time step } St[j] = St[j]/edxd if ( St[j] > H ) then C[j] = dpd*c[j] + dpu*c[j+1] { Apply the early exercise condition } else C[j] = max( C[j], St[j] K ) C[j] = 0.0 next i next j American_Down_and_Out_Call = C[0]
14 10.3 MULTI-DIM. BINOMIAL TREES * Applications: - Options on maximum or minimum value of a basket of equity indices - American spread option on the difference between two assets with payoff: max(0, S S K) 1, T 2, T * Clewlow and Strickland (1998) pp
15 Two asset case: Need two correlated GBM s ds = ( r q ) S dt +σ S dz ds = ( r q ) S dt +σ S dz corr( dz1, dz2) = ρdt Can do with a two-variable binomial lattice, but the convergence is very slow. Better to use other methods, e.g., implicit finite difference method of solving PDE. * Clewlow and Strickland (1998) pp
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