fec financial engineering club. an intro to options

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 todayK is the strike price

INTRINSIC VALUE

The intrinsic value is the value of an option if it expires right now.

, ,

S is the price of the underlying todayK 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.


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.


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 Current price of underlying

Strike price on option

current date

Expiration date

Value of European call option

Value of American call option

Value of European put option

Value of American put option

COMMON SENSE BOUNDS

Options cannot have negative value:

Why?

COMMON SENSE BOUNDS

American options are at least as valuable as European options:

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:

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:

COMMON SENSE BOUNDS

Calls with lower strikes are more valuable. Puts with higher strikes are more valuable:

If

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

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)

When S < K, what happens?

We exercise our short put options(s)

Profit/Loss Diagram is:

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 Long Call Long Put Long Strangle0 5

10 15 20 25 30 35 40 45 50 55 60 65 70 75

STRANGLE

Underlying

Long Call

Long Put

Long Strangle

0 -3 34 315 -3 29 26

10 -3 24 2115 -3 19 1620 -3 14 1125 -3 9 630 -3 4 135 -3 -1 -440 -3 -1 -445 2 -1 150 7 -1 655 12 -1 1160 17 -1 1665 22 -1 2170 27 -1 2675 32 -1 31


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)


Underlying

Long Call (K=35)

Position 1

0 0 05 0 0

10 0 015 0 020 0 025 0 030 0 035 0 040 5 1045 10 2050 15 3055 20 4060 25 5065 30 6070 35 7075 40 80

Underlying

Long Call (K = 30)

Long Call (K = 40)

Position 2

0 0 0 05 0 0 0

10 0 0 015 0 0 020 0 0 025 0 0 030 0 0 035 5 0 540 10 0 1045 15 5 2050 20 10 3055 25 15 4060 30 20 5065 35 25 6070 40 30 7075 45 35 80

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:

GENERAL APPROACHES TO SPREADS

38

50

51

52

37

REPLICATION WITH CALLS

Evaluate positions from left to right

38

50

51

52

37

1) Slope must be 10—buy 10 Calls at 37

2) Slope from 38 to 50 must be 0—sell 10 Calls at 38 to get flat

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

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

38

50

51

52

37

5) Slope must be 0—buy 10 Puts at 37 to get flat

4) Slope from 38 to 37 must be 10—sell 10 Puts at 38

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

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 , its price will move by

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

What is ?

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

Now, the underlying decreases to $39.00. What happens?

Sell 1220 Shares

DELTA-HEDGING

;

;

You make money even though your calls become less valuable

Now = 0.43 Repeat the process:

Why did delta change? Answer: Gamma

Sell 860 Shares

NEXT LECTURE

Continuous models for option valuation Stochastic calculus Black-Scholes-Merton More greeks

THANK YOU!

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fec financial engineering club. an intro to options

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