Static Replication of Exotics

A number of exotics can be perfectly hedged by vanillas in a semi-static way (semi because unwinding is usually required when barriers are triggered).

P(K) stands for vanilla put option with strike K
C(K) stands for vanilla call option with strike K
BP(K) stands for binary put option with strike K
BDOT(K) stands for binary down one-touch option with strike K and postponed rebate R

The following assumes that spot equals forward equals zero.

Down-Out Call (H = K)
Initial hedge: charge S0-K premium and borrow K, buy a unit of stock
When barrier is hit: sell stock to receive K and pay back loan; option terminates worthless
If barrier never hit: sell stock to receive ST ; pay out max(S-K,0) and pay back loan
Note: in this case where H = K, there is no optionality. 

Binary Up One-Touch with Rebate
The contract pays a fixed rebate R when stock price hits H from below before expiry.
Initial hedge: charge S0*R/H premium and use it to buy stock
When barrier is hit: sell stock to receive R and pay it out
If barrier never hit: sell stock and keep ST as profit

The following assumes that spot equals forward (for equity options this means interest rate = dividend yield).*
 
Binary Down One-Touch with Postponed Rebate
The hedge above is an overhedge in the sense that if the barrier is never breached, the writer gets to pocket the stock liquidation value at expiry. Here we try to more accurately hedge a contract that pays a fixed rebate R at expiry if stock price hits H from below before expiry.
Initial hedge: charge -R*P(H)/H+2*R*BP(H) premium to buy 2R units of binary put and sell R/H units of vanilla put with strike H
When barrier is hit: unwind to receive exp(-r(T-t))*R*[N(-d2)-N(-d1)], which, because S = H, equals exp(-r(T-t))*R. Put the amount into a money market account until expiry.
If barrier never hit: everything expires to give zero value

Down-In Call (H = K)
Initial hedge: charge P(H) premium and buy one put with strike H
When barrier is hit: sell put and buy call with strike H (self-financing by put-call parity)
Expires with barrier hit: pay out max(S-K,0), which is financed by the hedging call
If barrier never hit: put worthless; pays nothing out

Down-In Call (H > K)
Initial hedge: charge P(K)+(H-K)BDOT(H) premium to buy one put with strike K and (H-K) BDOT with strike H
When barrier is hit: sell put and BDOT to buy a call with strike K**
Expires with barrier hit: pay out max(S-K,0), which is financed by the hedging call
If barrier never hit: put and BDOT worthless; pays nothing out

Down-In Call (H < K) 
Initial hedge: charge (K/H)*P(H^2/K) and buy (K/H) units of put with strike (H^2/K)
When barrier is hit: sell put and buy call with strike K (self-financing by put-call symmetry)
Expires with barrier hit: pay out max(S-K,0), which is financed by the hedging call
If barrier never hit: put worthless (it is OTM because the strike H^2/K < H and we know that stock never went below H); pays nothing out


* If this assumption is relaxed, we will have no perfect hedge but only upper and lower bounds. See reference.

** Sketch of proof: (first argument is stock price; second argument is strike)
When barrier is hit (S = H),
    P(H,K)+(H-K)BDOT(H,H)
= P(H,K)+(H-K)[2BP(H,H)-P(H,H)/H]
= exp(-r(T-t))KN(...)-exp(-y(T-t))HN(...)+2exp(-r(T-t))HN(...)-2exp(-r(T-t))KN(...)
  - exp(-r(T-t))HN(...)+exp(-y(T-t))HN(...)+exp(-r(T-t))KN(...)-exp(-y(T-t))KN(...)
= C(H,K)

Reference:
Gatheral "The Volatility Surface", Ch. 9
Bowie and Carr 1994 "Static Simplicity"