options: valuation. arbitrage options?terminologyarbitragebinomialblack-scholes we have introduced...

Options: valuation

Arbitrage

Options? Terminology Arbitrage Binomial Black-Scholes

We have introduced Put-Call Parity. In fact, Put-call parity uses the concept of no arbitrage.

What is arbitrage?

Definition:

“An arbitrage opportunity arises when an investor can construct a zero investment portfolio that will yield a sure profit.”

• Effectively, if the law of one price is violated, arbitrage opportunity emerges. If a identical product is selling at different prices in two very close locations, you will buy it from the lower-priced location and immediately sell it at the higher-priced location and realize a gain. Again, arbitrage appears because the law of one price is violated.

Arbitrage


• It is easy to think of a physical item selling at different prices in close locations would stimulate huge amount of people taking advantage of it. It is the fact that so many people would jump on it that essentially keeps the law of one price holds because the huge demand for a lower cost product drives up its price until it equalizes to the higher price.

But how are we going to apply the concept to risky assets?

• Imagine you are holding two portfolios. They have different assets inside each of them. But ultimately, their payoffs across different possible states are EXACTLY the same. Under the law of one price, the two portfolios should have the same value. (e.g., if IBM or Hilton stocks offer the same payoff structure to you, you should expect them to have the same stock price, irrespective of them being different companies. The bottom line – payoff structure, not company name)

• If they are not valued the same, there is an arbitrage opportunity. Anyone can construct the lower-cost portfolio. And sell it at a higher price and earn immediate profit. Such forces of trying to take advantage of the mis-price will eliminate the arbitrage opportunity.

No Arbitrage: An Example


• Similarly put-call parity employs the concept of no arbitrage. A risk-less portfolio should be priced as another risk-less portfolio.

Possible states

Good Normal Bad

Risky assets

x 100 80 70

y 15 25 30

z 70 30 10• It may not be that obvious, but imagine a portfolio with (2y + 1z) would

have a payoff exactly the same as if you hold 1x alone.• No arbitrage means, Px = 2Py + Pz

• Payoff structure being the same = payoffs at EVERY possible state are the same

No Arbitrage: Put-Call Parity


• We set up a similar table as the previous slides. Payoffs at maturity are listed in the table cells.

• Same idea here. A portfolio consisting of the bottom 3 items would have a payoff exactly the same as if you hold the top risk-free investment alone.

• No arbitrage means, P1 = P2 - P3 + P4

Thus,

S0 + P – C = X/(1+Rf)T

Possible states

ST >X ST ≤X

Investments Risk-free Investment with an amount equal to X/(1+R)T

X X

Long a share of Stock ST ST

Short 1 Call option -(ST - X) 0

Long 1 put option 0 (X-ST)

No Arbitrage: Put-Call Parity


• The graph of combining different options and assets is such that the payoffs of all assets are added up vertically.

ST

x

Total Payoff

ST

x

ST

x

ST

x

x

$

<= Long 1 put

<= Long 1 stock

<= Short 1 call

<= Total payoffs

Financial Engineering


• One of the many attractions of options is the ability they provide to create investment positions with payoffs depend in a variety of ways on the values of other securities.

ST

x

Total Payoff

ST

x

ST

x

ST

x

$

<= Long 1 put

<= Long 1 call

<= Short 1 put

<= Short 1 call

• Imagine the 4 different payoffs patterns:• Long Put• Long Call• Short Put• Short Call

• And imagine options with many different levels of exercise prices.

• Wisely and creatively combines options and you can build up different types of payoff structure tailored towards your investment needs.

Example: Protective Put


• As an example. You would like to invest in Google but unwilling to bear potential losses beyond some given levels. What you can do is the following:• Invest in the Google stock• Buy one put per share of Google stock

• Such an option strategy is called protective put.

Total Payoff

ST

x

ST

x

ST

x

<= Long 1 stock

<= Long 1 put

<= Total Payoffs

x

x

X-(S0+P)

Profit

• The final payoff structure is such that no matter how much Google’s share drops in price, your overall loss is limited to a fixed amount, whereas if Google’s share increases in price, you will still gain from it.

• The precise exercise price you choose will dictate how much the maximum loss you are willing to bear.

• Again, it is a protective way of holding a stock, that’s why it’s called Protective Put.

Example: Covered Call


• What if you're neutral on Google’s performance? (i.e., you think its stock price will remain relatively unchanged) To potentially profit from such expectation:• Invest in the Google stock• Sell one call per share of Google stock

• Such an option strategy is called covered call.

Total Payoff

ST

x

ST

x

ST

x

<= Long 1 stock

<= Short 1 call

<= Total Payoffs

x

x

C

Profit

• The final payoff structure is such that no matter how much Google’s share drops in price, your overall loss is limited to its price today, plus the amount you acquired from selling a call.

• If Google’s price increases, and the call holder exercises its right to buy from you, you have a stock to cover your obligtion.

• If Google’s stock does not vary much in price, for example, it remains at X on the maturity date, then you’ve gained C, the sales price of the call you sold.

Example: Straddle


• Imagine another scenario. A pharmaceutical company just release a drug which is soon to be approved or disapproved by the FDA. You anticipate either a big jump of its stock price if FDA approves, or a big drop otherwise. To profit from it:• Buy one call of that company’s stock.• Buy one put of that company’s stock

• Such an option strategy is called Straddle.

Total Payoff

ST

x

ST

x

ST

x

<= Long 1 call

<= Long 1 put

<= Total Payoffs

x

x

-(P+C)

Profit

• The final payoff structure is such that if that company’s stock price varies a lot, you will benefit the most.

• If instead, the company’s stock price doesn’t vary a lot because of the news, you will likely make a loss.

http://www.fda.gov/

http://www.fda.gov/

Option strategies


• Of course, there are unlimited number of ways for how you combine different options to form a specific payoff structure that you want.

• To appreciate the power of using options, you need to be very familiar with the payoff structures of options.

• To be a successful financial controller, fund manager, pension fund manager, investment banker, etc., or purely to get the most out of your personal investments, you have to be creative in using options wisely.

Valuation: Option definitions revisited


• There are 2 basic types of options: CALLs & PUTs

• A CALL option gives the holder the right, but not the obligation

• To buy an asset

• By a certain date

• For a certain price

• A PUT option gives the holder the right, but not the obligation

• To sell an asset

• By a certain date

• For a certain price

• an asset – underlying asset

• Certain date – Maturity date/Expiration date

• Certain price – strike price/exercise price

Valuation: No arbitrage


• We have mentioned that if the law of one price be violated, people will jump into the opportunity of making pure profit out of nothing.

• In equilibrium, such opportunity should have been eliminated.

• The no arbitrage condition serves as one of the most basic unifying principles in the study of financial markets

• An application of that is given out in the previous slides to illustrate the put-call parity.

• And we’ll keep on using no arbitrage condition in order to derive the equilibrium option prices.

Range of possible call option values


• Let us first look at the boundary for a call option first. Assuming the stock itself doesn’t payout dividend during before the call option matures.

First, its value cannot be negative. Because the holder of a call option need not be obligated to exercise it if its value of exercise is less than zero.

C≥0 [1 – lower bound]

Second, its value cannot be higher than the present stock price. Because Stock price – exercise price is the payoff of the call.

C≤S0 [2 – Upper bound]

Third, its value cannot be lower than the present stock price minus the present value of the exercise price.

C≥S0 - Present value of Xor C≥S0 – X/(1+R)T [3 – lower bound]

• Reason for [3]: if you compare 2 different portfolios:

• {a} buy a stock now at S0 and borrow X/(1+R)T

• {b} buy a call option with exercise price X.



C≥S0 – X/(1+R)T [3 – lower bound]

• Reason for [3]: if you compare 2 different portfolios:

• {a} buy a stock now at S0 and borrow X/(1+R)T

• {b} buy a call option with exercise price X.

• Payoff of {a} at maturity is ST – X (i.e, the stock price at time T - the amount that you have to repay to your lender) NOTE: this payoff can be +ve or –ve.

• Payoff of {b} at maturity is either 0 if you don’t exercise, or ST – X if you choose to exercise.

• What we see is {b} has a payoff structure actually more favorable than {a}, if constructing {a} requires S0 – X/(1+R)T amount of money, than to construct {b}, you need at least more than that amount.

• Thus we have the lower bound of the value of call as C≥S0 – X/(1+R)T



• C≥0 [1 – lower bound]

• C≤S0 [2 – Upper bound]

• C≥S0 – X/(1+R)T [3 – lower bound]

• With all 3 boundary conditions, we get the following graph:

Call Value (C)

S0X/(1+R)T

Upper

bou

nd =

S 0

Lower

Bou

nd =

S 0 -

X/(1+R)T

Call option value as a function of stock price


• The value of call as a function of the current stock price is given in the following red line.

Call Value (C)

S0X/(1+R)T

Upper

bou

nd =

S 0

Lower

Bou

nd =

S 0 -

X/(1+R)T

Factors affecting the call option value


• We identify 5 factors that affect an option’s value

1) Stock price (S)

2) Exercise Price (X)

3) Volatility of the underlying stock price (σ)

4) Time to Maturity/expiration (T)

5) Interest rate (Rf)

• You should familiarize yourself with the following table:

Factor Effect on Call value Effect on Put value

Stock price increases decreases

Exercise price decreases increases

Volatility of stock price increases increases

Time to expiration increases increases

Interest rate increases decreases



• Stock price

• Recall the payoff for call and put. Call: max{0,S-X}, Put: max{0, X-S}

• The higher the stock price, the more likely that a call option will be exercised in-the-money to get profit. Thus C ↑ if S0 ↑

• The higher the stock price, the less likely that a put option will be exercised in-the-money to get profit. Thus P ↓ if S0 ↑









• Exercise price


• The higher the exercise price, the less likely that a call option will be exercised in-the-money to get profit. Thus C ↓ if X ↑

• The higher the exercise price, the more likely that a put option will be exercised in-the-money to get profit. Thus P ↑ if X ↑









• Volatility of stock price


• The higher the volatility of stock price , the higher the probability of S being higher than X and thus the more likely the call will be exercised in-the-money to get profit. Thus C ↑ if σ ↑

• Surprisingly, it is also true for put. The higher the volatility of stock price , the higher the probability of S being lower than X and thus the more likely the put will be exercised in-the-money to get profit. Thus P ↑ if σ ↑









• Time to expiration


• The longer the time to expiration, the more time allowed for the stock price to climb above the exercise price and thus the more likely the call will be exercised in-the-money to get profit. Thus C ↑ if T ↑

• Surprisingly, it is also true for put. The longer the time to expiration, the more time allowed for the stock price to fall below the exercise price and thus the more likely the put will be exercised in-the-money to get profit. Thus P ↑ if T ↑









• Interest rate (risk-free) - the least intuitive

• Recall the put-call parity. S0 + P – C = X/(1+Rf)T

• Keeping every other variables fixed, the higher the interest rate, the smaller the RHS, and thus C has to increase to lower the LHS too. Thus C ↑ if Rf ↑

• Keeping every other variables fixed, the higher the interest rate, the smaller the RHS, and thus P has to decrease to lower the LHS too. Thus P ↓ if Rf ↑







Binomial option pricing


• With all the insights you have acquired. Let’s go to a formal option pricing model.

• Assumption: The stock price can take only 2 possible values at option expiration.

• An example to illustrate, let’s consider a call, exercise price = $125. Stock price is now $100. It will either go up to $200 or down to $50. (Note: NO probability is given)

$100$200

$50Stock price

C$200 - $125 = $75

0Call option value

• Consider a portfolio that consists of short 1 option and long m shares of this stock.

• Payoff of this portfolio is:

[1] $200m - $75 if the stock price rises to $200 or [2] $50m if the stock price drops to $50.



$100m$200m

$50mLong m Stocks +

-C-$75

0Short 1 Call =

• Choose a specific m* to make the combined portfolio risk-less. (i.e., payoffs are the same for the two different states)

Set $200m - $75 = $50m, solving, we have m* = 75/150 = 0.5

• The ratio is what we needed. That means, if a portfolio consists of longing 1/2 share of the stock and shorting 1 call option, or if a portfolio consists of longing 1 shares of the stock and shorting 2 call options, the portfolio is risk-less.

$100m-C$200m-$75

$50mThe combined portfolio

$100m*-C = $50 - C $200m*-$75 = $25

$50m* = $25

The combined portfolio with m*



• So, the combined portfolio gives me $25 no matter which of the two states is realized. That means, the portfolio is risk-less. The present value of this $25 at maturity should be equal to the value of the combined portfolio that you pay now (i.e., no arbitrage condition). Thus:

100m* - C = $50 – C = 25/(1+Rf)T

• If time to expiration = 1 year, annual risk-free interest rate = 8%, then the Call option should have a value equal to:

C = $50- 25 /(1+8%)1 = $26.85 (round up 2 significant decimal places)

• Using put-call parity, we can find the put option value with the same exercise price.

$100m*-C=$50 - C$200m*-$75 = $25

$50m* = $25

The combined portfolio with m*

Black-Scholes option pricing forumla


• Generalizing the binomial option pricing, we have the Black-Scholes formula, which is the Nobel prize winner Prof. Scholes’ main contribution leading to his 1997 Nobel prize.

• Black-Scholes formula:

C = S0N(d1) – X•e-RfT•N(d2)

Where d1 = [ln(S0/X) + (Rf + σ2/2)T] / σ√T

And d2 = d1 - σ√TC = Call Option Price

S0 = Current Stock Price

N(d1) = Cumulative normal density function of (d1)

X = Strike or Exercise price

N(d2) = Cumulative normal density function of (d2)

Rf = discount rate (risk free rate)

T = time to maturity of option (as % of year)

σ = volatility or annualized standard deviation of daily stock returns





And d2 = d1 - σ√T

-0.5 –0.2 0 0.2 0.5

N(d1)= cumulative area below d1 for a standard normal distribution.

If d1 = 0, N(d1) = 0.50 If d1 = 0.5, N(d1) = 0.69

Standard NormalDensity Function~ N(0,1)



Some of the important assumptions are as follows:

• 1) The stock will pay no dividends until after the option expiration date.

• 2) Both the interest rate and the standard deviation of daily return on the stock are constant.

• 3) Stock prices are continuous, meaning that sudden extreme jumps such as those in the aftermath of an announcement of a take-over attempt are ruled out.




Black-Scholes: An example





ExampleWhat is the price of a call option given the

following?

S0 = 30, Rf = 5%, σ2 = 0.0305, X = $30, T = 1 year

d2 = 0.198978 N(d2) = 0.57886

d1 = 0.37362 N(d1) = 0.645657

C = S0[N(d1)] – Xe-rt[N(d2)]C = $ 2.85, using put-call parity, we can calculate the corresponding put option price.

Some more insights on options


• American Options can be exercised at anytime before maturity

• European Options can be exercised at maturity• It is never optimal to exercise an American call option

early: Thus, American and European calls should have the

same price• But it may be optimal to exercise an American put

option earlier than maturity• Empirical evidence:

– Black-Scholes option pricing model does well at pricing options that are at the money, but do much worse as the options go deeper into or out of the money

On the ECMC49F Exam


• You will not need to remember the Black-Scholes formula.• You will need to try using the Black-Scholes formula before the

exam because the final exam will for sure have a question concerning the Black-Scholes. That means you have to know how to use a Cumulative normal distribution table.

• You have to be familiar with the put-call parity and no arbitrage condition.

• You have to know the Binomial option pricing too. Work it out at least once.

• You should try to get yourself familiar with how to quote an option price from CBOE. And you should be able to understand the meaning of a table you see from a CBOE option quote, like the one posted on lecture note 06.

• I strongly encourage you to do the exercises on options posted on the course webpage. Solutions will be provided next week. But try it out before you look into the solutions.

http://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm

http://www.cboe.com/DelayedQuote/QuoteTable.aspx

options: valuation. arbitrage options?terminologyarbitragebinomialblack-scholes we have introduced...

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