Options: Call and Put. Option Contract Option contract – is an agreement to buy or sell an underlying asset within a specified time period (exercise period)

Options:Call and Put

Option Contract

Option contract – is an agreement to buy or sell an underlying asset within a specified time period (exercise period) for a price (strike price or exercise price) specified today.

2 Types of Options:1. Call option – gives the investor the right to buy the underlying asset at the strike price (XP).2. Put option - gives the investor the right to sell the underlying asset at the strike price (XP).

Key Concepts• The buyer of an option is termed as owner,

holder, or buyer.• The seller of an option is referred to as grantor,

seller, or writer. Strike or Exercise Price (XP) – the price per

dollar that must be paid if the option is exercised; the pre-specified price indicated in the option contract.

Profit (or k) – is the monetary gain derived from buying and/or selling option contracts. Formula: k = Sale price–(Purchase price + other costs)

k/unit = is the profit per the number of units pre-specified in the contract:

Formula: k/unit = profit ÷ no. of units

• Intrinsic value (I.V.) – the financial gain if the option is exercised immediately.

Formula: I.V. Call option = (S1 – XP) (# of units)

I.V. Put option = (XP – S1) (# of units)• Time value – exists because the price of the

underlying currency, the spot rate, can move further and further into the money between the present time and the option’s expiration date. Time value is generally equal to the difference between the premium and the intrinsic value.

• Internal rate of return (IRR) – calculation of the true interest yield expected from an investment.

Formula: IRR = (k÷Total Investment)(n÷360)

Key Concepts

Hedger – wants to lock the price; will actually buy and/or sell the pre-specified currencies in the contract.

Speculator – bettor on the direction of the value of the contract for profit.

Note: The seller is simply another investor; he receives the option premium and in return must stand ready to sell or buy the underlying asset for the pre-specified price he has written for the put or call option.

The option writer does not have an option; only the “owner” who has purchased the option can decide whether the option will be exercised, or he can elect to let his option expire.

Key Concepts

Sample – Call Option• Call Option:

No. of units: €50,000Strike Price/Exercise Price: $1.49/€Premium: $0.02/€Maturity: 1 yearS0: $1.47/€Settlement: In US$

Details of the call option:1. What is the contract size (per strike price or XP)?

$74,500 (50,000 x 1.49)2. How much is the premium cost? $1,000 (50,000

x 0.02)3. What is the dollar value of the contract (per spot

rate)? $73,500 (50,000 x 1.47)

• Interpretation:No. of units (currency): €50,000Strike Price/Exercise Price: $1.49/€$74,500Premium: $0.02/€$1,000Maturity: 1 yearS0: $1.47/€$73,500Settlement: In US$

The holder of this “call option contract” can buy €50,000 one year from now at $1.49/ € (or $74,500). The contract fee amounts to $1,000. Presently, €50,000 would cost only $73,500.

Call Option

“Exercising the contract”

• Call Option:No. of units (currency): €50,000Strike Price/Exercise Price: $1.49/€$74,500Premium: $0.02/€ $1,000S0: $1.47/€ $73,500

Scenario 1/Question:One day before the contract expires, the spot rate closed at $1.53/€. Will the option buyer exercise the contract or let it expire?

Answer: He will exercise the contract, since exercising the contract means buying €50,000 at only $1.49/€ (or $74,500) rather than buying it from the market at $1.53/€ (or $76,500).

“Letting the contract expire”

• Call Option:No. of units (currency): €50,000Strike Price/Exercise Price: $1.49/€$74,500Premium: $0.02/€ $1,000 S0: $1.47/€ $73,500

Scenario 2/Question:One day before the contract expires, the spot rate closed at $1.45/€. Will the option buyer exercise the contract or let it expire?

Answer: He will let the contract expire, since he would be a fool to buy €50,000 at $1.49/€ (or $74,500) if the market’s price is much cheaper at $1.45/€ (or $72,500).

The Decision Rule in a Call Option

• The decision rule:

Market rate vs. Contract rate

1. In the money (ITM): S1 > XP

2. Out of the money (OTM): S1 < XP

3. At the money (ATM): S1 = XP

Three Scenarios on maturity date:

Example:€50,000 call option with an XP of $1.49/€ while S0=$1.45/€

Three (3) scenarios Parameter Decision Rule

a. S1 = $1.50/ € 1.50 > 1.49 ITM

b. S1 = $1.48/ € 1.48 < 1.49 OTM

c. S1 = $1.49/ € 1.49 = 1.49 ATM

Intrinsic value of a call option and the hedger’s “k”

• In a call option, if the strike price is cheaper (meaning more favorable) compared with the market rate, the option is said to possess an intrinsic value.

Example: Call Option:

€50,000; XP= $1.49/€ ($74,500)If at maturity, the S1 = $1.53/€ ($76,500), the contract is said to possess an intrinsic value since the contract is a better buy (of euro) when compared with the market.

Selling the option at its intrinsic value

Example: €50,000; XP= $1.49/€ ($74,500); Prem=$0.02/€ ($1,000) If at maturity, the S1 = $1.53/€ ($76,500), what is the contract’s intrinsic value? Answer: $2,000Solution: Intrinsic Value (I.V.) = (S1 – XP) (No. of units)

= (1.53 – 1.49) (50,000) = $2,000

• How much is the “k” of the holder (who may be referred to as a “speculator” if he sells the contract at its intrinsic value)?Answer: $1,000

Solution: k = I.V. – Total investment = 2,000 – 1,000 = $1,000

His total investment is only the premium fee of $1,000

Decision: Exercise the contract Decision: Exercise the contractk= Selling price - (Puchase Price + Premium) I.V.= (S1 - XP) (# of units) = $76,500 - ($74,500 + $1,000) = ($1.53 - $1.49) (50,000) = $1,000 = $2,000

k/ unit = 1,000 / 50,000 k= I.V. - Total Investment = $0.02/€ = $2,000 - $1,000

= $1,000

IRR = k/ Total investment IRR = k/ Total investment = $1,000/ ($74,500+$1,000) = $1,000/ $1,000 = 1.32% = 100%

Hedger: Speculator:Owner, Holder or Buyer

SUMMARYCall Option. €50,000; XP=$1.49/€; Prem=$0.02 ;S0=$1.47/€; S1=$1.53/€. Decision rule is ITM

The writer of the call option contract

• There are two (2) of “writers” - the “naked writer” and the “covered writer”.

The Naked Writer

• The“naked writer” will obtain currencies at S1 (only if the contract is exercised). Ex. €50,000; XP= $1.49/€ ($74,500); Prem=$0.02/€ ($1,000)If S1= $1.53/€ (or $76,500), how much would be the writer’s “k”, if the option is exercised?Answer: –$1,000.Sol. k = Selling Price – (Purchase Price – Premium)

= $74,500 – ($76,500 – $1,000) = –$1,000

The writer of the call option contract

The Covered Writer

• The “covered writer” will obtain currencies at S0 to have a stock or inventory just in case the contract is exercised.

Ex. Using the previous example €50,000; XP= $1.49/€ ($74,500); Prem=$0.02/€ ($1,000); S0= $1.47/€ ($73,500);

If S1= $1.53/€ (or $76,500), how much would be his “k”, if the option is exercised?Answer: $2,000.Sol. k = Selling Price – (Purchase Price – Premium)

= $74,500 – ($73,500 – $1,000) = $2,000

Decision: To comply with the contract Decision: To comply with the contractk= Sell price - (Puchase Price - Premium) k= Sell price - (Puchase Price - Premium) = $74,500 - ($76,500 - $1,000) = $74,500 - ($73,500 - $1,000) = -$1,000 = $2,000

k/unit = -1,000 / 50,000 k/unit = 2,000 / 50,000 = -$0.02/ € = $0.04/ €

IRR = k/ Total investment IRR = k/ Total investment = -$1,000/ $76,500 = $2,000/ $73,500 = -1.31% = 2.72%

CoveredWriter, Grantor, Seller

Naked

SUMMARYCall Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.53/€. Decision rule is ITM

Gains/Losses in an OTM Decision Rule

Example: Using the previous example

€50,000, XP= $1.49/€ ($74,500); Prem=$0.02/€ ($1,000)If at maturity, the S1 = $1.45/€, ($72,500), what could be the decision and “k” of the option holder as hedger or speculator?Answer: Let the contract expire and lose the $1,000 premium fee.

Solution: Profit (k) = - $1,000

Note: There will be no “k/unit” for both the hedger or speculator since both of them would let the contract expire.

Important: The intrinsic value in an OTM is always zero (0). The contract is either “with value” or “no value”.

Decision: Let the contract expire Decision: Let the contract expirek= -$1,000 I.V.= (S1 - XP) (# of units)IRR = -100% = ($1.45 - $1.49) (50,000)

= $0 (No intrinsic value)

k= I.V. - Total investment = $0 - $1,000 = -$1,000

IRR = k/ Total investment = -$1,000/ $1,000 = -100%

Hedger: Speculator:Owner, Holder or Buyer

SUMMARYCall Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.45/€. Decision rule is OTM

Example:

Using the previous example

€50,000, XP= $1.49/€ ($74,500; Prem=$0.02/€ ($1,000)

If at maturity, the S1 = $1.45/€, ($72,500), what could be the “k” of the writer?

Answer: It depends on whether the writer is “naked” or “covered”.

Naked Writer:

k = $1,000

Note: His IRR would be “N/A or Not Applicable” since there was no investment made.

Covered Writer:

k = Sell Price – (Purchase Price – Premium)

= $72,500 – ($73,500 – $1,000)

= $0

Gains/Losses in an OTM Decision Rule

Decision: Keep the options fee Decision: Sell position in the marketk= $1,000 k= Sell price - (Puchase Price - Premium)IRR = Not applicable = $72,500 - ($73,500 - $1,000)

= $0

k/unit = 0 / 50,000 = 0

IRR = k/ Total investment = 0/ $73,500 = 0

Writer, Grantor, SellerNaked Covered

SUMMARYCall Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.45/€. Decision rule is OTM

Put Option• Put Option:

No. of units: €50,000Strike Price/Exercise Price: $1.49/€Premium: $0.02/€Maturity: 1 yearS0: $1.47/€Settlement: In US$

Details of the call option:1. What is the contract size (per strike price or XP)?

$74,500 (50,000 x 1.49)2. How much is the cost (Premium)? $1,000

(50,000 x 0.02)3. What is the dollar value of the contract (per spot

rate)? $73,500 (50,000 x 1.47)

• Interpretation:No. of units (currency): €50,000Strike Price/Exercise Price: $1.49/€$74,500Premium: $0.02/€$1,000Maturity: 1 yearS0: $1.47/€$73,500Settlement: In US$

The holder of this “put option contract” can sell €50,000 one year from now at $1.49/€ (or $74,500). The contract fee amounts to $1,000. Presently, €50,000 can be exchanged at $1.47/€ (or $73,500).

Put Option

• The decision rule:

Contract rate vs. Market rate

1. In the money (ITM): XP > S1

2. Out of the money (OTM): XP < S1

3. At the money (ATM): XP = S1

Put Option

Three Scenarios on maturity date:

Example:€50,000 put option with an XP of $1.49/€ while S0 = $1.47/€

Three (3) scenarios Decision Rule Decision

a. S1 = $1.45/ € 1.49 > 1.45 ITM Exercise contract.

b. S1 = $1.50/ € 1.49 < 1.50 OTM Let contract expire.

c. S1 = $1.49/ € 1.49 = 1.49 ATM Exercise the contract.

Intrinsic value of a put option and the hedger’s “k”

• In a put option, if the strike price is more favorable (i.e., a better “sell”) compared with the market price, the option is said to possess an intrinsic value.

Example: Using the previous example

€50,000; XP=$1.49/€ ($74,500); Prem= $0.02/€ ($1,000); S0=$1.47/€($73,500)

If at maturity, the S1 = $1.45/€, ($72,500), what could be the decision and “k” of the put option holder if he bought €50,000 at S0.Answer: Exercise the option and breaks-even.Solution: Profit (k) = Selling price – (Purchase Price + Premium)

= $74,500 – ($73,500 + $1,000) = $0

Note: In this particular instance, the “owner” acts as “hedger”.

Selling the option at its intrinsic value

Example: Using the previous example€50,000; XP=$1.49/€ ($74,500); Prem= $0.02/€ ($1,000);

S0=$1.47/€($73,500)

If at maturity, the S1 = $1.45/€, ($72,500), what is the contract’s intrinsic value? Answer: $2,000

Intrinsic Value = (XP – S1) (No. of units) = (1.49 – 1.45) (50,000) = $2,000

• How much is the “k” of the holder (who may be referred to as a “speculator” if he sells the contract at its intrinsic value)?Answer: $1,000

Solution: k = I.V. – Total investment = 2,000 – 1,000 = $1,000

His total investment is only the premium fee of $1,000

Decision: Exercise the contract Decision: Exercise the contractk= Selling price - (Puchase Price + Premium) I.V.= (XP - S1) (# of units) = $74,500 - ($73,500 + $1,000) = ($1.49 - $1.45) (50,000) = $0 = $2,000

k/ unit = 0 / 50,000 k= I.V. - Total Investment = $0 = $2,000 - $1,000

= $1,000

IRR = k/ Total investment IRR = k/ Total investment = $0/ ($73,500+$1,000) = $1,000/ $1,000 = 0% = 100%

Owner, Holder or BuyerHedger: Speculator:

SUMMARYPut Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.45/€. Decision rule is ITM

The writer of a put option contract

• The seller of the put option contract is called the “writer”. He buys the currency stipulated in the contract (from the holder) and sells it at S1 (only if the contract is exercised). Example. Using the previous example €50,000; XP=$1.49/€ ($74,500); Prem= $0.02/€ ($1,000); S0=$1.47/€($73,500)If at maturity, the S1 = $1.45/€ ($72,500), what is the writer’s “k” in an ITM decision rule? Answer: -$1,000.Sol. k = Selling Price – (Purchase Price – Premium)

= $72,500 – ($74,500 – $1,000) = -$1,000

Computing for “k” when no put option is purchased

• In cases, no put options were availed/purchased, the investor of €50,000 who bought it at S0 would exchange it at S1.

Example: If one bought €50,000 at S0= $1.47/€ ($73,500), if S1= $1.45/€ ($72,500), how much would be the investor’s “k”, if no option was availed or purchased?

Answer: –$1,000Sol. k = Selling Price – Purchase Price

= $72,500 – $73,500 = –$1,000

Decision: To comply with the contract Decision: To comply with the contractk= Sell price - (Puchase Price - Premium) k= Sell price - Puchase Price = $72,500 - ($74,500 - $1,000) = $72,500 - $73,500 = -$1,000 = -$1,000

k/unit = -1,000 / 50,000 k/unit = -1,000 / 50,000 = -$0.02/ € = -$0.02/ €

IRR = k/ Total investment IRR = k/ Total investment = -$1,000/ $74,500 = -$1,000/ $73,500 = -1.35% = 1.36%

No OptionWriter

SummaryPut Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.45/€. Decision rule is ITM

An OTM situation and the holder’s “k”

Example: Using the previous example €50,000, XP= $1.49/€ ($74,500); Prem=$0.02/€ ($1,000)If at maturity, the S1 = $1.53/€, ($76,500), what could be the decision and “k” of the option holder as hedger or speculator?Answer: Let the contract expire and lose the $1,000 premium fee.

Solution: k = - $1,000

Note: There will be no “k/unit” for both the hedger or speculator since both of them would let the contract expire.

Important: The intrinsic value in an OTM is always zero (0). The contract is either “with value” or “no value”.

Decision: Let the contract expire Decision: Let the contract expirek= -$1,000 I.V.= (XP-S1) (# of units)IRR = -100% = ($1.49 - $1.53) (50,000)

= $0 (No intrinsic value)

k= I.V. - Total investment = $0 - $1,000 = -$1,000

IRR = (k/TI) = -$1,000/$1,000 = -100%

Hedger: Speculator:

SummaryPut Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.53/€. Decision rule is OTM

Example:

Using the previous example

€50,000, XP= $1.49/€ ($74,500); Prem=$0.02/€ ($1,000)

If at maturity, the S1 = $1.53/€, ($76,500), what could be the “k” of the writer?

Answer: $1,000

Writer:

k = $1,000

Note: His IRR would be “N/A or Not Applicable” since there was no investment made.

An OTM situation and the writer’s “k”

Example:

Using the previous example, the investor bought €50,000 at S0 =$1.47/€ ($73,500) and waits at the next S1. If at S1 = $1.53/€, ($76,500), what could be the “k” of the investor?

Answer: $3,000

Solution:

k = S1 – S0

= $76,500 - $73,500

= $3,000

No Option Availed

Decision: To comply with the contract Decision: To comply with the contractk= $1,000 k= Sell price - Puchase PriceIRR = N/A = $76,500 - $73,500

= $3,000

k/unit = 3,000 / 50,000 = $0.06/ €

IRR = k/ Total investment = $3,000/ $73,500 = 4.09%

Writer No Option

SummaryPut Option. €50,000; XP=$1.49/€; Prem=$0.02; S0=$1.47/€; S1=$1.53/€. Decision rule is OTM

Currency Option Quotation and Prices

Option and Strike Calls - LastUnderlying Price Aug. Sept.

Dec.62,500 Swiss francs-cents per unit58.51 56 - - 2.7658.51 56 ½ - - -58.51 57 1.13 - 1.7458.51 57 ½ 0.75 - -58.51 58 0.71 1.05 1.2858.51 58 ½ 0.50 - -58.51 59 0.30 0.66 1.2158.51 59 ½ 0.15 0.40 -58.51 60 - 0.31 -

Currency Option Quotation and Prices

Option and Strike Puts - LastUnderlying Price Aug. Sept.

Dec.62,500 Swiss francs-cents per unit58.51 56 0.04 0.22 1.1658.51 56 ½ 0.06 0.30 -58.51 57 0.10 0.38 1.2758.51 57 ½ 0.17 0.55 -58.51 58 0.27 0.89 1.8158.51 58 ½ 0.50 0.99 -58.51 59 0.90 1.36 -58.51 59 ½ 2.32 - -58.51 60 2.32 2.62 3.30

Forward Rate Sensitivity Standard foreign currency options are priced around

the forward rate (which is central to valuation) because the current spot rate and both the domestic and foreign interest rates are included in the option premium calculation.

The option pricing formula calculates a subjective probability distribution centered on the forward rate.

(Note: This approach does not mean that the market expects the forward rate to be equal to the future spot rate; it is simply the result of the arbitrage pricing structure of options).

The forward rate focus also provides a helpful information for the trader managing a position.

Spot Rate Sensitivity (Delta) As long as the option has time remaining before

expiration, the option will possess time-value element.

This characteristic is one of the primary reasons why an American-style option, which can be exercised on any day up to and including the expiration date, is seldom actually exercised prior to expiration.

Over-the-Counter Options

OTC options – are most frequently written by bank for US dollars vs. pounds, Swiss francs, yen, Canadian dollars, and euro.

Its main advantage is that they are tailored to specific needs of firms.

Financial institutions are willing to buy/write that vary by amount, XPs, and maturities.

Firms normally “place a call” to banks, specify the currencies, maturity, strike rates, and ask for an indication – a bid/offer quote. Banks take a few minutes to a few hours to price the option and return the call.

Advantages1. Flexible

Disadvantages1. Difficult to trust a

counter-party2. May not get the

price it wants3. Liquidity problem*4. Default risk*

*To solve this problem, enter a currency futures contract

Over-the-Counter Options

Option Pricing and Valuation The value of a call option is actually the sum of two

(2) components:Total value (premium) = I.V. + Time Value

Intrinsic value is the financial gain if the option is exercised immediately. Intrinsic value will be zero when the option is OTM – i.e. when the XP is above the market price – since no gain can be derived from exercising the option.

The time value of an option exists because the price of the underlying currency, the spot rate, can move further and further into the money between the present time and the option’s expiration date.

(Note: An investor will pay something today for an OTM option, i.e., it has zero I.V.) on the chance that the spot rate will move far enough before maturity to move the option into the money.)

Components of Option Pricing The pricing of a currency option combines six

elements.1. Present spot rate, $1.70/£2. Time to maturity, 90 days3. Forward rate for matching maturity (90 days), $1.70/£4. US dollar int. rate, 8.00% p.a.5. British pound int. rate, 8.00% p.a.6. Volatility, the standard deviation of daily spot price movement, 10% p.a.

(Note: These assumptions are all that is needed to calculate the option premium)

Currency Option Pricing Sensitivity

If currency options are to be used effectively, either for the purposes of speculation or risk management, the individual traders need to know how option values – premiums – react to their various components.1. The impact of changing forward rates2. The impact of changing spot rates3. The impact of time to maturity4. The impact of changing volatility5. The impact of changing interest differentials6. The impact of alternative option XPs

History of Option Pricing

1877 – Charles Castelli wrote “The Theory of Options in Stocks and Shares” introducing the hedging and speculations aspects of options.

1900 – Louis Bachelier wrote “Theory of Speculation”

1955 – Paul Samuelson wrote “Brownian Motion in the Stock Market. Richard Kruizenga (student of Samuelson) wrote “Put and Call Options: A Theoretical and Market Analysis”

1962 – James Boness developed “A Theory and Measurement of Stock Option Value” which served as a model for Black, Scholes, and Merton.

The Black-Scholes-Merton model is a formula used to assign prices to option contracts. The model is named after Fischer Black and Myron Scholes, who developed it in 1973. Robert Merton also participated in the model creation, and this is why the model is sometimes referred to as the Black-Scholes-Merton model. All three men were college professors working at both the University of Chicago and MIT at the time.

An option valuation formula is used to estimate a fair price for an option contract.

The calculation takes into account the elements of time value, stock price variation, an assumed market interest rate, and the time left until expiration.

The Black-Scholes-Merton Pricing Model


The easiest way to understand the Black-Scholes-Merton formula intuitively is to consider what happens if a derivative (e.g., European call or put option) is exercised. It has two parts:

1. Value of the Cash to Buy the Option

Firstly, if the option is exercised the strike price is paid (say, $50). The strike price is paid only if the underlying asset (e.g., stock price) is above the strike at maturity. To work out the expected value, the probability should state that the stock price is above the strike at maturity. This probability = N(d2), and the strike price X. The expected value of this is just XN(d2) or the value of the cash flow at maturity.

To get the value of this cash flow today we need to discount it, and the discount factor is . So the value of the cash to buy the option today is XN(d2) .

2. Value of the Stock Received, if any

Secondly, if the option is exercised we get a unit of the stock. This is worth whatever the stock price is in the market at maturity. (Note: This only happens if the underlying stock price is above the strike at maturity).

The expected value should then be proportional to S, the stock price today, and can be written as SN(d1). That is to say, SN(d1) is the expected value that is equal to the final stock price if the final stock price is above the strike, and equal to zero if the final stock price is below the strike.


The Black-Scholes-Merton Formula CO = SN(d1) - Xe-rTN(d2)

Where:

d1 = [ln(S/X) + (r + σ2/2)T]/ σ √T and

d2 = d1 - σ √TWhere:C0 = current option valueS = current stock priceN(d) = the cumulative probability function that a random draw

from a standard normal distribution, φ(0,1) will be less than (d).

X = exercise pricee = 2.71828, the base of the natural log functionr = risk-free interest rateT = time to option maturity, in yearsln = natural logarithm functionσ = standard deviation of the annualized continuously

compounded rate of return on the stock

The equation can be dissected into two parts. The first part (SN(d1)) is simply the expected benefit on outright purchase of stock multiplied by change in the call premium with respect to the change in price.

The second part (Xe-rTN(d2)) is the present value of the option price on the time it expires.

When you subtract the latter from the former, that then becomes the value of a fair market call option (CO) premium.



Let us assume you would like to know the value of an option to purchase one share of XYZ Company stock for $95. The current price of the shares is $100, and the option expires in three months (one-quarter year). Assuming that the stock pays no dividends, the standard deviation of the stock’s returns is 50% per year, and the risk-free rate is 10% per year, we can calculate that the value of the option, even though it is out of the money right now, is as follows:d1 = [ln($100/$95) + (.10 + .52/2).25]/ .5√.25 = .43d2 = .43 - .5√.25 = .18N(.43) = .6664N(.18) = .5714 Thus, the value of the call options is: CO = 100 x .6664 – 95e-(.10)(.25) x .5714 = 66.64 – 52.94 = $13.70

Note that this formula values a call option. The Black-Scholes-Merton model can be used to price other derivatives, including puts. To value a put option (PO), use the value of the call option to solve for the value of the put option, as follows:

PO = CO + PV(X) - S = CO + X-rT - S = $13.70 + $95e-(.10)(.25) - $100 = $6.35


Assumptions Underlying the Black-Scholes-Merton Model

1. Stock price behavior corresponds to the lognormal model with μ and σ constant.

2. There are no transaction costs or taxes. All securities are perfectly divisible.

3. There are no dividends on the stock during the life of the option.

4. There are no riskless arbitrage opportunities.

5. Security trading is continuous.6. Investors can borrow or lend at the same

risk-free rate of interest.7. The short-term risk-free rate of interest, r,

is constant.

As one can see the annual volatility of the stock price is of tremendous importance in calculating the value of a put or call option.

This is be because the holder of the option wants the option price to reflect the probability that the underlying asset, the stock, will reach the option strike price.

In more volatile times, option sellers will charge more because the probability is greater that the stock might reach the option strike price than in quiet times.


1.The model is very predictive. The model tends to overvalue deep OTM calls and undervalue ITM calls. It tends to misprice options that involve high-dividend stocks.

2.Several of the model’s assumptions make it less than 100% accurate, since it assumes that the risk-free rate and the stock’s volatility are constant. It also assumes that stock prices are continuous and that large changes (ex. mergers) do not occur. Likewise, it assumes stock pays no dividends until after expiration. Lastly, analysts can only estimate a stock’s volatility instead of directly observing it, as they can for other inputs.

Criticisms to the Model

1.The model represents a major contribution to the efficiency of the options and stock markets and it is still one of the most widely used financial tools on Wall Street.

2.Besides providing a dependable way to price options, it helps investors understand how sensitive an option’s price is to stock price movements. This in turn help investors maximize the efficiency of their portfolios by giving them a way to calculate hedge ratios and implement portfolio insurance.

3.Many financial theorists claim the model’s introduction indirectly increased the volatility of the stock and options markets by encouraging more trading. Others claim the model steadies the markets because of its ability to measure equilibrium pricing relationships. When these relationships are violated, arbitrageurs are usually the first to discover and exploit mispriced options.

Defense to the Model

Documents

Options: Call and Put. Option Contract Option contract – is an agreement to buy or sell an underlying asset within a specified time period (exercise period)