**Short Selling**

Short selling means the selling of an asset that is not owned by the seller or is borrowed.

For Example, you can ask a broker to short 1000 shares of company A. The broker in turn will borrow 1000 shares from a client and sell them in the market. Eventually you would want to close out the position by buying back 1000 shares. The broker will then buy shares from the market and give them back to the client he/she had originally borrowed from. You will earn profits if the price of the share has decreased and suffer losses if the price has increased.

Let’s say that the price of one share is $110 in April when the investor ordered a short selling. Now, the shares pay dividend of $1000. The short position was finally closed out in July when the price of a share was $100. (since you are selling the shares, you will have to pay the dividend to the party taking the long position when the dividends are announced)

So, the total cash flow is (110 × 1000) ─ 1000 ─ (100 × 1000) = $9000

**Forward Price for an Investment asset**

An investment asset come with no income. Non-dividend paying stock and zero-coupon bonds are examples.

**When forward price of a stock is not in line with the spot price**

Consider a long forward contract to purchase on a non-dividend paying stock in 6-months.

Spot Price of the stock = $50

Risk free interest rate = 6% p.a.

forward price is $54.

Spot Price of the stock = $50
Risk free interest rate = 6% p.a. forward price is $54. |
Spot Price of the stock = $50
Risk free interest rate = 6% p.a. forward price is $47. |

1. A person can borrow $50 at the risk-free interest rate | 1. A person can short one unit (short selling) and receive $50 today |

2. Buy the stock for $50 | 2. Invest the $50 at the risk-free interest rate for 6 months |

3. Enter into a short position to sell the stock at $54 after 6 months | 3. Enter into a long position to buy the share at $47 |

4. After 6 months, sell the stock for 54 and closes the position | 4. After 6 months, receive 50e^{0.06* 0.5} = 51.5227 |

5. pay off the debt which is now 50 e^{0.06* 0.5} = 51.5227 |
5. Closes the position and pay $47 |

6. Arbitrage = $54 – 51.5227 = 2.4773 | 6. Arbitrage = $51.5227 – 47 = 4.5227 |

Thus, it is evident that the forward price has to be $51.5227 for there to be no arbitrage.

**Notations**

** T **: time until delivery in a forward or futures contract

** S_{0}** : Price of the asset underlying the forward or the futures contract

** F_{0} **: forward or futures price today

** r **: zero-coupon risk-free rate of interest per annum (continuous compounding)

The forward price **F _{0} = S_{0}e ^{r T}**

- If F
_{0}> S_{0}*e*^{r T}, arbitrageurs can buy the asset and short forward contracts on the asset. (you can sell it at a price higher than the market price later) - If F
_{0}< S_{0}*e*^{r T}, arbitrageurs can short the asset and enter into long forward contracts on the asset. (You can buy it at a price lower than the market price later)

Now, **for assets with known income** (coupon paying bonds or dividend paying stock)

The forward price **F _{0} = (S_{0 }─ I) e ^{r T}**

Where I is the income (coupon or dividend).

**Assets with known yield**

There are assets which provide a known yield. This means that the fixed income is in terms of percentage of the asset’s price when the income is paid.

If ** q **is the average yield per annum on an asset during the life of a forward contract, then

The forward price is **F _{0} = S_{0}e ^{(r ─ q) T}**

**Valuation of a forward contract**

**Notations**

** T **: time until delivery in a forward or futures contract

** S_{0}** : Price of the asset underlying the forward or the futures contract

** F_{0} **: forward or futures price today

** r **: zero-coupon risk-free rate of interest per annum (continuous compounding)

*K ***: **delivery price for a contract at the delivery time

** f** : value of the forward contract today

*f*** = (F _{0} ─ K) e ^{─rT}**

**Forward and Future contracts on currencies**

Let’s say you have 100 units of foreign currency. In how many ways can you convert it into dollars? Let’s say F_{0} is the rate of exchange at time T and S_{0} is the rate of exchange today

- Investing it for T years at r
_{f}(risk free rate in the home country) and enter into a short position in a forward contract to sell the amount realised at time T.

The amount (in dollar value) realised at time T is [100e^{r}_{f}^{T} F_{0}] (because the currency is exchanged at time T)

- Convert it into dollars and invest it for T years at rate r (risk free rate in US market)

The amount (in dollar value) realised at time T is [100e^{rT} S_{0}]

Now both the strategies should have the same result and hence,

100e^{rfT} F_{0} = 100e^{rT} S_{0}

**F _{0} = S_{0 }e ^{(r-r}_{f}^{) T}**

Which is also known as the interest rate parity.