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日期：2023-03-11 08:28

BFF5340 – Applied Derivatives

Topic 2 – Binomial Trees and Black Scholes Merton model

No-arbitrage approach

An option is a derivative - it ‘derives’ its value from the underlying asset given certain

conditions

When the underlying stock changes in value, so does the derivative

By combining the ‘right’ amount of the stock with an opposite investment in the option, we

can create a momentary no-arbitrage portfolio that must earn the risk-free rate

The ‘right’ amount is the sensitivity of the option value to a change in the underlying asset

price, known as the option’s delta.

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Delta - example

Assume a call option has a delta of 0.6

Therefore, for a small change in the stock price (say $1), the option’s value will

change by approximately $0.60

It is approximate because the relationship between the call price and stock price is curved (non-linear).

Delta is only the linear part of the total change that corresponds to the slope of the call option’s price

function (i.e. the gradient of the price function.) with respect to the stock price

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No arbitrage portfolio

If Δc = 0.6, then a portfolio that is long 0.6 shares and short 1 call option, will have

approximately no net change in value for a small change in the stock price

For example:

Let’s say S increases from $50 to $51. The option will change in value from say $4.00

to $4.60.

The change in value of our stock holding is:

+0.6(51  50) = $0.60

The change in value of the option holding is:

(4.60  4.00) = ?$0.60.

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No arbitrage portfolio

The portfolio is riskless and must therefore earn the risk-free rate. Why?

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No arbitrage portfolio

The portfolio is riskless and must therefore earn the risk-free rate. Why?

If there is no-arbitrage opportunities in the market, a portfolio that has no change in value

(riskless) must earn the same rate of return as an asset that also has no risk, i.e. ‘the

Law of One Price’.

Since delta depends on the value of the stock price itself, it will change from moment to

moment, so the no-arbitrage risk free rate of return is only earned over a very short

period of time when delta is constant.

(But, remember that delta is not constant – its change is measured as gamma)

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No arbitrage portfolio – a.k.a. “replication approach”

To value an option using the no-arbitrage approach:

Construct a riskless portfolio using delta and a known change in future

stock prices

Discount the riskless future cash flow at the risk free rate. (The risky

discount rate y is not required to be specified.)

To implement this model we need to specify a process for future stock prices

The simplest model of future stock prices is the binomial tree

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Binomial model – assumptions/definitions:

Stock prices only appear at discrete points in time (and at no other times) defined by a large number

of equally spaced time intervals (or steps)

S0 is known

Future stock price can only take one of two alternative values at the end of the time interval - an ‘up’

or a ‘down’ value relative to the current value. No other values are possible.

The stock price assumes a “random walk” path (i.e. equal probability of rising or falling)

A node is defined by a unique stock price and time. (Nodes can be described by the sequence of ups

and downs applied to the initial stock price over the time steps from the initial price)

A node represents a probability, and not a certainty, of observing a given stock price at a point in time

[Clearly this is a highly stylised and unrealistic representation of real stock price changes but it will

help illustrate the construction of no-arbitrage portfolios through time (time intervals) and space

(stock prices)]

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One step model:

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Consider a one-step 3-month binomial model (ΔT = 0.25)

The ‘up’ factor is given here as u = 1.1 and the ‘down’ factor is d = 0.9

What is the price of a call option with strike price K = 21?

One step model:

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Construct a no-arbitrage hedge portfolio

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Construct a no-arbitrage hedge portfolio

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To make the hedge portfolio riskless we need to solve for the value of Δf

such that Πu = Πd .

Activity #1 – determine the portfolio delta

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Assume the following:

So: $19.72

Call K: So+$2.00 (ie $2.00 OTM relative to So)

The up factor ("u") in the underlying price is 1.25 while the down factor is 2-u

Calculate the portfolio delta (displayed as a percentage, to 1 decimal place).

Activity #1 – determine the portfolio delta

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Assume the following:

So: $19.72

Call K: So+$2.00 (ie $2.00 OTM relative to So)

The up factor ("u") in the underlying price is 1.25 while the down factor is 2-u

Calculate the portfolio delta (displayed as a percentage, to 1 decimal place).

Answer: Delta is 29.7%

Pricing a call option with a one-period binomial tree

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Activity #2 – determine the future value of a portfolio

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A one-step binomial call option has a delta of 0.2

The value of the asset in its "up" position is $24.20. Further assume that the "up"

position is $2.27 in-the-money relative to the current So.

Calculate the future value of the portfolio. (Please display your answer in $ to 2

decimal places)

Activity #2 – determine the future value of a portfolio

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A one-step binomial call option has a delta of 0.2

The value of the asset in its "up" position is $24.20. Further assume that the "up"

position is $2.27 in-the-money relative to the current So.

Calculate the future value of the portfolio. (Please display your answer in $ to 2

decimal places)

Answer: $2.57

Activity #3 – determine the today’s value of a portfolio

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If the value of a riskless portfolio at time (t) is: $46.37 and:

- t (years): 0.26

- Risk free rate (% p.a.): 6.50

Calculate today's value of the portfolio (displayed in $, to 2 decimal places)

Activity #3 – determine the today’s value of a portfolio

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If the value of a riskless portfolio at time (t) is: $46.37 and:

- t (years): 0.26

- Risk free rate (% p.a.): 6.50

Calculate today's value of the portfolio (displayed in $, to 2 decimal places)

Answer: $45.59

Determine the option price

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Activity #4 – determine the value of a binomial call

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Determine the current value of a binomial call option (displayed in $ rounded to 2

decimal places) that has the following characteristics:

call delta: 0.28

So: $57.24

value of riskless portfolio: $13.72

Activity #4 – determine the value of a binomial call

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Determine the current value of a binomial call option (displayed in $ rounded to 2

decimal places) that has the following characteristics:

call delta: 0.28

So: $57.24

value of riskless portfolio: $13.72

Answer: $2.31

Convergence of model with BSM/Binomial

As time steps get smaller, the probability distribution of terminal stock prices

approach the continuous lognormal probability distribution of the BSM/CRR

model.

Therefore, pricing under the two models (BSM/Binomial) converges

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Outputs from the one-step binomial model:

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Developed in 1973 by Fischer Black, Myron Scholes with subsequent assistance by

Robert Merton

(.. drawing on earlier work by A. James Boness, 1962 Ph.D dissertation, Bachelier,

Samuelson, others)

(Myron Scholes was a Nobel Prize winner and one of the founders of LTCM, which

subsequently imploded in 1998, forcing the US govt to intervene)

It has been said that it is actually not a model – rather, it’s a converter of option prices to

implied volatility

A mathematically mechanism of valuing a predicting of future market movement

In early days, it was a means of identifying arbitrage

The Black-Scholes model

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An equation that estimates the theoretical value of an option considering a range

of inputs and assumptions

Determines the price of an option using risk adjusted probabilities and the cost of

funding the premium

It is the calculation of the value of a delayed decision

A replication equation

An equation that many believed was “the devil” represented as a formula

(… mainly driven by users taking all assumptions upon which it was based

prima-facie and not questioning what the outcome would be if the assumptions

failed ?)

Smiles differ in structure between asset types – i.e. FX vs. equities (i.e. left

skew/right skew)

Assets display differing outlier price characteristics after a “jump” in price

Option prices (and therefore implied volatility) will change according to

demand/supply

Some assets move suddenly over a sustained price range, while others can

suddenly price adjust with no sustained move – every asset has its own

price profile personality

Some assets often display mean-reversion (energy, agriculture), others do

not (investment assets)

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The volatility smile – the inside job

They are a manifestation of years of experience by options traders that things can get very ugly if there

is a sudden move in the underlying

Normal distributions of price returns are often challenged – with multi-sigma moves occurring more

frequently than the BSM suggests

The VS has substantial implications for portfolio management, especially in regard to gamma (This will

be discussed in greater depth later in the unit.)

Seasoned portfolio managers will wish to be “long the wings” – resulting in greater demand for DITM or

DOTM options than the BSM model might suggest

Some more contemporary pricing models incorporate smiles through the use of jump diffusion (i.e. a

normal distribution with random jumps)

What is the two-way price of an ATM option (K=$20,000)?

What is the two-way price and delta of this option (K=$17,000)?

What is the probability of exercise for this option?

What (two-way) value of σ would be recommended for this option?