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A binary option pays an amount of money if an event takes place and zero otherwise. Binary options are usually used to insure portfolios against large drops in the stock market. On March 25, 2021 the price of a binary option that pays one dollar if the S&P500 falls by more than 10% (e.g., -10% and below) within one year from today is 0.30. At the same time, the price of a binary option that pays one dollar if the S&P500 increases by more than 10% (e.g., +10% and above) within one year from today is 0.20. Using a non-arbitrage argument, derive the price of a binary option that pays $1 if the S&P500 is within [-10%, 10%] one year from today.

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  • $\begingroup$ And what have you tried so far? $\endgroup$ – simsalabim Apr 3 at 18:58
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I think you can simply construct a portfolio equivalent to the double digital option (let's call it $DO$) you want to price, that qualitatively will look like this (dotted lines):

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The replicating portfolio should contain:

  • a zero-coupon bond expiring in one year (current value ($t=0$) = \$ $\exp(-r \cdot (1 - t)\text{ years})$);
  • a shorted digital valid if S&P falls by 10% or more (current value ($t=0$) = \$ $-0.30$);
  • a shorted digital valid if S&P increases by 10% or more (current value ($t=0$) = \$ $-0.20$).

So your option $DO$ will be worth as much as this portfolio, that is $$ DO(t) = \exp(-r \cdot (1-t)) - 0.5 $$

In a year, if the S&P stays within $[-10 \%, \, 10\%]$ of its current value, then the two digitals in your portfolio will have gone out of money, and the portfolio will yield \$ $ \exp(0) - 0 - 0 = 1 $. On the contrary, if the S&P finishes outside of that range, then one of the digitals will have to pay \$ $ 1 $ in full, so your portfolio will be worth \$ $ \exp(0) - 1 = 0 $.

This is exactly the behaviour you want to simulate for the option you are asked to replicate by the exercise.

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I will assume that the interest rate is 0. The price of a binary option is then the same as the risk-neutral probability that the event will occur $$\mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{S(T)\geq K}\right]=\mathbb{Q}\left[S(T) \geq K\right]$$ Denote the current spot price $s$. You need to find $$\mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{0.9 s< S(T) <1.1 s}\right]=\mathbb{Q}\left[0.9s < S(T) <1.1s\right]$$ You know that $$\mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{S(T)\leq 0.9s}\right]=\mathbb{Q}\left[S(T) \leq 0.9s \right]=0.2$$ and $$\mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{S(T)\geq 1.1s}\right]=\mathbb{Q}\left[S(T) \geq 1.1s \right]=0.3$$ and in general we have that $$\mathbb{Q}\left[0.9s < S(T) < 1.1s\right]=1-\mathbb{Q}\left[S(T) \leq 0.9s \right]-\mathbb{Q}\left[S(T) \geq 1.1s \right]$$ So in this case we have $$\mathbb{Q}\left[0.9s < S(T) < 1.1s\right]=1-0.2-0.3=0.5$$

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