MONASH
BUSINESS
SCHOOL
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.
2
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
3
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.
4
No arbitrage portfolio
• The portfolio is riskless and must therefore earn the risk-free rate. Why?
5
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)
6
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
7
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)]
8
One step model:
9
• 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:
10
Construct a no-arbitrage hedge portfolio
11
Construct a no-arbitrage hedge portfolio
12
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
13
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
14
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
15
Activity #2 – determine the future value of a portfolio
16
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
17
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
18
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
19
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
20
Activity #4 – determine the value of a binomial call
21
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
22
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
23
Outputs from the one-step binomial model:
24
25
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
26
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 ☺)
The Black-Scholes model
27
Size of derivative market
Source: BIS
https://www.bis.org/publ/otc_hy2205.htm#:~:text=The%20gross%20market%20value%20of%20derivatives%20contracts
%20%E2%80%93%20a%20measure%20of,%2C%20right%2Dhand%20panel).
28
Derivatives markets
Source: STATISTA
https://www.statista.com/statistics/272832/largest-international-futures-exchanges-by-number-of-contracts-traded/“
29
BSM
30
BSM - inputs
Type Input
Contract • Maturity, t
• Strike, K
Market (observable) • Current spot rate, S0
• RFR, r
Implied (non-observable) • Volatility, σ
31
BSM – assumption groups
Assumption
group
Expanded assumption
The risky
asset
- the underlying price follows GBM/Wiener process
- constant σ (across all t and K)
- returns are normally distributed (per GBM)
- a future stock price is lognormally distributed around μ
- no dividends over the life of the option
The riskless
asset
- the risk-free medium used as an investment alternative
or source of funds is constant in price, with no restrictions
The option - European style
The market - prices are infinitely continuous, and do not gap
- no limits on liquidity
- no transaction costs, taxes
- null bid/offer spreads
- capacity to effortlessly short-sell
32
BSM – limitations to assumptions
Assumption …. referred to as …..
Extreme price moves in S Tail risk
Continuous, non-discrete trading Gap risk
Existence of market frictions (taxes,
commissions etc)
Liquidity risk
Constant volatility Volatility risk
33
The key to using BSM is to abandon its use
when market conditions were outside of these
assumptions
Assumptions made in BSM
34
Option
delta, Δ
The Black-Scholes model – value of a call option
Today’s
price
Borrowing
requirement
in
replicating
portfolio
… and
Probability
adjusted
expected
payment
Value of
long
position in
underlying
Probability
of St > K
35
• A conditional probability (based on a normal distribution) based on two factors:
1. St > K
2. Magnitude of St - K
- If K = 10 and St = 11, #1 is satisfied
- Return is $1
- If K = 10 and St = 12, #1 is satisfied
- Return is $2
• N(d1) therefore determines the expected value if St > K
• N(d1) represents the hedge applicable for the option
N(d1)
36
• N(d2) is the probability of the option expiring ITM
• N(d2) is the expected value of paying the strike price
• Since d2 = d1 – σ * sqrt(t), as σ and/or t approach zero, d1 and d2 converge
• As σ rises, d1 increases and d2 decreases
N(d2)
37
• N(d2) is the probability of the option expiring ITM
• N(d2) is the expected value of paying the strike price
• Since d2 = d1 – σ * sqrt(t), as σ and/or t approach zero, d1 and d2 converge
• As σ rises, d1 increases and d2 decreases
N(d2)
39
Interpreting the BSM formula (as the value of a
replicating portfolio)
• Price of a call is the sum of two positions:
- Value of long position in underlying asset: S0N(d1)
- Value of short position in risk-free borrowing: e−rTKN(d2)
c = S0N(d1) − e
−rTKN(d2)
• Price of a put is the sum of two positions:
- Value of long position in risk-free investment: e−rTKN(−d2)
- Value of short position in underlying asset: S0N(−d1)
p = e−rTKN(−d2) − S0N(−d1)
40
• We have seen that its predictions accord with put-call parity and the expected behavior
of option prices.
• But, how realistic are its assumptions?
- Recall that we have assumed GBM - daily returns in stock prices should be
normally distributed
- We have also assumed that volatility of daily stock returns is constant
- In reality, daily returns are skewed with a greater frequency of negative
returns observed than predicted by the normal distribution (termed a negative
skew).
- Volatility is also not constant - increasing in periods of market ‘crashes’.
• All of this can sometimes cause negative skew.
Is the BSM model realistic?
41
C + PVK = P + So
- Where:
- C = price of European call option
- PVK = present value of K (discounted at r)
- P = price of European put option
- So = current price of the underlying
Put-call parity review
42
C + PVK = P + So
- Where:
- C = $8.59
- PVK = 100e
(-0.05*1)
- P = $3.71
- So = $100.00
$8.59 + $95.12 = $3.71 + $100.00
Therefore, put-call parity holds
Put-call parity example
Activity #5 – Put-call parity
43
You are presented with the following data relating to options which have a strike rate of $98.48
The risk free rate, r, is: 8.7% p.a.
Time, t, is: 0.5 years
Underlying price, So, is: $101.08
Price of call option, c, is: $9.92
In order for put-call parity to hold, the price of the put option must be:
(Please display your answer to 2 decimal places.)
Activity #5 – Put-call parity
44
You are presented with the following data relating to options which have a strike rate of $98.48
The risk free rate, r, is: 8.7% p.a.
Time, t, is: 0.5 years
Underlying price, So, is: $101.08
Price of call option, c, is: $9.92
In order for put-call parity to hold, the price of the put option must be:
(Please display your answer to 2 decimal places.)
Answer: $3.13
45
Stock returns distribution
46
Stock volatility distribution
47
• Volatility (sigma, σ) is a measure of the uncertainty
about future returns of a stock
• The BSM formula tells us how to compute option
prices given the volatility of the underlying (and vice-
versa)
Volatility – historic/implied
Volatility – historic volatility
calculation
Day
Closing
stock
price (St) St/St-1
Daily return
ln(St/St-1)
0 20.00$
1 20.10$ 1.005000 0.004988 0.000024875570324339
2 19.90$ 0.990050 -0.010000 0.000100001666698615
3 20.00$ 1.005025 0.005013 0.000025125575532782
4 20.50$ 1.025000 0.024693 0.000609725116538169
5 20.25$ 0.987805 -0.012270 0.000150555172211698
6 20.90$ 1.032099 0.031594 0.000998203926179838
7 20.90$ 1.000000 0.000000 0.000000000000000000
8 20.90$ 1.000000 0.000000 0.000000000000000000
9 20.75$ 0.992823 -0.007203 0.000051881945515892
10 20.75$ 1.000000 0.000000 0.000000000000000000
11 21.00$ 1.012048 0.011976 0.000143429151987431
12 21.10$ 1.004762 0.004751 0.000022568226569997
13 20.90$ 0.990521 -0.009524 0.000090704319040436
14 20.90$ 1.000000 0.000000 0.000000000000000000
15 21.25$ 1.016746 0.016608 0.000275816908320610
16 21.40$ 1.007059 0.007034 0.000049477531016732
17 21.40$ 1.000000 0.000000 0.000000000000000000
18 21.25$ 0.992991 -0.007034 0.000049477531016732
19 21.75$ 1.023529 0.023257 0.000540881637727722
20 22.00$ 1.011494 0.011429 0.000130615088228894
20.000000 0.095310 0.003263
Std Dev of daily return: 0.01216
Trading days: 252
Volatility. estimate: 19.30%
Std. error: 3.05%
Volatility – historic volatility
calculation – activity #1 Date Closing price
3/10/2022 $20.00
4/10/2022 $20.12
5/10/2022 $19.80
6/10/2022 $19.66
7/10/2022 $19.75
8/10/2022 $19.78
9/10/2022 $19.60
10/10/2022 $19.77
11/10/2022 $19.78
12/10/2022 $19.86
13/10/2022 $19.83
14/10/2022 $19.77
15/10/2022 $19.58
16/10/2022 $19.49
17/10/2022 $19.45
18/10/2022 $19.43
19/10/2022 $19.53
20/10/2022 $19.64
21/10/2022 $19.59
22/10/2022 $19.60
23/10/2022 $19.68
Using the adjoining data, calculate
the following:
1) Annualised (historic) volatility
using a 252 day basis
2) Annualised (historic) volatility
using a 365 day basis
3) Daily Std. Dev (252 day bsis)
4) Based on #4, your prediction for
the upper and lower boundary
for price on 24/10/22
Volatility – historic volatility
calculation – activity #1
Using the adjoining data, calculate
the following:
1) Annualised (historic) volatility
using a 252 day basis
2) Annualised (historic) volatility
using a 365 day basis
3) Daily Std. Dev (252 day basis)
4) Based on #4, your prediction for
the upper and lower boundary
for price on 24/10/22
Date
Closing
price
Inter-day
return Ln Ln^2
3/10/XX 20.00$ 20.00000 Vol: 10%
4/10/XX 20.12$ 1.00600 0.00598 0.0000 Daily vol: 0.006299
5/10/XX 19.80$ 0.98410 -0.01603 0.0003
6/10/XX 19.66$ 0.99293 -0.00710 0.0001
7/10/XX 19.75$ 1.00458 0.00457 0.0000
8/10/XX 19.78$ 1.00152 0.00152 0.0000
9/10/XX 19.60$ 0.99090 -0.00914 0.0001
10/10/XX 19.77$ 1.00867 0.00864 0.0001
11/10/XX 19.78$ 1.00051 0.00051 0.0000
12/10/XX 19.86$ 1.00404 0.00404 0.0000
13/10/XX 19.83$ 0.99849 -0.00151 0.0000
14/10/XX 19.77$ 0.99697 -0.00303 0.0000
15/10/XX 19.58$ 0.99039 -0.00966 0.0001
15/10/XX 19.49$ 0.99540 -0.00461 0.0000
17/10/XX 19.45$ 0.99795 -0.00205 0.0000
18/10/XX 19.43$ 0.99897 -0.00103 0.0000
19/10/XX 19.53$ 1.00515 0.00513 0.0000
20/10/XX 19.64$ 1.00563 0.00562 0.0000
21/10/XX 19.59$ 0.99745 -0.00255 0.0000
22/10/XX 19.60$ 1.00051 0.00051 0.0000
23/10/XX 19.68$ 1.00408 0.00407 0.0000
-0.01613 0.00075
Daily Std. Dev:0.624%
Annual vol @252 days 9.91%
Std. Error: @252 days 1.57%
Annual vol @365 days 11.93%
Std. Error: @365 days 1.89%
51
• Daily volatility is unobservable if using daily price data. (If intraday volatility is required,
intraday returns must be captured.)
• Yet, we can observe (intraday) option prices
• Therefore, given an observable option price, what level of volatility is implied by the
observed price?
• This level of volatility is called the implied volatility (IV)
• Example: Consider a call option with S0 = 30, K = 30, r = 0.05. t = 0.5 and c = $2.89.
What is the value of σ in the BSM model that gives the observed price?
Volatility – historic/implied
52
• According to the BSM model, the price of any option (put or call), on a stock has the same
underling GBM process and therefore the same σ
• Therefore, the implied volatility (IV) inferred from any option on an asset should equal the IV
inferred from any other option on the same asset
• However, in every market, the BSM model fails this test
• The pattern observed in equity option markets suggests that traders do not price options
assuming stock returns are normal nor is volatility constant
• If we do not take this into account, we could incorrectly price options relative to market using
BSM
An indirect test of BSM model
53
The volatility smile
• Volatility smile/skew describes the empirical relationship observed between IV and strike
price across different options holding all else equal
• In equity option markets:
- Stock prices are lognormally distributed, but returns are normally distributed
- Stocks often exhibit sharp price decreases more frequently than predicted by BSM
- i.e. a price ‘skew’ is commonly observed
- a ‘smile’ is sometimes observed where options that are deep ITM and OTM
have increasing IV as the strike price moves away from the current stock
price
- Conclusion: Options traders demand additional compensation for risk, since BSM assumes a
NORMAL distribution – and extreme outliers are not “normal”
• Therefore, DITM or DOTM are valued higher by the market than BSM predicts
• The smile does not differentiate between DITM calls/DOTM puts of the same strike (due to put-call
parity)
54
The volatility smile
• Example:
• S0 = $10.00, K = $5.00, σ = 20%, t = 1, r = 4.00%
Using BSM, c = $0.0000 !!!
55
The volatility smile – what is a practical price for these?
• Example:
• S0 = $50.00, K = $50.00, σ = 20.0/21.0%, t = 1, r = 4.00%
» C = $4.96/5.15
» P= $3.00/3.19 = $0.19 spread
• S0 = $50.00, K = $45.00, σ = 20.0/21.0%, t = 1, r = 4.00%
» C = $8.03/8.17
» P = $1.27/1.41 = $0.14 spread
• S0 = $50.00, K = $35.00, σ = 20.0/21.0%, t = 1, r = 4.00%
» C = $16.44/16.47
» P = $0.07/0.10 = $0.03 spread
56
The volatility smile – what could possibly go wrong ?
• Potential issues if market makes sudden move, as described earlier
• Non-continuous market prices – inability to smoothly re-hedge
• Underlying market depth depletes
• Spreads (implied vol, underlying) widen
• Absence of gamma sellers
• Practical considerations ;
• A fundamental change in assumptions for the value
of the underlying
• Therefore, the assumption of a lognormal distribution
no longer apply
57
The volatility smile
• 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)
58
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)
59
The volatility smile
60
The volatility smile – TSLA stock options
Volatility skew/smile as at 10/3/22 for expiry 20/5/22
61
The volatility smile – AUD/USD
AUD/USD
S0
S
ig
m
a
,
re
la
ti
v
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t
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A
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62
The volatility smile – exercise
Given:
S0: $20,000.00
t (years): 0.042 years
r (% p.a.): 6.50
K: $17,000.00
σ (for ATM): 25.00/27.00%
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?