One of the most elegant formulations in the field of derivatives has been the risk symmetry (or simply the symmetry) between a put option and a call option. For every European call option there is a symmetrical European put option that matches its risk, i.e. identical and offsetting gammas and vegas. The risk symmetry is established vis-à-vis the forward market (and not the cash market). This means that a put option can be transformed into a call option by operating it with a transformation in the strike price.

What does this actually mean?

It means that theoretically speaking, if F is the forward price of an asset then there exists a call option on one side of F with a strike price X that is equidistant (the distance of X from F) from F as that of a put option with a strike price on the other side of F. Please note that this statement pre-supposes that there is no skew (not a realistic assumption), i.e. symmetrical volatility on both sides of the forward F. Mathematically speaking,

Though the skew assumption is important this risk symmetry is extensively used by exotic option traders to do static hedging of barrier options. The above relationship also has a big implications in risk management of barrier options and risk reversals.

The above symmetry was first put forward in a formal treatment by Carr and Bowie in 1994 and Carr, Ellis and Gaupta in 1998 using quite a bit of sophisticated mathematics. To understand the above one can take refuge in Girsanov’s theorem, reflection principle of a geometric Brownian motion and other esoteric fields of mathematics.

But one can understand option risk symmetry from another angle as well. That is from a payoff function point of view. Let us look at the payoff function of a put option and a call option.

Call = max (0,S - X)

Put = max (0,X - S)

Thus the value of a European (as well as an American) call option with an asset price equal to S and strike price equal to X is equal to the value of an European put option with an asset price equal to X and a strike price equal to S. Using a payoff function we can therefore say,

Call(S,X) = Put(X,S)

A more detailed analysis of this method of analysis can be found in Bjerksund and Stensland (1993).

But realistically speaking, one cannot buy a put option with an asset price equal to X when the asset price is actually S (assuming X<>S, i.e. asset price different from strike price). Therefore, the payoff function above has limited appeal for traders and financial engineers.

However, we can rewrite the call payoff function in slightly different way to get a very powerful insight into the symmetry of a call and a put option.

Now, it is perfectly possible to buy a put option with a strike price equal to S^{2}/X given a certain asset price of S and a strike price of X (see Espen Gaarder Haug (1999) for a more detailed analysis).

Therefore, rewriting the payoff functions of Call option to match that of the put option creates the Put-Call transformation equation that we get using Carr's analysis (much more sophisticated mathematical analysis). Also, written in this way, it becomes a very useful tool for traders and practitioners to value many exotic options and perform risk management of those options.

__Note__: No matter how complex the mathematics or how difficult the concept the last word in option trading and valuation, according to us, is Nassim Taleb's book *Dynamic Hedging*. If you want to forget the math and yet understand all that is there to know about options valuation and trading then Nassim Taleb is the last word.

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