Digital Options Pricing
Understand about digital options in less than 5 mins
In this article, we will understand about digital options, which are also known as binary options. A key feature of digital options is that they provide a discontinuous payoff to the holder.
There are 2 types of digital options:
- Cash-or-nothing
- Asset-or-nothing
We will first understand about the cash-or-nothing call option definition, its payoff and its pricing. Next, we will take a look at an asset-or nothing call option, its payoff and how it can be priced using portfolio replication methodology, by using a vanilla call and a cash-or-nothing call).
In this article, we will only consider European call options which do not provide any dividends.
Cash-or-nothing call option
Cash-or-nothing calls provide a fixed amount (say Q) at expiration if the asset ends up above the strike price and 0 otherwise.
Payoff of a cash-or-nothing call
The payoff profile for a cash-or-nothing call option looks like below:
Mathematically, it can be represented as:
Pricing a cash-or-nothing call
The price of cash-and-carry options is simple. Based on the payoff equation, we need to first calculate the expected cash amount received at expiration and then discount it back to its present value. To calculate expected cash amount at expiration, we need to multiply below terms:
- Cash amount received at maturity, Q
- Probability of receiving cash amount at expiration, N(d2)
Multiplying above terms gives us expected cash amount received at maturity, given by Q*N(d2). Next, we multiply this expected cash amount by discounting factor, given by exp{-rt}, to obtain the price of cash-and-carry option as below:
Asset-or-nothing call option
Asset-or-nothing calls provide the value of a stock at expiration if the asset ends up above the strike price and 0 otherwise.
Payoff of an asset-or-nothing call
The payoff profile for an asset-or-nothing call option looks like below:
Mathematically, it can be represented as:
Pricing an asset-or-nothing call
To price an asset-or-nothing call, we notice that the payoff of an asset-or-nothing call is the sum of a vanilla call and a cash-or-nothing call, given that the cash amount Q for cash-or-nothing call is equal to its strike price K. This is illustrated in the figure below.
So, the price of an asset-or-nothing call must equal the sum of cash-or-nothing call and vanilla call, with the cash price of cash-and-carry being K. This simple technique is also known as portfolio replication.
