Basel IV Framework Equity Derivatives Risk

Managing Market Risk Under The Basel IV Framework
Copyright © 2016 CapitaLogic Limited
Chapter 9
Equity Derivatives
Managing Market Risk Under The Basel IV Framework
The Presentation Slides
Website : https://sites.google.com/site/quanrisk
E-mail : quanrisk@gmail.com

Copyright © 2016 CapitaLogic Limited 2
Declaration
Copyright © 2016 CapitaLogic Limited.
All rights reserved. No part of this presentation file may be
reproduced, in any form or by any means, without written
permission from CapitaLogic Limited.
Authored by Dr. LAM Yat-fai (林日林日林日林日辉辉辉辉),
Principal, Structured Products Analytics, CapitaLogic Limited,
Adjunct Professor of Finance, City University of Hong Kong,
Doctor of Business Administration (Finance),
CFA, CAIA, FRM, PRM.

Equity futures
Equity options
Equity index futures
Equity index options
Trinomial tree
Outline

Equity futures
A forward like agreement
Equity as underlying
Exchange as counterparty
Standard strike price
Standard maturity date
Standard quantity
Physical settlement
Margin deposits from investor

Equity futures exchange
OneChicago
http://www.onechicago.com
Equity futures specification
http://www.onechicago.com/?page_id=751
Equity futures prices
http://www.onechicago.com/?page_id=20781
Example 9a.1

Functional purposes
To speculate on
the down trend of
equity price
To speculate on
the up trend of
equity price
Speculation
To hedge the
long position in
an equity
To hedge the
short position in
an equity
Hedging
ShortLongPosition

Dividend effect
Dividends
The interim cash inflows from holding an equity
investment
Once announced, becomes a deterministic part of equity
price
Virtually no risk
Present value of dividends (“PVD”)
Sum of present values of dividends
Do not contribute to the payoff of equity derivatives

Cash flows
EquityK
Out flow at
maturity
K = (S0 - PVD)exp(rdT)
Equity
None
Long
Strike rate
K
In flow at
maturity
Cash flow at
origination
ShortPosition

Valuation
f
Payoff
Strike rate
S0 - PVD - Kexp(-rdT)
= (F - K) exp(-rdT)
Value
ST - KBenefit
K
Cash outflow
at maturity
ST
Equity price at
maturity

Future price
The strike price at which the value of an equity future
Interest rate parity
In general, future price moves in the same direction as current price
Equity futures are entered at future price to avoid cash outflow from either
party
( ) ( )
0 d
0 d
f = S - PVD - Fexp(-r T) = 0
F = S - PVD exp r T
Future price = Dividend adjusted current price
× Interest rate effect

Equity risk
Replicating portfolio
A long position in ONE share of equity
at dividend adjusted equity price = S0 - PVD
A short position in exp(-rdT) units of domestic currency
Equity risk
Equivalent to ONE share of equity at dividend adjusted
equity price
0 df = S - PVD - Kexp(-r T)

Equity risk factors for
international equity futures portfolio
Equity risk
Value
Quantity
Holding period
dispersion
Dividend adjusted
equity price
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency
FX rate

Equity futures
Equity options
Trinomial tree
Outline

Equity options
An option
Equity price as underlying
Standard strike price
Standard quantity
Physical settlement
Margin deposits from investor entering short positions
Exercise style
European – can be exercised at maturity only
American – can be exercised any time at or before maturity

Equity options exchange
Chicago Board Options Exchange
http://www.cboe.com
Equity options specification
http://www.cboe.com/products/equityoptionspecs
.aspx
Equity options prices
http://www.cboe.com/delayedquote/quotetable.as
px

Functional purposes
To speculate on the
down trend of equity
price
To speculate on the up
trend of equity price
Speculation
To construct trading strategies and structured
products
Product
development
To hedge the long
position in an equity
To hedge the short
position in an equity
Hedging
PutCallPosition

Cash flows
KSτ
In flow upon
exercise
Max[K - Sτ, 0]Max[Sτ - K, 0]Payoff
S τ < KSτ > KExercise at time τ
S τK
Out flow upon
exercise
Premium
Call
Cash flow at
origination
PutPosition

Valuation
Mark-to-market
Value = Option price from exchange
Mark-to-model
European style
Black-Scholes formulas
America style
Binomial tree
Trinomial tree
Special case
For an American call option with underlying equity paying no
dividend on and before maturity
Value of American call = Value of European call

Black-Scholes formula – Call
European equity call option
( ) ( ) ( )
1
2
*
0 0
*
0 1 d 2
*
0
d
2
d
1 1 -
2
d
2 1 2
-
*
0
S = S - PVD
c = S Φ d - Kexp -r T Φ d
2S σ
ln + r + T
K 2 1 t
where d = Φ(d ) = exp - dt
2σ T 2π
1 t
d = d - σ T Φ(d ) = exp - dt
22π
S : Dividen
∞
∞
  
        
 
 
 
 
 
∫
∫
d adjusted equity price
PVD: Present value of dividends

Black-Scholes formula – Put
European equity put option
( ) ( ) ( )
1
2
*
0 0
*
d 2 0 1
*
0
d
2
d
1 1
-
2
d
2 1 2 -
S = S - PVD
p = Kexp -r T Φ - d - S Φ -d
2S σ
ln + r + T
K 2 1 t
where d = Φ(-d ) = 1 - exp - dt
2σ T 2π
1 t
d = d - σ T Φ(-d ) = 1 - exp - dt
22π
∞
∞
  
        
 
 
 
 
 
∫
∫
*
0S : Dividend adjusted equity price
PVD: Present value of dividends Example 9a.2

User defined VBA function
for equity options valuation
BSValueEQ(Option type,
Strike rate,
Maturity,
Current equity price,
Volatility,
Domestic risk-free rate,
Present value of dividend)

Equity risk
A position in Delta shares of equity
at dividend adjusted equity price = S0 - PVD
A position in Delta’ units of domestic currency
Equity risk
Equivalent to Delta share of equity at dividend adjusted
equity price
*
0Call/Put = Delta × S + Delta' × K

Equity risk factors for
international equity options portfolio
Equity risk
Value
Quantity
Holding period
dispersion
Dividend adjusted
equity price
Standard
deviation
Holding period
Diversification
effect
Concentration
of equities
% change
dependency
FX rate
Volatility

Equity futures
Equity options
Trinomial tree
Outline

An equity futures like agreement
Equity index level as underlying
Standard strike level
Standard quantity
Cash settlement
Cash value = USD 250 per index point
Margin deposits from investor

Equity index futures exchange
CME Group
http://www.cmegroup.com
Equity index futures specification
http://www.cmegroup.com/trading/equity-
index/us-index/sandp-
500_contract_specifications.html
Equity index futures prices
http://www.cmegroup.com/trading/equity-
index/us-index/sandp-500.html

Dividend yield
Dividend yield (q)
The natural annual growth rate of equity index
In continuous compounding
Assume that dividends from the component equities will
be invested immediately back in the corresponding
equities
Future value
ST = S0exp(qT)
Present value
S0 = STexp(-qT)

Valuation
f
Payoff
Strike rate
S0exp(-qT) - Kexp(-rdT)
(F - K) exp(-rdT)
Value
ST - KBenefit
K
Cash outflow
at maturity
ST
Equity price at
maturity

Future price
The strike price at which the value of an equity index future
Interest rate parity
In general, future level moves in the same direction as current index level
Equity index futures are entered at future level to avoid cash outflow from
either party
( )
0 d
0 d
f = S exp(-qT) - Fexp(-r T) = 0
F = S exp r - q T
Future level = Current level
× Interest rate-dividend yield differential effect
  

Equity index risk factors for international
equity indices and equity index futures portfolio
Equity index risk
Value
Quantity
Holding period
dispersion
Equity index level
Standard
deviation
Holding period
Diversification
effect
Concentration
of equity indices
% change
dependency
FX rate

Equity futures
Equity options
Trinomial tree
Outline

An equity option like agreement
Equity index level as underlying
Standard strike level
Standard quantity
European style
Cash settlement
Cash value = USD 100 × Equity index points
Margin deposits from investor entering short positions

Equity index options exchange
Chicago Board Options Exchange
http://www.cboe.com
Equity index options specification
http://www.cboe.com/framed/pdfframed.aspx?co
ntent=/micro/spx/pdf/spx_qrg2.pdf&section=SEC
T_MINI_SITE&title=SPX+Fact+Sheet
Equity index options prices
http://www.cboe.com/delayedquote/quotetable.as
px?ticker=SPX

Cash flows
KST
In flow upon
exercise
Max[K - ST, 0]Max[ST - K, 0]Payoff
ST < KST > KExercise at maturity
STK
Out flow upon
exercise
Premium
Call
Cash flow at
origination
PutPosition

Valuation – Call
Black-Scholes formula
( ) ( ) ( ) ( )
1
2
0 1 d 2
0
d
2
d
1 1
-
2
d
2 1 2 -
c = S exp -qT Φ d - Kexp -r T Φ d
2S σ
ln + r - q + T
K 2 1 t
where d = Φ(d ) = exp - dt
2σ T 2π
1 t
d = d - σ T Φ(d ) = exp - dt
22π
∞
∞
           
 
 
 
 
 
∫
∫

Valuation – Put
Black-Scholes formula
( ) ( ) ( ) ( )
1
2
d 2 0 1
0
d
2
d
1 1 -
2
d
2 1 2
-
p = Kexp -r T Φ -d - S exp qT Φ -d
2S σ
ln + r - q + T
K 2 1 t
where d = Φ(-d ) = 1 - exp - dt
2σ T 2π
1 t
d = d - σ T Φ(-d ) = 1 - exp - dt
22π
∞
∞
           
 
 
 
 
 
∫
∫

Equity index risk
A position in Delta points of equity index
A position in Delta’ units of domestic currency
Equity risk
Equivalent to Delta points of equity index
0Call/Put = Delta × S + Delta' × K

Equity risk factors for international
equity indices and equity index options portfolio
Equity index risk
Value
Quantity
Holding period
dispersion
Equity index level
Standard
deviation
Holding period
Diversification
effect
Concentration
of equity indices
% change
dependency
FX rate
Volatility

Equity futures
Equity options
Trinomial tree
Outline

Approximation to
standard normal distribution
Standard normal distribution
Symmetric
Peak at the middle
Thin on both sides
Average = 0
Standard deviation = 1
Simple discrete approximation
(√2, ¼), (0, ½), (-√2, ¼)

Approximation to
log-normal equity price
Log-normal
Trinomial
[ ]
( )
( )
( )
2
T 0 d
2
+ T 0 d
2
0 T 0 d
2
- T 0 d
σ
S = S exp r - T + σ T × Normal 0,1
2
σ 1
S = S exp r - T + σ T × 2 Probability =
2 4
σ 1
S = S exp r - T + σ T × 0 Probability =
2 2
σ
S = S exp r - T + σ T × - 2
2
   
  
   
  
  
  
  
  
  
 
 
 
1
Probability =
4

 
 

Approximation to
European call option value
Payoff
Value
[ ]
[ ]
[ ]
( )
+ + T
0 0 T
- - T
d + 0 -
1
Payoff = Max S - K, 0 Probability =
4
1
2
1
4
1 1 1
Value = exp -r T × Payoff × + Payoff × + Payoff ×
4 2 4
 
 
 
Example 9a.3

Trinomial tree

Approximation to
log-normal equity price
No. of steps N
Time between steps ∆t = T / N
Equity price
( )
( )
( )
2
+ k+1 k d
2
0 k+1 k d
2
- k+1 k d
σ 1
S = S exp r - t + σ t × 2 Probability =
2 4
σ 1
S = S exp r - t + σ t × 0 Probability =
2 2
σ 1
S = S exp r - t + σ t × - 2 Probability =
2 4
  
∆ ∆  
  
  
∆ ∆  
  
  
∆ ∆  
  
Example 9b.2

Approximation to
European call option value
Payoff
Value
[ ]
[ ]
[ ]
( )
+ N + + T
0 N 0 0 T
- N - - T
k-1 d + k 0 k -
1
Value = Payoff = Max S - K, 0 Probability =
4
1
2
1
4
1 1
Value = exp -r t × Value × + Value × +
4 2
∆
( )
k
0 d + 1 0 1 - 1
1
Value ×
4
1 1 1
Value = exp -r t × Value × + Value × + Value ×
4 2 4
 
 
 
 
∆  
 
⋮
Example 9b.3

Approximation to
European put option value
Payoff
Value
[ ]
[ ]
[ ]
( )
+ N + + T
0 N 0 0 T
- N - - T
k-1 d + k 0 k -
1
Value = Payoff = Max K - S , 0 Probability =
4
1
2
1
4
1 1
Value = exp -r t × Value × + Value × +
4 2
∆
( )
k
0 d + 1 0 1 - 1
1
Value ×
4
1 1 1
Value = exp -r t × Value × + Value × + Value ×
4 2 4
 
 
 
 
∆  
 
⋮
Example 9b.4

Approximation to
American call option value
Payoff
Value
[ ]
[ ]
[ ]
( )
+ N + + T
0 N 0 0 T
- N - - T
k-1 d + k 0 k
1
4
1
2
1
4
1 1
Value = Max exp -r ∆t × Value × + Value ×
4 2
( )
- k k-1
0 d + 1 0 1 - 1 1
1
+ Value × , S - K
4
1 1 1
Value = Max exp -r t × Value × + Value × + Value × , S - K
4 2 4
  
  
  
  
∆   
  
⋮
Example 9b.5

Approximation to
American put option value
Payoff
Value
[ ]
[ ]
[ ]
( )
+ N + + T
0 N 0 0 T
- N - - T
k-1 d + k 0 k
1
4
1
2
1
4
1 1
Value = Max exp -r ∆t × Value × + Value ×
4 2
( )
- k k-1
0 d + 1 0 1 - 1 1
1
+ Value × , K - S
4
1 1 1
Value = Max exp -r t × Value × + Value × + Value × , K - S
4 2 4
  
  
  
  
∆   
  
⋮
Example 9b.6

Equity paying dividend
Equity price
S0 = S*
0 + PVD
Dividend adjusted equity price
Follow the equity price tree
Present value of dividends
Grow at risk free rate
Examples 9c

Basel IV Framework Equity Derivatives Risk

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