The dynamics of option pricing

options trading option pricing

Like we established in the introductory article, an option represents a right with no obligations for the option holder. Also, we saw that the seller of an option runs a much greater risk than the buyer, which is why a premium passes from buyer to seller. To liken it to our everyday lives, we can observe that the valuations the options seller must make are comparable to the considerations an insurance company makes when pricing an insurance policy – the two parties have to agree on the size of the premium.  This is where it gets challenging, so let’s have a look at the considerations that must be made.

We can start by establishing that the premium can never be negative, since payoff at maturity is zero or positive. Digging deeper, the size of the premium must naturally depend on the choice of strike, the time to maturity and the expected movements in the underlying FX rate. However, the price of options also has much to do with our abilities to replicate the payoff.

Starting off very simply, if we choose to buy a call option and sell a put option with the same strike and maturity we end up with a payout profile at maturity equal to that of a long FX forward outright contract, with a difference attributable to funding cost on the option premium which is paid up front (see Figure 1).

options trading option pricing premium

So a call option can always be replicated by the comparable put option combined with a forward contract and vice versa. This relationship is known as the put–call parity. Given the price of an equivalent option, it is possible to replicate the payoff thereby inducing a price on the option being valued. Replication using the underlying forward or spot is the main cornerstone used to price options and is the foundation of the famous Black-Scholes pricing formula. The Black-Scholes model and its extension to FX markets, the German-Kohlhagen model, is based on trading the underlying. We will not go into the details about this here, but we will look into it in upcoming articles regarding hedging. Great intuition can, however, be made of the Black-Scholes model and the general pricing of options. The Black-Scholes model prices the option using strike, spot, time to maturity, interest rates of the domestic and foreign currencies in the underlying, and finally volatility.

Starting with the placement of the strike, if we look at a call option, the higher the strike the less likely it is that the option yields a payoff at maturity. The call option is therefore cheaper for higher strikes. In the option market, the measure of how likely the option is to have a positive payout at maturity is called moneyness. In general, call options are said to be in-the-money if the strike is below the forward of the same maturity and out-of-the-money if the strike is above. The opposite is true for put options. The option with strike equal to the forward is said to be at-the-money.  Notice that the in- and out-of-the-money term is with respect to the FX forward outright and therefore is implicitly a function of the interest rates of the domestic and foreign currencies in the underlying (given that the FX forward outright rate is calculated simply as a product of the FX spot rate and the ratio of discount factors from each of the two currency yields over the respective time period).

The value of being in - or out-of-the-money of course also depends on the time to maturity. If we think of options like insurance, it becomes easy to understand that the longer you have to expiry, the more the option will cost, just like with car insurance:  if you take it out for one day, it costs a little, if you take it out for one year, it will cost a lot more. Holding an out-of-the-money option is not a problem if maturity is far away, since the price will have time to move the option in the money. In the case of in-the-money options, it is also a benefit to have a long time to maturity, since downside risk is limited and there is a potential for larger gains. An interesting consequence of this is that everything being equal an option loses its value as time goes by, known as time decay or bleed. This is how earnings can be made by selling options.

Volatility, or the movement in the underlying FX rate, the last component of the Black-Scholes price, is therefore important for the value of the option. FX spot traders will know that some currency crosses such as AUDUSD tend to exhibit much larger moves up or down than others, for example EURCHF. Say we look at a call option in both the AUDUSD and EURCHF with the strike placed 10% higher than the current forward price. It is much more likely that the AUDUSD would have moved up more than 10% by the time of maturity. Furthermore, since the option has limited down side, the increased likelihood for a fall in underlying from higher volatility does not increase the risk in the option, therefore a higher volatility of the underlying always translates into a higher price of the option.  The same holds true for the put option. So again, to liken the theory of option pricing to our everyday lives, we could say that AUDUSD cross is like an 18 old boy taking out car insurance – he will have to pay a high premium, because he is perceived as “risky” to the seller of the insurance (option). Conversely, EURCHF might be likened to a middle aged female driver who has never had an accident – naturally, a lower premium will be asked for! Actually going back to the put-call parity, from which we know that a short put and long call with same strike and maturity should be equal to the forward plus premium funding, the price of the put and call with same strike and maturity must therefore increase with the same value from an increase in volatility.

Looking at the parameters used to price the option, we see that volatility is the great unknown. Therefore, the options market has adopted the terminology of trading “vols”.  If the market participants anticipate larger moves in the price of the underlying in the future, the volatility parameter is raised, raising the price of the traded options. Conversely if the market participants expect price to be more stable in future, then the volatility parameter is lowered. Note that the volatility parameter used is participants’ expectation about future volatility, not the historic volatility measured from past FX rate changes, as can be seen in Figure 2. The volatility parameter is referred to in the markets as implied volatility, since it is the volatility the markets imply will be observed in the underlying over the life time of the option.

options trading option pricing volatility

Readers may be familiar with the fact that the Black-Scholes model is considered by some to be flawed in that some of the assumptions don’t always strictly hold, e.g. that interest rates and volatility are known and constant over the term of the option or that the price changes in the underlying follow a normal distribution as opposed to a model which incorporates kurtosis, or the gap risk of significant immediate changes/jumps of prices from one level to another.  However, the pricing dynamics remain broadly the same and the Black-Scholes model is very robust compared to fancier academic models. Therefore, the Black-Scholes model remains the dominant way of pricing and quoting options for many reasons, not least because it provides all market participants a well-understood common ground reference framework for agreeing prices, which therefore lends support to a market like fx options with transparency and decent depth of liquidity.

Steffen Gregersen