*Note: An error in the formula in Approach I in the earlier version of this article was pointed out to us by Soumaydip Sarkar of ICICI Bank, Mumbai , India . *

A pay later option is one of the simplest structured products that exist in the market today. However, given the nature of the product and the high premium it is not a popular product. Yet, it may be interesting to look at this product (we recently encountered a similar type of product in one of the term sheets).

Though there are quite a few variations in this product itself, in its basic form a pay later option is one where the buyer of the option pays the premium (cost of the option) to the seller only if the option finishes in the money. If the option finishes out of the money then the buyer pays nothing to the seller. Given this kind of a payoff, a pay later option displays the characteristics of both a vanilla as well as digital (binary) option. Even though a pay later option may look complex at a first glance, its pricing is pretty trivial in a closed form framework.

**Approach I **

Say the price of a pay later option is . In a Black-Scholes framework, we have the value of a vanilla option (either a call or a put) as a function of spot, strike, volatility of the underlying, time to expiry and the interest rate.

We are pricing the pay later option at time t = 0 and making the statement that if the vanilla option has any value at maturity, i.e. then at the buyer will pay the price otherwise he will not pay anything. The vanilla option has a certain probability of finishing in the money, which is given by in the Black-Scholes formula. And the probability (of finishing in the money) times the price of the pay later option at t = 0 is the value of the vanilla option at t = 0. But since the seller of the option has to wait until to get the premium (only if the option finishes in the money) the value needs to be adjusted by the discounting factor. Therefore,

Where, **S ** is the underlying, **K ** is the strike price, is the interest rate and is the volatility of the underlying. Given the above,

And in standard Black-Scholes notation is the probability of finishing in the money.
The premium of the pay later option, the price, is more than that would be paid by the buyer of the corresponding (underlying) vanilla option. The increase in the price is given by the factor of **exp(r*T) **. This makes intuitive sense because the price of a pay later option is paid by the buyer of the option at the end of the period (upon maturity).

We can alternatively write the price of the pay later option as:

**Approach II **

The way the payoff of a pay later is constructed it can be easily deduced that a PayLater option is a long Vanilla option and a short times a digital option, where is the price of the PayLater option. At any point in time t, the payoff of a PayLater option is given by:

At initiation of the contract, i.e. t = 0, the value of a PayLater option is zero. One needs to differentiate between the option value and the option price. At t = 0, the value of the option is zero. If we set the payoff at t = 0 as zero then we get

From the above it follows that:

Once again we get the same value for the pay later option.

*Reference: ** Approach II is taken from ***Uwe Wystup's ** excellent work **FX Structured Products **.

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