Put-call parity arbitrage I

say stock XYZ is trading at $31 we have a call option on stock XYZ with a $35 strike price it's trading at $8 we have a put option on stock XYZ with a $35 strike prices have the same strike price trading at $12 and they both have the same expiration over here and then finally there's a bond and this bond is unrelated to stock XYZ it's going to be a risk-free bond so it could be some type of a Treasury bill worth $35 at option expiration and you can buy it right now for $30 and the reason where you can buy it for less is you pay $30 you're going to get $35 in the future at option expiration so you're essentially getting interest on that bond so with these numbers is there a way to make risk-free money and to think about that let's think about the the put-call parity we learned that a stock plus a put at a given strike price and the put is a put on that stock is equal to it's going to have the same value at expiration as a call with the same strike price a call with the same underlying stock plus a bond a risk-free bond that's going to be worth that strike price at the expiration of these two options so since these are going to have since this is going to have the same value the same payoff in any circumstance as this at expiration they really should be worth the same thing but when you look at the numbers over here let's see if that works out the stock is trading at $31 so the stock is 31 the put option is trading at $12 so that that's plus 12 so this on the left-hand side right now if you had to buy it it's trading at 43 it's trading at $43 on the right hand side you have the call option it's trading at $8 so that's at $8 and then the bond is trading at $30 the bond is trading at 30 so this combination is trading at $38 at $38 so even though they have the exact same payoff at option expiration at in exploration the call plus the bond is cheaper than the stock plus the put so you have an arbitrage opportunity you have an opportunity to make profit from a discrepancy in price from two things that are essentially equal and what you always want to do is you always want to buy the cheaper thing you always want to buy the cheaper thing and you want to sell the more expensive thing especially when they are the same thing when they're going to have the exact same payoff in the future so you want to sell this so buying is pretty straightforward what does it mean to sell this over here well you could short the stock so you would short the stock you would short the stock that's essentially you're selling the stock and then you would you essentially are shorting a put option or another way to think it you could write a put option so you would short the stock plus plus right right a put and so what would happen there shorting the stock you're borrowing the stock and you are selling it so you're going to get 31 you're going to get 31 dollars from shorting the stock and writing the put means you literally are essentially creating a put option and then selling it to someone else and so you're going to get 12 dollars for that so you're going to get your $43 and then you're going to buy the call and the bond so you're going to spend $8 on the call $30 on the bonds you're going to spend $38 so you're going to spend $38 and so you're going to make a profit of $5 and what we're going to see in the next video is you make this profit upfront and no matter what happens to the stock price going forward you're able to rearrange things so that you everything else just cancels out and you can just keep your $5