Replicating a One-Step Binomial Call to Find Arbitrage Trades

# Arbitrage strategies in Rubinstein's binomial tree one-step


# Arbitrage strategies in Rubinstein's binomial tree one-step












Suppose that the current stock price is $S_0=20$ and the call option price with no arbitrage is $c=0.633$. Knowing that the expiry stock price can be $S_T=22$ with call option price $1$ or $S_T=18$ with call option price $0$, which strategies can be realized if the current call option is $>0.633$ and $<0.633$?

I thought that if it was $c=0.62$, now, I could:

- buy the call at $c$

- sell $∆=0.25$ shares at $S_0$

- invest $S_0\Delta-c$ at the risk free rate $r=$ 12% for $3$ months.

But what actions do I have to take at expiry? Considering that I have to return the shares and I can cash the profit $(S_0\Delta-c)e^{rT}$.

And if it was $c=0.65$?

## Answer by user217285 (score 2)

https://quant.stackexchange.com/a/37410

Let's first check that 0.633 is the call option price. The risk-neutral probability $p^*$ of an up-tick in the stock is computed by assuming the stock earns a risk-free rate of return: $$20 = e^{-0.03}(22p^* + 18(1-p^*)) \qquad \Rightarrow \qquad p^* = 0.65227.$$ The price of the option is the risk-neutral expected payoff, discounted at the risk-free rate. In this case, the option has payoff 1 in an up-tick and payoff 0 in a down-tick, so $$c = e^{-0.03}p^* = 0.633.$$ The delta-hedge ratio needed for a replicating portfolio is the ratio of the change in the option price to the change in the stock price: $$\Delta = \frac{1 - 0}{22-18} = 0.25.$$ If the option is trading at $C < 0.633$, then you should be able to capture the $0.633 - C$ spread by employing the buy-cheap sell-expensive strategy:

- Buy the option at $C$

- Sell $\Delta = 0.25$ shares of the stock

If $C = 0.62$, then selling $\Delta$ shares of the stock and buying the option gives you an initial cash position of $$0.25*20 - 0.62 = 4.38,$$ which grows to $4.38e^{0.03} = 4.5133$ at expiry. We now consider two cases:

- $S_T = 22$: the option has payoff 1, and the short position has payoff $-0.25*22 = -5.50$, so the total payoff is $1 - 5.50 + 4.5133 = 0.0133$.

- $S_T = 18$: the option has payoff 0, and the short position has payoff $-0.25*18 = -4.50$, so the total payoff is $0 - 4.50 + 4.5133 = 0.0133$.

If the option is trading at $C > 0.633$, then again employ the buy-cheat sell-expensive strategy. The main point is that you can completely replicate the payoff of the option using just a portfolio of stock and cash; by the law of one price, the cash required to set up the replicating portfolio is the price of the option.