Introduction to Derivatives Dr. Islam Azzam Assistant Professor of Finance Department of Management American University in Cairo iazzam@aucegypt.edu

Examples of Derivatives Futures Contracts Forward Contracts Options 3

Terminology Call Put Long Right to buy Pay premium Can Cancel Right to sell Short Obligation to sell Receive premium Obligation to buy 10

Terminology Future/Forward Spot Long Obligation to buy Buying the stock Short Obligation to sell Short Selling 10

Long Call on eBay Profit from buying one eBay European call option: option price = $5, strike price = $100, option life = 2 months 30 20 10 -5 70 80 90 100 110 120 130 Profit ($) Terminal stock price ($)

Short Call on eBay Profit from writing one eBay European call option: option price = $5, strike price = $100 -30 -20 -10 5 70 80 90 100 110 120 130 Profit ($) Terminal stock price ($)

Long Put on IBM Profit from buying an IBM European put option: option price = $7, strike price = $70 30 20 10 -7 70 60 50 40 80 90 100 Profit ($) Terminal stock price ($)

Short Put on IBM Profit from writing an IBM European put option: option price = $7, strike price = $70 -30 -20 -10 7 70 60 50 40 80 90 100 Profit ($) Terminal stock price ($)

American vs European Options An American option can be exercised at any time during its life A European option can be exercised only at maturity

Intel Option Prices (May 29, 2003; Stock Price=20.83) Strike Price June Call July Call Oct Call June Put July Put Oct Put 20.00 1.25 1.60 2.40 0.45 0.85 1.50 22.50 0.20 1.15 1.85 2.20 2.85

Options vs Futures/Forwards A futures/forward contract gives the holder the obligation to buy or sell at a certain price An option gives the holder the right to buy or sell at a certain price

Derivatives Markets Exchange traded Over-the-counter (OTC) Traditionally exchanges have used the open-outcry system, but increasingly they are switching to electronic trading Contracts are standard there is virtually no credit risk Over-the-counter (OTC) A computer- and telephone-linked network of dealers at financial institutions, corporations, and fund managers Contracts can be non-standard and there is some small amount of credit risk

Forward Contracts vs Futures Contracts FORWARDS FUTURES Private contract between 2 parties Exchange traded Non-standard contract Standard contract Usually 1 specified delivery date Range of delivery dates Settled at end of contract Settled daily Delivery or final cash Contract usually closed out settlement usually occurs prior to maturity Some credit risk Virtually no credit risk 16

Futures Contracts Available on a wide range of underlyings Exchange traded Specifications need to be defined: What can be delivered, Where it can be delivered, & When it can be delivered Settled daily

Margins A margin is cash or marketable securities deposited by an investor with his or her broker The balance in the margin account is adjusted to reflect daily settlement Margins minimize the possibility of a loss through a default on a contract

Ways Derivatives are Used To hedge risks To speculate (take a view on the future direction of the market) To lock in an arbitrage profit 4

Optimal Hedge Ratio Proportion of the exposure that should optimally be hedged is where sS is the standard deviation of DS, the change in the spot price during the hedging period, sF is the standard deviation of DF, the change in the futures price during the hedging period r is the coefficient of correlation between DS and DF.

Hedging Using Index Futures To hedge the risk in a portfolio the number of contracts that should be shorted is where P is the value of the portfolio, b is its beta, and A is the value of the assets underlying one futures contract

Reasons for Hedging an Equity Portfolio Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.) Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outpeform the market.)

Example Value of S&P 500 is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?

Determination of Forward and Futures Prices

Pricing Forward/Futures when Interest Rates are Measured with Continuous Compounding F0 = S0erT This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs

When an Investment Asset Provides a Known Dollar Income F0 = (S0 – I )erT where I is the present value of the income during life of forward contract

Futures and Forwards on Currencies A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if rf is the foreign risk-free interest rate

Options Strategies

Types of Strategies Take a position in the option and the underlying Take a position in 2 or more options of the same type (A spread) Combination: Take a position in a mixture of calls & puts (A combination)

Positions in an Option & the Underlying Long Stock Profit Profit Long Call K K ST ST (a) Short Call (b) Short Stock Profit Profit Long Stock Short Put K ST K ST Long Put (c) (d) Short Stock

Bull Spread Using Calls Long Call at K1 Profit ST K1 K2 Short Call at K2

Bull Spread Using Puts K1 K2 Profit ST Short Put at K2 Long Put at K1

Bear Spread Using Calls Profit Long Call at K2 K1 K2 ST Short Call at K1

Bear Spread Using Puts K1 K2 Profit ST Short Put at K1 Long Put at K2

Butterfly Spread Using Calls Long Call at K1 Profit Long Call at K3 K1 K2 K3 ST Short 2 Calls at K2

Butterfly Spread Using Puts Profit K1 K2 K3 ST Long put at K1 Long put at K3 Short 2 puts at K2

A Straddle Combination Long Call at K Profit K ST Long put at K

A Strangle Combination Profit Long a Call at K2 with same T K1 K2 ST Long a put at K1 with same T

Binomial Trees

Generalization A derivative lasts for time T and is dependent on a stock S0u ƒu S0d ƒd S0 ƒ

Generalization ƒ = [ pƒu + (1 – p)ƒd ]e–rT where

Choosing u and d One way of matching the volatility is to set where s is the volatility of the stock price and Dt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein

p as a Probability S0u ƒu p S0 ƒ S0d (1 – p ) ƒd It is natural to interpret p and 1-p as probabilities of up and down movements The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate S0u ƒu S0d ƒd S0 ƒ p (1 – p )

Example S0u = 22 ƒu = 1 p S0 ƒ S0d = 18 (1 – p ) ƒd = 0 A stock price is currently $20 and it is known that at the end of 3 months it will be either $22 or $18. We are interested in valuing an option to buy the stock for K= $21 in 3 months. (u = 1.1 & d = 0.9 & r = 0.12) Alternatively, we can use the formula p S0 ƒ S0d = 18 ƒd = 0 (1 – p )

Valuing the Option Using Risk-Neutral Valuation The value of the option is e–0.12´0.25 (0.6523´1 + 0.3477´0) = 0.633 S0u = 22 ƒu = 1 S0d = 18 ƒd = 0 S0 ƒ 0.6523 0.3477

Valuing a Call Option Value at node B = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257 Value at node A = e–0.12´0.25(0.6523´2.0257 + 0.3477´0) = 1.2823 24.2 3.2 D 22 B 20 1.2823 19.8 0.0 2.0257 A E 18 C 0.0 16.2 0.0 F

A Put Option Example; K=52 K = 52, a 2 year European Option with time step = 1yr r = 5%, u = 1.2, d = 0.8 Market price = 50 50 4.1923 60 40 72 48 4 32 20 1.4147 9.4636 A B C D E F

What Happens When an Option is American 50 5.0894 60 40 72 48 4 32 20 1.4147 12.0 A B C D E F

V = P[N(d1)] - Xe -rRFt[N(d2)]. d1 = .  t d2 = d1 -  t. Black-Scholes Model Call Option P: Spot Price X: Strike Price r = Risk-Free Rate t = Time to Maturity σ = Risk of the stock V = P[N(d1)] - Xe -rRFt[N(d2)]. d1 = .  t d2 = d1 -  t. ln(P/X) + [rRF + (2/2)]t

V = Xe -rRFt[N(-d2)] - P[N(-d1)] d1 = .  t d2 = d1 -  t. Put Option Black-Scholes Model V = Xe -rRFt[N(-d2)] - P[N(-d1)] d1 = .  t d2 = d1 -  t. ln(P/X) + [rRF + (2/2)]t

What is the value of the following call option? Assume: P = $27; X = $25; rRF = 6%; t = 0.5 years: 2 = 0.11 V = $27[N(d1)] - $25e-(0.06)(0.5)[N(d2)]. ln($27/$25) + [(0.06 + 0.11/2)](0.5) (0.3317)(0.7071) = 0.5736. d2 = d1 - (0.3317)(0.7071) = d1 - 0.2345 = 0.5736 - 0.2345 = 0.3391. d1 =

N(d1) = N(0.5736) = 0.5000 + 0.2168 = 0.7168. N(d2) = N(0.3391) = 0.5000 + 0.1327 = 0.6327. Note: Values obtained from Excel using NORMSDIST function. V = $27(0.7168) - $25e-0.03(0.6327) = $19.3536 - $25(0.97045)(0.6327) = $4.0036. 16