BOUNDS, AND ROBUST HEDGING
OF THE AMERICAN OPTION
Anthony Neuberger
(University of Warwick)
University of Warwick, 11/12 July 2008
Objective
• What is the value of American as opposed to Europeanstyle rights?
– when are they particularly valuable?
– the essence of being American is the right to chose the timing of
the exercise decision
– value depends on what information you will get
– most past work in the area assumes that asset price process is
Markov – so only useful information is current stock price
– very restrictive; eg future option prices likely to contain additional
information
2
Problem Formulated
• Assume we have a complete set of
European options (all strikes, maturities)
– frictionless markets, known interest rates, no
dividends on asset
• What is the upper bound on the price of
the American option?
– under what process is that bound achieved?
3
Supplementary Questions
• What is the trading strategy that enforces
the bounds?
– does it provide a useful hedging strategy?
– when does it make money?
– can it be refined without losing robustness?
• How do robust hedging strategies
compare with conventional dynamic
hedging strategies?
4
Outline of Seminar
•
•
•
•
Two upper bounds for American put
The generalised European portfolio
Rational bounds
Numeric examples
– how wide are the bounds?
– what do the bounding portfolios look like?
– what do the bounding processes look like?
• Tightening the bounds
– imposing a floor on implied volatility
– comparison with dynamic hedging
• Conclusions
Will speak only about the American put option, but
argument works for any American option
5
Upper Bounds on Amcan Option
• Work with discounted prices
• American option is function A(S,t)
– if exercised at t, receive A(St,t)
• Portfolio of European options V(S,t)
– pays V(St,t)- V(St,t+dt) over interval (t, t+dt)
T
– pays  0
V  St , t 
dt over [0,T]
t
6
Proposition
• If V is convex in S, decreasing in t, with V(.,T)=0 and V 
A, then V dominates A
Proof:
Strategy is to do nothing until exercise (at time t*), then
delta hedge. Terminal cash is:
t * V  S , t 
T  V  S , t 

V St , t 
t
t

dt  A  St * , t *    
dt 
dS 
0
t*
t

t
S

T  V  S , t 

V  St , t 
V St , t 
t
  A  St * , t *    
dt 
dS  since
0
t*

t

S

t


T
 2V  St , t 
  A  St * , t *    dV since
0
t*
S 2
  A  St * , t *   V  St * , t *  since V (.,T )  0
 0.
7
Example – European Put
• A = [Ke-rt – S]+ ; take V = A
– satisfies assumptions
easiest to work in nominal terms
– European portfolio pays at nominal rate rK so long as
put in the money, and [K – S]+ at T
– do nothing until exercise
• then pay K – S, borrow K and buy share;
• cash flow from European portfolio pays interest on loan so
long as S < K
• if S > K, sell share and repay debt
• if S remains below K liquidate at T
8
Look for cheapest such strategy
• Use discrete space/time formulation
• Price of {V} is a linear function of {vj,t} in each state
• Monotonicity with t, convexity with S, domination of put
pay-off are all linear inequalities
• Search for cheapest {V} is an LP - call it LP1
• Readily show that the feasible set is non-empty and
bounded
• The solution is an upper bound on the value of the
American put
– but is it the least upper bound?
9
The dual problem
• Consider a regime switching model:
– two regimes I = 1 (initial state) and 2
• no switching from 1 to 0
– consider processes where (It , St) is Markov
– specify transition matrix P
• make sure EP[Max{0, St – x} ¦S0] = CE(t, x) for all nodes (t, x)
• also EP [St+1¦St, It] = St at all nodes
• Value American option assuming it is exercised when
regime switches v(P) = EP[(Kt – St)dIt]
– v is a feasible price for the American
10
LP2
• Find feasible process P to maximise v(P)
– can also be formulated as an LP
– call it LP2
• Main result: LP1 and LP2 are primal/dual,
so solution to LP1 is not only an upper
bound but the upper bound
11
How wide are the bounds?
• Formulate as an LP in discrete space/time
– use geometrically spaced price nodes Sj = S0uj
– use equally spaced time nodes
– assume all European options priced on same implied
volatility, ie as if:
St u
with probability  1+u 

St 1  St u with probability u  1+u 
S
otherwise.
 t
where  log u    2 t
2
and   0,1.
12
S0 = 100, T = 1 year,  = 10%
dt = 1/40 years
Strike Upper Amercn PE(K, ≤T) PE(K, T)
Ratio
(K) Bound diffusion
(2-3)/(1-3)
(1)
(2)
(3)
(4)
(5)
r = 2%
95
1.67
1.42
1.35
1.35
20%
100
3.76
3.22
3.03
3.03
26%
105
6.89
6.13
5.68
5.68
37%
110
10.84
10.12
10.00
9.17
15%
r = 5%
95
1.34
0.91
0.77
0.77
25%
100
3.44
2.43
1.94
1.94
33%
105
6.63
5.33
5.00
3.92
20%
110
10.68
10.00
10.00
6.82
0%
r = 10%
95
0.83
0.44
0.26
0.26
31%
100
2.90
1.61
1.01
0.79
32%
105
6.25
5.00
5.00
1.90
0%
110
10.46
10.00
10.00
3.76
0%
d0
dT
(6)
(7)
0.51
0.00
-0.49
-0.95
0.71
0.20
-0.29
-0.75
0.51
0.00
-0.49
-0.95
1.01
0.50
0.01
-0.45
0.51
0.00
-0.49
-0.95
1.51
1.00
0.51
0.05
13
Bound is very high…
(same data as in table)
12
r = 2%
r = 5%
r = 10%
10
Bound
8
Am
6
1 yr Eu
4
2
1 yr Eu
0
95
100
105
110
95
100
105
110
95
100
105
110
Strike
14
What do the bounding portfolios
look like?
• For low strike put
–
–
–
–
–
–
S = 100,
K = 95,
r = 10%,
T = 1 year,
 = 10%,
11 ex dates
15
What do the bounding portfolios
look like?
• For high strike put
–
–
–
–
–
–
S = 100,
K = 105,
r = 10%,
T = 1 year,
 = 10%,
11 ex dates
16
Nature of the hedging strategy that
enforces rational bounds
1.
2.
3.
4.
Write the American option at time 0
Buy the dominating European portfolio
Do nothing until option is exercised
Then delta hedge European portfolio to release
intrinsic value
but step 4 is not necessary – if European options are
traded can liquidate portfolio provided they trade at
least at intrinsic value
–
then you have a static hedging strategy
How well does it work?
17
The Horse Race
• Take a “true” returns generating model
– all options are priced according to the true model
– American option is exercised optimally
– one bank writes an American option at fair value,
buys the European option and liquidates at
exercise/expiry
– other bank does same but hedges dynamically using
incorrect model
• Race outcome depends on
– how incorrect the model of the dynamic trader
– how much weight we put on extreme losses
18
The Race
• Assume the world is Heston:
dSt St  v t dzts ; dv t     v t  dt   v t dztv ; E dzts .dztv    dt .
– dynamic trader uses underlying and the European option with same
strike and maturity to hedge
– assumes the world is Black-Scholes, but uses the European option
price to impute the current volatility
– constructs a portfolio that is delta-gamma neutral
– rebalances every period
– implement on a lattice (exact tri x tri – nomial process) with 100,000
simulations
• Note that since all transactions occur at fair prices and since we
assume no risk premia, all strategies are mean zero
19
The Result
Hedge Error
Strike
r=2%
95
100
105
r=10%
95
100
105
Std dev and 99%'ile
DeltaRational
gamma
bounds
0.09
(0.34)
0.18
(0.47)
0.30
(0.67)
0.36
(0.28)
0.54
(0.59)
0.63
(0.87)
0.36
(1.29)
0.78
(1.79)
0.00
(0.00)
0.79
(0.38)
1.04
(1.25)
0.00
(0.00)
•
•
•
•
•
•
•
Parameters
1 year maturity
rms volatility 10%
coefficient of variation of
variance = 1
mean reversion rate  = 2/yr
correlation  = 0
Sensitivities
DG hedge improves if vol of vol
declines and mean reversion
rate increases
Results not very sensitive to
initial vol or to correlation
20
Tightening the bounds
• Allowing for option to trade on intrinsic is
pessimistic
• Implied volatility is volatile, but does not go to
zero
• Suppose we put a floor on implied volatility …
S&P VIX (Volatility Index) (CBOE)
40
30
20
10
0
1993
1995
1997
1999
2001
2003
2005
2007
21
A volatility floor
• Consider an “instantaneous volatility contract”
(IVC)
– buy it at time t
– pays $1 if price next period is different from price
today
– price of contract is implied jump probability
• Assume a permanent floor on the price of IVC
– implies a minimum level of implied volatility for all
options
– means trader can sell IVCs against her portfolio
– easy to incorporate this constraint in LP
22
The outcome with a floor
Strike
(K )
Hedge Error - Standard deviation and 99%'ile
Delta-gamma
Robust hedge with floor on imp vol of
hedge
0%
5%
8%
10%
r=2%
95
100
105
r=10%
95
100
105
0.09
(0.34)
0.18
(0.47)
0.30
(0.67)
0.36
(0.28)
0.54
(0.59)
0.63
(0.87)
0.20
(0.14)
0.33
(0.38)
0.41
(0.66)
0.13
(0.24)
0.20
(0.40)
0.25
(0.45)
0.11
(0.35)
0.19
(0.43)
0.25
(0.47)
0.36
(1.29)
0.78
(1.79)
0.00
(0.00)
0.79
(0.38)
1.04
(1.25)
0.00
(0.00)
0.55
(0.25)
0.94
(0.91)
0.00
(0.00)
0.30
(0.30)
0.50
(0.77)
0.00
(0.00)
0.21
(0.73)
0.40
(1.19)
0.00
(0.00)
23
Conclusions from Horse race
• Rational bounds hedge generally has larger
standard error, but lower VAR than delta-gamma
hedge
• Floor on implied volatility greatly reduces
standard error and VAR, and retains substantial
robustness
• Conclusions depend on how far true process
departs from assumed model
• Robust hedges are not only robust but involve
no intermediate trading
24
Conclusions
• Demonstrated how to find rational bounds on the value of an
American option, and also robust hedges
• For reasonable parameters, possible value of being American
several times the value assuming a Markov diffusion
• Have characterized the processes that lead to extreme high values
– great uncertainty over future volatility
• Rational bounds allow for possibility of implausibly low volatilities –
can tighten bounds and get better robust hedging strategies through
restrictions on implied volatility
– appear to have considerable advantages over conventional dynamic
strategies when true process unknown
• General approach can be applied to other hedging problems
25