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We know that we can use the below equation in the link to replicate the payoff of a call option using stocks and bonds.

I am wondering what the equation would look like for a put option instead. In addition, if the price of the put option is not equal to this value, how would you arbitrage it to make riskless profit?

call option pricing equations here

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This is basic binary tree modelling which works for puts and calls. What you don't show is how to get Cu and Cd - the price of the call in the "up" and "down" underlying price cases. If you know how calls work and how to evaluate those, you can do roughly the same for puts (hint: for calls, the payoff value is MAX(0, S-K), for puts, the value is MAX(0, K-S))

if the price of the put option is not equal to this value, how would you arbitrage it to make riskless profit?

You can't always arbitrage just from the mispricing of a put. You also have to have another derivative (usually a call at the same strike and expiry) in order to pull off an arb.

But let's say a put is priced higher than its value, and a call at the same strike and expiry is fairly priced. You would sell the put, short the stock, and buy the call. Then your price exposure is zero:

  • If the stock goes below the strike, you make money on the stock, lose it on the put, and your call is worthless.
  • If the stock goes above the strike, you'd make money on the stock, lose it on the call, and the put would be worthless.

Your profit would be the difference between the put you sold and the call you bought, less any borrowing costs for shorting the stock and the difference between the strike and the current stock price.

In reality, though, if a put is "mispriced" in your model, it's because the market is using different values for u and d (and/or different discounting rates), and calls will also be "mispriced", negating this arbitrage opportunity. It's called "put-call parity".

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  • Good explanation. I'd add some minor details. (1) In order to do the put arb, you need a 'same series' call (same strike and expiry). (2) If the put is 'mispriced' and the same series call is 'mispriced' (with no possible arb) then are they really mispriced? (3) When you do the reversal (sell the put, short the stock, and buy the call), there are some infrequent events that alter what appears to be a risklsss arb: Change in rates, special dividend (you're short the stock), early assignment and Pin Risk (expiration at the strike). – Bob Baerker Feb 26 at 23:05
  • Do you think the same approach can be applied to path-dependent options like lookback options or ATM Asian options? Consider a 3-period binary tree model for example for a lookback put option. Pay off at L3 = max of Sn - S3, but I am not sure how the formula would apply to this case... – Birtan Feb 27 at 0:07

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