# Pricing of Derivatives

## Futures

There are two basic concepts in finance: time-value of money and uncertainty about expectations. The two concepts are the core of financial valuations, including futures contracts.

cost-of-carry model is the most widely accepted and used for pricing futures contract

Cost-of-carry Model

Cost-of-carry model is an arbitrage-free pricing model. Its central theme is that futures contract is so priced as to preclude arbitrage profit. In other words, investors will be indifferent to spot and futures market to execute their buying and selling of underlying asset because the prices they obtain are effectively the same. Expectations do influence the price, but they influence the spot price and, through it, the futures price. They do not directly influence the futures price. According to the cost-of-carry model, the futures price is given by Futures price(Fp) = Spot Price(Sp) + Carry Cost(Cc) - Carry Return(Cr) (1)

Carry cost (CC) is the interest cost of holding the underlying asset (purchased in spot market) until the maturity of futures contract. Carry return (CR) is the income (e.g., dividend) derived from underlying asset during holding period. Thus, the futures price (F) should be equal to spot price (S) plus carry cost minus carry return. If it is otherwise, there will be arbitrage opportunities as follows

When F > (S + CC - CR): Sell the (overpriced) futures contract, buy the underlying asset in spot market and carry it until the maturity of futures contract. This is called "cash-and-carry" arbitrage. When F < (S + CC - CR): Buy the (under priced) futures contract, short-sell the underlying asset in spot market and invest the proceeds of short-sale until the maturity of futures contract. This is called "reverse cash-and-carry" arbitrage

Written by Surendra Agrawal

## Options

### Pricing in Discrete Time

#### One-Period Model

##### Pricing via Duplication

Since for all options, put-call-parity must hold, if three of the terms are known, the last one can also be found using the formula:

Put-Call-Parity: C + PV(X) = P + S

where

C = Price of Call Option

PV(X) = Present Value of Strike Price

P = Price of Put Option

S = Current value of the underlying asset

The put-call parity is only relevant for European options.