FEC
FINANCIAL ENGINEERING CLUB
AN INTRO TO OPTIONS
AGENDA
 What are options?
 Bounds on prices
 Spread strategies
 Greeks
OPTIONS CONTRACTS
 An option contract is a right to buy (call option) or sell (put option) an underlying
security at a pre-specified date in the future and at a pre-specified price.
 Date is called the maturity or expiration date
 Pre-specified price is called the strike price
 Ex) AAPL is currently at (about) $509.00
You want to buy a call option with a strike of $505.00 whose expiration is March 21,
2014.
This means you can ‘exercise’ your option to buy AAPL at $505.00 at the maturity date
(European-style option) or before (American-style option)
OPTION VALUE
 What is the value of such an option?
 Depends on many things—most importantly, the underlying (AAPL) price.
 Suppose this call option expired today and AAPL was at $509.00. How much would you
be willing to pay for it?
 Right to buy AAPL (worth $509.00) for $505.00
CallIntrinsic = (S-K)+ = max{S-K,0}, S is the price of the underlying today
K is the strike price
INTRINSIC VALUE
 The intrinsic value is the value of an option if it expires right now.
𝐶𝑎𝑙𝑙𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = (𝑆 − 𝐾)+ = max{𝑆 − 𝐾, 0},
𝑃𝑢𝑡𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = (𝐾 − 𝑆)+ = max{𝐾 − 𝑆, 0},
S is the price of the underlying today
K is the strike price
For AAPL at 509.00, what is the intrinsic value of?
 Call(K=520)?
$0.00
Out of the money (trading lingo)
 Put(K=520)?
$11.00
In the money
 Call(K=500)?
$9.00
In the money
 Put(K=500)?
$0.00
Out of the money
INTRINSIC VALUE
Intrinsic Value of a Call Option (Green)
Intrinsic Value of a Put Option (Green)
TIME VALUE
 However, if there is time left until expiration, the stock price (at time t) St, could change
and thus the value of the option would change.
 This component of price that changes over time is known as time value.
 Time value is the value associated with the likelihood that the option will become in the
money (valuable) by a favorable move in the underlying price
 Some determinants of time value:
 How volatile is the underlying stock
 What is your borrowing rate, are there any dividends from the underlying stock
 How much time is left until maturity
BUYING VS SELLING OPTIONS
 If you buy an option you have the option to exercise it:
Long Call option: You may pay K to receive the stock.
Long Put option: You may sell the stock for K.
BUYING VS SELLING OPTIONS
 When you sell an option, you give the buyer the right to exercise the option:
Short Call option: Buyer may buy the stock from you for K.
Short Put option: Buyer may sell the stock to you for K.
When will the buyer buy the stock from you?
In the same situations you would exercise the call option—when S > K. They would not
exercise when S < K.
BUYING VS SELLING OPTIONS
 Same logic applies for short puts.
 When you sell (called writing) an option and it is exercised by the buyer, it is said to
be assigned against you.
OPTION CONTRACT STYLES
 European—Option may be exercised at maturity only.
 American—Option may be exercised at any time preceding maturity.
 Others—Asian, Bermudan, Barrier options.
 For this lecture, we will discuss the simplest and most common cases—European
and American options.
COMMON SENSE BOUNDS
 Without making any assumptions about the returns of the underlying or imposing any
model, what can we say about option prices?
Denote
𝑆
Current price of underlying
𝐾
Strike price on option
𝑡
current date
𝑇
Expiration date
𝑐(𝑆, 𝐾, 𝑡, 𝑇)
Value of European call option
𝐶(𝑆, 𝐾, 𝑡, 𝑇)
Value of American call option
𝑝(𝑆, 𝐾, 𝑡, 𝑇)
Value of European put option
𝑃(𝑆, 𝐾, 𝑡, 𝑇)
Value of American put option
COMMON SENSE BOUNDS
Options cannot have negative value:
𝑐 𝑆, 𝐾, 𝑡, 𝑇 ≥ 0
𝐶 𝑆, 𝐾, 𝑡, 𝑇 ≥ 0
𝑝 𝑆, 𝐾, 𝑡, 𝑇 ≥ 0
𝑃 𝑆, 𝐾, 𝑡, 𝑇 ≥ 0
Why?
COMMON SENSE BOUNDS
American options are at least as valuable as European options:
𝐶 𝑆, 𝐾, 𝑡, 𝑇 ≥ 𝑐 𝑆, 𝐾, 𝑡, 𝑇
𝑃 𝑆, 𝐾, 𝑡, 𝑇 ≥ 𝑝 𝑆, 𝐾, 𝑡, 𝑇
Why?
You can exercise American options in more (potentially more profitable) situations.
COMMON SENSE BOUNDS
Call options never worth more than underlying stock; puts never worth more than exercise
price:
𝑐 𝑆, 𝐾, 𝑡, 𝑇 ≤ 𝑆𝑡
𝐶 𝑆, 𝐾, 𝑡, 𝑇 ≤ 𝑆𝑡
𝑝 𝑆, 𝐾, 𝑡, 𝑇 ≤ 𝑃𝑉(𝐾)
𝑃 𝑆, 𝐾, 𝑡, 𝑇 ≤ 𝐾
Why?
If P > K, sell the put for P and invest K of the proceeds. In the worst scenario, the stock will be
worthless and you will pay K and receive the stock. However, you earned P > K.
COMMON SENSE BOUNDS
American options are worth at least their intrinsic value:
𝐶 𝑆, 𝐾, 𝑡, 𝑇 ≥ max{0, 𝑆𝑡 − 𝐾}
𝑃 𝑆, 𝐾, 𝑡, 𝑇 ≥ max{0, 𝐾 − 𝑆𝑡 }
COMMON SENSE BOUNDS
Calls with lower strikes are more valuable. Puts with higher strikes are more valuable:
If 𝐾1 > 𝐾2
𝑐 𝑆, 𝐾2 , 𝑡, 𝑇 > 𝑐 𝑆, 𝐾1 , 𝑡, 𝑇
𝐶 𝑆, 𝐾2 , 𝑡, 𝑇 > 𝐶 𝑆, 𝐾1 , 𝑡, 𝑇
𝑝 𝑆, 𝐾2 , 𝑡, 𝑇 < 𝑝 𝑆, 𝐾1 , 𝑡, 𝑇
𝑃 𝑆, 𝐾2 , 𝑡, 𝑇 < 𝑃 𝑆, 𝐾1 , 𝑡, 𝑇
Why?
The strike price is what you pay to exercise a call—smaller K means larger payoff. Converse is
true for puts.
COMMON SENSE BOUNDS
Options are worth more when there is additional time until maturity:
If 𝑇2 > 𝑇1
𝑐 𝑆, 𝐾, 𝑡, 𝑇2 ≥ 𝑐 𝑆, 𝐾, 𝑡, 𝑇1
𝐶 𝑆, 𝐾, 𝑡, 𝑇2 ≥ 𝐶 𝑆, 𝐾, 𝑡, 𝑇1
𝑝 𝑆, 𝐾, 𝑡, 𝑇2 ≥ 𝑝 𝑆, 𝐾, 𝑡, 𝑇1
𝑃 𝑆, 𝐾, 𝑡, 𝑇2 ≥ 𝑃 𝑆, 𝐾, 𝑡, 𝑇1
Why?
PUT-CALL PARITY
 What happens if I sell a put at strike price K, buy an identical call, and lend PV(K)?
 Has the same payoff as a long position in the underlying!
 Therefore costs of our combined position must equal that of the underlying.
PUT-CALL PARITY
PUT-CALL PARITY
−𝑆𝑡 = −𝑐 𝑆, 𝐾, 𝑡, 𝑇1 + 𝑝 𝑆, 𝐾, 𝑡, 𝑇1 − 𝑃𝑉(𝐾)
Cost to be long
underlying
Cost to be long
call option
Cost to be short
put
Cash outflow from
lending PV(K)
This, and other bounds may be used in the Black-Scholes PDE
(next lecture)
SPREAD STRATEGIES
SPREADS
 Spread strategies are multi-legged option positions
 What is a leg?
 A position using one type of options contract
 Example: What if we buy a call option and buy an identical put option (same strike,
time until maturity, etc)?
 One leg is the call option
 One leg is the put option
 What does our position look like?
SPREADS
When S > K, what happens?
We exercise our long call option(s)
Profit/Loss Diagram is:
When S < K, what happens?
We exercise our short put options(s)
LONG STRADDLE
 This is known as a long Straddle position.
 One of the simpler spread positions
 When would one want to trade a straddle?
A ) When volatility is high ?
B) When volatility is low?
C) When we are certain the underlying will increase?
LONG STRADDLE
 This is known as a long Straddle position.
 One of the simpler spread positions
 When would one want to trade a straddle?
A ) When volatility is high ?
B) When volatility is low?
C) When we are certain the underlying will increase?
MORE SPREAD STRATEGIES
 Underlying is 37
 Strategy: Long call (Strike = 40); Long a put (Strike = 35). The call is worth $3. The put is worth $1.
Underlying
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
Long Call
Long Put
Long Strangle
STRANGLE
Underlying Long Call Long Put Long Strangle
0
-3
34
31
5
-3
29
26
10
-3
24
21
15
-3
19
16
20
-3
14
11
25
-3
9
6
30
-3
4
1
35
-3
-1
-4
40
-3
-1
-4
45
2
-1
1
50
7
-1
6
55
12
-1
11
60
17
-1
16
65
22
-1
21
70
27
-1
26
75
32
-1
31
MORE SPREAD STRATEGIES
 From a payoff standpoint (ignore costs), would you prefer to be long
 Position 1: two call options (K = 35) , or
 Position 2: one call option (K = 30) and another call option (K = 40)
MORE SPREAD STRATEGIES
Underlying Long Call (K=35) Position 1
0
0
0
5
0
0
10
0
0
15
0
0
20
0
0
25
0
0
30
0
0
35
0
0
40
5
10
45
10
20
50
15
30
55
20
40
60
25
50
65
30
60
70
35
70
75
40
80
Underlying Long Call (K = 30) Long Call (K = 40) Position 2
0
0
0
0
5
0
0
0
10
0
0
0
15
0
0
0
20
0
0
0
25
0
0
0
30
0
0
0
35
5
0
5
40
10
0
10
45
15
5
20
50
20
10
30
55
25
15
40
60
30
20
50
65
35
25
60
70
40
30
70
75
45
35
80
MORE SPREAD STRATEGIES
GENERAL APPROACH TO SPREADS
 Options can replicate any risk profile at maturity with exclusively puts or calls.
 That is, you can construct a position like this:
37
52
51
50
38
GENERAL APPROACHES TO SPREADS
REPLICATION WITH CALLS
 Evaluate positions from left to right
3) Slope from 50 to 51 must
be -5—sell 5 Calls at 50
4) Slope from 51 to 52 must be
-3—buy 2 Calls at 51
52
51
1) Slope must be 10—
buy 10 Calls at 37
50
38
2) Slope from 38 to 50 must be 0—
sell 10 Calls at 38 to get flat
37
5) Slope after 52 is 0—buy 3
Calls at 52 to get flat
+10 Calls(37)
-10 Calls(38)
-5 Calls(50)
+2 Calls(51)
+3 Calls(52)
REPLICATION WITH PUTS
 Evaluate positions from right to left
3) Slope from 50 to 38 must be
0—sell 5 Puts at 50 to get flat
2) Slope from 51 to 50 must be
-5—buy 2 Puts at 51
52
51
5) Slope must be 0—buy 10
Puts at 37 to get flat
50
38
4) Slope from 38 to 37 must be
10—sell 10 Puts at 38
37
1) Slope from 52 to 51 is -3—
buy 3 Puts at 52
+3 Put(52)
+2 Put(51)
-5 Put(50)
-10 Put(38)
+10 Put(37)
GREEKS
GREEKS
 Recall that there are five drivers of an option’s price:





Price of the underlying
Volatility of the returns on the underlying
Interest rates
Strike price
Time until maturity
 What is the risk of an option? How does the price of an option change as the
underlying factors change?
 These are the greeks.
DELTA
 The sensitivity of an option with respect to a change in the underlying’s price.
 Ex) Suppose that the underlying is at 60. A call option with strike has a delta of .5
(usually quoted as 50).
 What happens if underlying moves to 65?
 Option price increases by .5*(65-50) = .5*15 = 7.50
DELTA-HEDGING
 In general, for an option with a delta of ∆𝑜𝑝𝑡𝑖𝑜𝑛 , its price will move by ∆𝑜𝑝𝑡𝑖𝑜𝑛 ∗
(𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔)
 Many traders like to be delta—neutral. That is, they prefer to be immune to the risk of
the underlying price trading.
 Delta-hedging is the process of offsetting the delta of your portfolio to 0, by selling the
underlying.
 You may want to do this in the trading competition.
DELTA-HEDGING
 Ex) Suppose you are long 20 call options with 0.3 years until maturity and strike is
$40. The risk-free interest rate is 0 and the expected standard deviation of returns
over the next 0.3 years is 0.2. The underlying is at $41.
 Delta is .61  We want to trade X shares of the underlying so that
100 ∗ .61 ∗ 20 + 𝑋 ∗ ∆𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 = 0
 What is ∆𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 ?
 ∆𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 = 1
DELTA-HEDGING
 Ex) Suppose you are long 20 call options with 0.3 years until maturity and strike is
$40. The risk-free interest rate is 0 and the expected standard deviation of returns
over the next 0.3 years is 0.2. The underlying is at $41.
 Delta is .61  We want to trade X shares of the underlying so that
100 ∗ .61 ∗ 20 + 𝑋 ∗ ∆𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 = 1220 + 𝑋 = 0
𝑋 = −1220
Sell 1220 Shares
 Now, the underlying decreases to $39.00. What happens?
DELTA-HEDGING
 𝑆𝑡−1 = 41.00; 𝑆𝑡 = 39.00
 𝐶𝑎𝑙𝑙𝑡−1 = 2.31; 𝐶𝑎𝑙𝑙𝑡 = 1.27
 Profit = 20 ∗ 100 ∗ 1.27 − 2.13 − 1220 ∗ 39 − 41 = 360.00
 You make money even though your calls become less valuable
 Now ∆𝑜𝑝𝑡𝑖𝑜𝑛 = 0.43
 Repeat the process: 100 ∗ .43 ∗ 20 + 𝑋2 ∗ ∆𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 = 860 + 𝑋 = 0
 𝑋 = −860
Sell 860 Shares
 Why did delta change?
 Answer: Gamma
NEXT LECTURE
 Continuous models for option valuation
 Stochastic calculus
 Black-Scholes-Merton
 More greeks
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