Model-Independent Barrier Option Pricing by Forward Replication
Summary
The document presents a no-arbitrage replication argument for a barrier option with strike equal to its barrier. Under continuous stock-price paths, zero rates, and no dividends, a forward struck at the barrier is worthless when the barrier is hit. The replicating strategy then closes the forward at that point; if the barrier is never reached, the forward is held to expiry and pays the same amount as the option.
Because the portfolio matches the option’s payoff in either case, the initial option value equals the forward’s initial value, stated as the difference between spot and strike. The argument can accommodate rates when the interest and dividend rates are equal, with the corresponding discount factor. It also allows some jumps if they cannot cross the barrier. These assumptions are essential; the document does not provide a general model-independent price without them.
Key ideas
- A forward struck at the barrier can replicate the option when the stock price moves continuously and rates and dividends are absent.
- If the barrier is hit, the forward is at the money and can be closed with zero value.
- If the barrier is never hit, the forward’s expiry payoff matches the option payoff.
- Equal interest and dividend rates allow a discounted version of the replication, while jumps must not cross the barrier.
Tags
Full text
# How to price barrier options with making in model-independent way? # How to price barrier options with making in model-independent way? I have to use simple no arbitrage arguments to find the price of a barrier option where initial stock price $S_0 = 100$ and barrier/strike $B = K = 80$. Here we don't assume geometric Brownian motion for the stock price. What should be the approach for this problem? The problem I'm facing is at time $T$ the stock price can be above the barrier or below the barrier. But in the second possibility, both the prices can be above the barrier. ## Answer by q.t.f. (score 1) https://quant.stackexchange.com/a/32775 You need a couple more assumptions and it becomes doable. (1) no arbitrage (2) no interest rates or dividends (3) spot price moves continuously Then there is a replication possible. Buy the forward struck at $K$ expiring at the same time as the barrier option. If the barrier is ever hit, the forward is at-the-money so has value zero. In that case sell the forward and have zero value at expiry. If the barrier is never hit, hold the forward until expiry, at which time it has value $S(T)-K$, the same as the barrier option. In every case the replicating portfolio gives the same final value as the barrier option, so it must have the same initial value as well. Thus the barrier option value is $S(0)-K $. Looking back thru the argument, we can actually weaken the assumptions slightly. There can be an interest rate, IF it is exactly equal to the dividend rate. Then the replication succeeds and the price is now $exp (-r T)(S (0)-K) $. And jumps in the spot are allowable, IF it is guaranteed that a jump will not cross the barrier. For instance, upward jumps would be fine.
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