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May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
CASM WORKSHOPBlack- Scholes and Beyond: Pricing Equity
DerivativesMay 16 -18 2014 @ LUMS
Day 1: LECTURE 1OPTIONS
Seema NandaTata Institute of Fundamental Research
Centre for Applicable MathematicsBangalore, India
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Financial Derivative
●What is a financial derivative (aka Contingent Claim)?
Ans: A derivative is a financial instrument whose value depends on the values of other underlying assets such as a stock, a bond, an interest rate etc
● Example 1 : forward contract (simplest derivative)
A forward contract is an agreement to buy or sell an asset at an agreed price (delivery price) at a certain future date. Two parties are involved:One takes a long position i.e. agrees to buy the underlying asset at agreed price and on agreed date. The other takes a short position and agrees to sell the asset on the same date and same price.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Forward Contract
● Value of forward contract to both parties is ZERO on date of writing the contract. Forward Price is chosen this way.
● Contract is settled at maturity
● At maturity Short position holder delivers asset to Long position holder in return for cash payment = delivery price
As time passes forward price is likely to change. Fwd price = delivery price only at start of contract. Delivery price is contractual so remains the same.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Forward Contract -example
An investor takes a long position in a forward contract on May 18, 1996 to buy 1 million pounds in 90 days at an exchange rate of $1.80/pound.
Foreign Exchange quotes on May 8, 1996Spot $1.8330 day forward $1.7990 day forward $1.80
What would happen if spot rate at 90 days becomes $1.82?(Spot rate is rate quoted for immediate settlement)
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Forward Contract
Problem 1
Draw a graph representing the payoff Vs underlying asset price for the long position.
Draw a graph representing the payoff Vs asset price for the short position
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Stocks and Bonds
A business can raise money for its operations in two ways:
●Borrow it from individuals or banks (sell bonds)●Sell an ownership interest i.e sell shares/stocks in the business to interested parties
Bond: a financial instrument issued by a borrower that obligates the issuer to make specified payments to the bond holder over a specified period. A coupon bond obligates the issuer to make coupon (interest) payments over the life of the bond, and then repay the face value at maturity.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Stocks and Bonds
Common Stock: Equities issued as ownership shares in a publicly held company. Shareholders have voting rights and may receive dividends in proportion to their ownership.
QuestionWhich one is riskier from the investor's perspective? a stock or a bond for the same company?
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Risk-Return Trade-off
● Risk is the chance that an actual return will be different than expected. Risk means you have the possibility of losing some, or even all, of the original investment.
● Technically, this is measured by standard deviation of the return
● The risk/return tradeoff tells us that the higher risk gives us the possibility of higher returns. There are no guarantees.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Some definitions
RISK:
RISK FREE INVESTMENT :
PORTFOLIO
Gives a guaranteed return (called the risk-free rate of return) with no chance of a default. T-bill rates are considered a proxy for risk-free rate.
Specific and Non-specific. Specific is associated with a Single asset. Non-specific is related to whole market
A collection of investments.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Present Value/discounting
How much would I pay now to receive a gauranteed amount E at a future time T?
●Ans: discount future value E using continuous compounding risk-free interest rate r. Money in a bank M(t) grows exponentially according to
dM/M = rdt => M=ce(rt) . Now M = E at final time T
The value of E (the gauranteed payoff) at a time t <T is M = E e-r(T-t)
If r is not fixed, write M = Ee- ∫ r(s)ds
M is the Present Value of E at time t.
T
t
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
ARBITRAGE
Example: Stock XYZ is traded in NY at $172 and simultaneoulsy in London at ₤100. Exchange rate is $1.75/₤. Simultaneously buying 100 shares in NY and selling them in
London gives the following net result:
100 x [(₤100x 1.75) – $172] = 100X $3 = $300
If instead the shares were traded at $176 in NY, buy in London and sell in NY to get: (100 x $176) – (₤100x 100 x$1.75) = $100.
Arbitrage is the locking in of a riskless profit by simultaneously entering into two or more transactions
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
OPTIONS
Options are derivatives on stocks, currencies, bonds, stock indices, commodities and futures contracts.
Call Option: is a contract which gives the holder (buyer) the right to buy the underlying asset by a certain date (expiration date) at a fixed price (expiration or strike price)
Put Option: gives the holder (buyer) the right to sell the underlying asset by a certain date at a fixed price (expiration or strike price).
There is a cost associated with each optionAmerican Options can be exercised at any time up to the exp. dateEuropean Option can only be exercised on expiration date
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
OPTIONS
Problem 2
Describe a Call Option on a stock from the point of view of the seller (writer)
Describe a Put Option from the point of view of a seller.
In this lecture assume that Options are on stocks. We only analyse European options.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Example – CALL OPTION
Consider a long position in a call option on IBM shares with a strike at $175 expiring June, 2014. The cost is $2.00 per option. On Aug 30, 2014 IBM shares are at $177.
●Draw the payoff diagram of this position
●Draw a payoff diagram for the short position
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Example – PUT OPTION
Consider an IBM put contract at $160 strike, expiring June 2014. This option costs the buyer $3.85 on Nov 4, 2013. ●Draw the payoff diagram of this position
●Draw a payoff diagram for the short position
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Example Put Option
Collecting $3.85 as the premium represents a 2.4% return against the $160 commitment, or a 3.8% annualized rate of return.
QUESTIONWhat % would the stock have to drop for the
put to be exercised? Spot price of IBM is $179.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
OPTIONS
Fill in the blanks
Purchaser of a Call Option is hoping that the stock price will ____________
Writer of a Put Option is hoping that the stock price will ________
Purchaser of a Put Option is hoping that the stock price will _________
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
OPTIONSWhy are options used?
- for hedging - for speculation
Speculator will typically buy an option
Hedger will typically write (sell) an option
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Hedging Hedging is the investing in an asset to reduce risk of the overall portfolio.Example: A portfolio contains a put option and a stock. What happens to the value of this portfolio as stock price falls?
stock price declines => the put price goes up.
If portfolio has only puts, value increases as stock price declines.
If it has only stocks, value decreases.
In between there is a ratio of put options and stocks where the portfolio is instantaneously risk-free.
Hedging is risk reduction by taking advantage of such correlations.
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
6 Factors affect price of an option● Current Stock price, S● The Strike Price, K● Time to expiration (T-t)● Volatiliy of S denoted by σ
● The risk-free interest rate rf
● Dividends expected during the life of an option
FACTORS AFFECTING OPTION PRICES
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Effect on the Price of a stock Option, changing one variable only
Variable European Call European Put
S + -
K - +
T-t
σ + +
rf
+ -
Div - +
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
ASSUMPTIONS
● No Transaction Costs
● Borrowing at rf rate exists
● All profits (net of losses) have same tax rate
● People behave rationally towards profit => No arbitrage opportunities exist in the mkt
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
BOUNDS FOR European OPTION PRICES
Let c and p be prices of European call & put options respectively.
UPPER BOUND FOR c :
UPPER BOUND FOR p :
c ≤ S
p ≤ Ke-r(T-t)
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
LOWER BOUND FOR European CALL NO DIVIDEND
LOWER BOUND FOR c
At time T, long call position will need ≤ K amt of cash => At time t holder will need Ke-r(T-t) amt of cashTherefore at time t, c+Ke-r(T-t) ≥ S => c ≥ S - Ke-r(T-t)
It is possible that the call expires at T and is worthless.Hence c > max {S - Ke-r(T-t) , 0}
PROBLEMIs the reverse inequality possible? Explain
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
LOWER BOUND FOR European CALL NO DIVIDEND
Example: S = $20, K = $18, r = 10% per annum T-t = 1 yr
Then S – Ke-r(T-t) = 20 -18(-0.1) = 3.71
Suppose European Call price is $3.00
Explore any arbitrage opportunity
Buy Call and short the stock: 20 -3 = 17Invest 17 in bank for 1 yr at 10% to get 18.79At expiration if S
T > 18 exercise.
Net profit = 18.79 – 18.00 = 0.79If S
T < 18, buy in mkt and close out short position
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
LOWER BOUND FOR European PUTNO DIVIDEND
LOWER BOUND FOR p :
At time T a long put position will get ≤ K cash
At time t the total worth of this cash = Ke-r(T-t)
At expiration, time T, holder gets max { K-ST, 0} as she must buy
ST
shares to exrcise the option if K > ST
Hence at time t, p ≥ Ke-r(T-t) - S => p > max {Ke-r(T-t) - S, 0}
PROBLEM
Is the reverse inequality possible? Explain
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
Bounds for prices of European options (no dividend)
Bounds for c:
S > c > S - Ke-r(T-t)
Bounds for p:
Ke-r(T-t) > p > max {Ke-r(T-t) - S, 0}
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
PUT-CALL PARITY
A relationship between call price c and put price p:
Portfolio A: One European call option plus cash amount of Ke-r(T-t)
Portfiolio C: One European put option plus one share
Both Portfolios are worth max(ST,K) at expiration. Therefore
they must have identical values today:
=> c + Ke-r(T-t) = p + S
Value of a European call can be deduced from a put and vice versa, with the same expiration T and strike K
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
PUT - CALL PARITY
Problem
Show that if put-call parity is violated then an arbitrage opportunity exists
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
TRADING STRATEGIES INVOLVING OPTIONS
Strategies involving one stock and one option● Long a stock, short a call● Long a stock, long a put● Short a stock, long a call● Short a stock, short a put
May 16-18, 2014Tata Institute of Funamental Research
CASM workshop - LUMS
TRADING STRATEGIES INVOLVING OPTIONS
Spreads: A trading strategy where you take a position in 2 or more options of same type
● Bull spread: long a call with k1 strike, short a
call with strike k2. k
2 > k
1
● Bear Spread: take k2 < k
1