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?