Options Valuation and Pricing Guide

This guide is accompanied by aninteractive spreadsheet(in order to modify the values, copy it with File -> Make a copy).

Options on Quedex should be valuated using Black’76 model. Since we are pricing the inverse option contracts, we have to switch option type and invert prices. The formula for the price of an option with a given Strike is then as follows:

Put Option Price = (1/Futures Price) * N(d1) - (1/Strike) * N(d2) Call Option Price = (1/Strike) * N(-d2) - (1/Futures Price) * N(-d1)

with

d1 = [ ln(Strike/Futures Price) + T*(Sigma ^2)/2 ] / [ Sigma * sqrt(T) ] d2 = d1 - Sigma * sqrt(T)

where

The good thing about having such a formula is that we can calculate the sensitivities (Greeks) of the price with respect to different market factors. Knowledge of these allows us to built up more sophisticated strategies.

Example

Assume we want to calculate the price of a call option with given parameters:

  • Futures Price = $10000,
  • Strike = $11000,
  • T = 7 days = 0.01917808219 of a year,
  • Quantity = 10000.

To make calculations easier assume that sigma = 100%. In general sigma is taken from the historical data as a standard deviation of historical log returns. We therefore have

d1 = 0.3681391023, d2 = 0.2296541493.

Now we plug these values into the main formula and receive

Option Price = 0.00000156 BTC, Total Option Premium = Option Price * Quantity = 0.01559730 BTC.

Delta

In the Web App you can find delta parameter for the option at the order book. Delta displays the sensitivity of changes of the option value with respect to the change of the Spot Price of underlying asset. Delta of the call option ranges in value from 0 to 1. If the option is out-of-the-money (current Spot Price is lower than Strike Price) than the values are smaller than 0.5. On the other hand, if the option is in-the-money (current Spot Price is lower than Strike Price) than values are bigger than 0.5. In case of put options delta has values from the interval (-1; 0) and if the option is out-of-the money, then values are smaller than -0.5. If the option is in-the-money, then values are bigger than -0.5.

Mathematically, delta is the derivative of the portfolio value function with respect to the Spot Price. It is popular to build portfolios, which are delta-neutral. This means, that the value of the portfolio doesn’t change, if the Spot Prices changes.

Example

Assume that there is a call option with:

  • Strike Price = $10,000,
  • Spot Price = $10,000,
  • Volatility = 100%,
  • Maturity = 1 Month

You take long position in 100 contracts (Quantity = 100). In the Black'76 Option Pricingmodel delta of the call option with inverse notation is computed as a - (N(d1) - 1), where N(d1) - 1 is delta of the ordinary put option. Hence, in our example delta of a call option is equal to 0.47.

In order to build delta-neutral portfolio you can use a futures contract. In the case of long futures contract delta is equal to 1 (and, consequently, short futures is equal to -1). Hence, you have to sell 47 futures contracts (Quantity = 47) with the same maturity and futures price. Indeed, we see that you hold delta-neutral portfolio:

Delta = Call_Quantity * 0.47 - Futures_Quantity * 1 = = 47 - 47 = 0.

Vega

Vega is another parameter which is of high importance. It indicates the sensitivity of the value of derivative with respect to the changes of volatility of the underlying asset. Mathematically, it is a derivative of value with respect to the volatility parameter. From Black-Scholes formula one can observe that option vega is always positive. This implies that the higher the volatility is, the higher the value of the portfolio is if you take long position. Conversely, if you take a short option position, the value of your portfolio decreases with the volatility. rsely, if you take a short option position, the value of your portfolio decreases with the volatility.

In the above example, after creating delta-neutral portfolio investor has exposure only on the changes of volatility of BTCUSD exchange rate.