MATH5350M-MATH5350M: Computations in Finance代写
时间:2023-04-11
Chapter 5
Greeks and hedging
strategies
Reading:
Hull (Fifth Edition) pp. 299–329
Hull (Sixth Edition) pp. 341–374
Hull (Seventh Edition) pp. 349–380
MATH5350M: Computations in Finance
Section 5.1: Dynamic hedging
Jonathan Ward, University of Leeds, 2023
Dynamical hedging
If an option pays h(ST ) at maturity, then the price of that option
at time t is Vt = V (St , t), where V (s, t) is given by the formula
V (s, t) = er(Tt) E
h(ST )
St = s.
The replicating trading strategy is to hold 't shares and t bonds,
where
't =
@V
@s
(St , t),
t = e
rtV (St , t) 'tSt.
The Delta of the option is
(s, t) =
@V
@s
(s, t).
Deriving the replicating strategy
Applying Ito¯’s lemma and the Black–Scholes SDE to Vt = V (St , t)
yields:
dVt =
✓
@V
@t
+ rSt
@V
@s
+
1
2
2S2t
@2V
@s2
◆
dt + St
@V
@s
dWt .
If Pt is the value of the portfolio, then by the self-financing
property we have:
dPt = tdBt + 'tdSt =
⇣
trBt + 'trSt
⌘
dt + 'tSt dWt .
Comparing the dWt terms gives:
't =
@V (St , t)
@s
.
tδdft= a ( Xt, t) dt tb cxt , ti dWt
then flxt , tlsatisfies
df ( a " t 1 =↓+ a ( t , t )哦 + th ( a, ( 对 ) ) ⽐ +b (… 登 dw
h的 dSte r Stodt
t σ Stdwt
thendo ( st , t ) = 毙+ r 外碌+ i[ o 欢 )at的
Value of post folio PE
= YEStt ψEBt
( prwcheot role ) dPt
= dE St t St dst t d ψ o dBo
self ,finan ing stdfo
+ 1Bt d ψt = w
Bond ot - ertBodst
: rBtdt
dpt = φt Ir Stdt t 5 bt dwt
) + ψt rBtdt
= φtzsttttrrJdttostfidu…
⼀
wcant PE = Et
s0 dUisa , t 1 = dPt
Delta for vanilla options
For European vanilla options:
C (s, t) =
d1(s, t)
, (call)
P(s, t) =
d1(s, t)
1, (put)
where
d1(s, t) =
log sK + (r +
1
2
2)(T t)
p
T t .
And how does it work in practice? . . .
MATH5350M: Computations in Finance
Section 5.2: Hedging in practice
Jonathan Ward, University of Leeds, 2023
Theoretical and practical hedging
I Theoretical hedging
I continuous trading (the number of shares changes at every
time instant),
I the strategy replicates the payo↵.
I Practical hedging (dynamic hedging)
I trades only in discrete time moments, e.g. weekly,
I the strategy does not replicate the payo↵,
I the strategy poses a risk of losses to the writer,
I the strategy sometimes yields profits.
The writer of an option may demand a premium (a price higher
than the theoretical price) to compensate for the hedging risks.
Example
A financial institution has sold a European call option on
100 shares which matures in 10 weeks. The parameters are
S0 = 100, K = 105, r = 0.05, = 0.20.
The theoretical price of the option is £192.
The financial institution trades weekly to rebalance the replicating
portfolio.
Closing out of the money
Shares Cost of Cum. cost Interest
Week Stock price Delta purchased purchase incl. interest cost
0 100.0 0.3435 34.35 3435 3435 3.3
1 99.2 0.2976 4.60 456 2983 2.9
2 99.5 0.2933 0.43 43 2943 2.8
3 100.2 0.3067 1.34 134 3080 3.0
4 97.7 0.1715 13.52 1320 1763 1.7
5 99.2 0.2093 3.78 375 2140 2.1
6 96.3 0.0727 13.67 1317 825 0.8
7 95.9 0.0361 3.66 351 475 0.5
8 98.0 0.0460 1.00 98 573 0.6
9 94.7 0.0001 4.59 435 139 0.1
10 92.2 0.0000 0.01 1 138
Call on 100 shares, S0 = 100, K = 105, r = 0.05, = 0.20.
Price of option plus interest is 192.
Closing in the money
Shares Cost of Cum. cost Interest
Week Stock price Delta purchased purchase incl. interest cost
0 100.0 0.3435 34.35 3435 3435 3.3
1 98.8 0.2807 6.28 621 2818 2.7
2 100.8 0.3498 6.91 696 3516 3.4
3 106.1 0.6082 25.84 2743 6263 6.0
4 99.8 0.2661 34.21 3415 2854 2.7
5 101.8 0.3492 8.31 846 3702 3.6
6 103.1 0.4108 6.16 635 4341 4.2
7 105.3 0.5595 14.87 1566 5912 5.7
8 107.9 0.7779 21.84 2357 8274 8.0
9 110.3 0.9666 18.87 2082 10364 10.0
10 107.4 1.0000 3.34 358 10733
Call on 100 shares, S0 = 100, K = 105, r = 0.05, = 0.20.
After selling shares for strike price, cost is 233.
Price of option plus interest is 194.
MATH5350M: Computations in Finance
Section 5.3: Computing delta
Part a: Algorithms
Jonathan Ward, University of Leeds, 2023
Recap
For European vanilla options:
C (s, t) =
d1(s, t)
, (call)
P(s, t) =
d1(s, t)
1, (put)
where
d1(s, t) =
log sK + (r +
1
2
2)(T t)
p
T t .
What can be done
if there is no formula for the delta of an option?
Forward di↵erencing
(s) ⇡ V (s +s) V (s)
s
Algorithm
Input: s, s, all other parameters
Run Monte Carlo to approximate V (s)
Run Monte Carlo to approximate V (s +s)
Set Delta =
V (s +s) V (s)
s
Output: Delta
Central di↵erencing
(s) ⇡ V (s +s) V (s s)
2s
Algorithm
Input: s, s, all other parameters
Run Monte Carlo to approximate V (s s)
Run Monte Carlo to approximate V (s +s)
Set Delta =
V (s +s) V (s s)
2s
Output: Delta
How to choose s?
If main source of error is round-o↵ error, take
s = "1/3 · s,
where " is the precision of double.
Notice that s depends on the stock price s at which the delta is
approximated.
MATH5350M: Computations in Finance
Section 5.4: Variate recycling
Part a: Algorithm
Jonathan Ward, University of Leeds, 2023
Central di↵erencing
(s) ⇡ V (s +s) V (s s)
2s
Algorithm
Input: s, s, all other parameters
Run Monte Carlo to approximate V (s s)
Run Monte Carlo to approximate V (s +s)
Set Delta =
V (s +s) V (s s)
2s
Output: Delta
Central di↵erencing with Monte Carlo
Input: r ,, t,T , St ,M,S
for i = 1 to M do
Generate Z+ ⇠ N (0, 1)
Set S+ = (St +S) e
(r 122)(Tt)+
p
Tt Z+
Set V+i = e
r(Tt)h(S+)
end for
Set a+ = 1M
PM
i=1 V
+
i
for i = 1 to M do
Generate Z ⇠ N (0, 1)
Set S = (St S) e(r 122)(Tt)+
p
Tt Z
Set Vi = e
r(Tt)h(S)
end for
Set a = 1M
PM
i=1 V
i
Set Delta =
a+ a
2S
Output: Delta
Combine the two loops
Input: r ,, t,T , St ,M,S
for i = 1 to M do
Generate Z+ ⇠ N (0, 1)
Set S+ = (St +S) e
(r 122)(Tt)+
p
Tt Z+
Set V+i = e
r(Tt)h(S+)
Generate Z ⇠ N (0, 1)
Set S = (S0 S) e(r 122)(Tt)+
p
Tt Z
Set Vi = e
r(Tt)h(S)
end for
Set a+ = 1M
PM
i=1 V
+
i
Set a = 1M
PM
i=1 V
i
Set Delta =
a+ a
2S
Output: Delta
Variate recycling
Input: r ,, t,T , St ,M,S
for i = 1 to M do
Generate Z ⇠ N (0, 1)
Set S+ = (St +S) e
(r 122)(Tt)+
p
Tt Z
Set V+i = e
r(Tt)h(S+)
Set S = (St S) e(r 122)(Tt)+
p
Tt Z
Set Vi = e
r(Tt)h(S)
end for
Set a+ = 1M
PM
i=1 V
+
i
Set a = 1M
PM
i=1 V
i
Set Delta =
a+ a
2S
Output: Delta
Variate recycling mathematically
Value of option is
V (S0) = e
r(Tt)E(h(StXt)) where
Xt = exp
(r 122)(T t) + WTt
.
Central di↵erencing:
(St) ⇡ V (St +s) V (St s)
2s
,
⇡ e
r(Tt)
2s
✓
E
h
h((St +s)Xt)
i
E
h
h((St s)Xt)
⌘
.
Variate recycling combines the two E terms:
(St) ⇡ e
r(Tt)
2s
E
h
h((St +s)Xt) h((St s)Xt)
i
.
No di↵erence mathematically, but variance much less now.
MATH5350M: Computations in Finance
Section 5.4: Variate recycling
Part b: Example
Jonathan Ward, University of Leeds, 2023
Numerical example
European call option with K = 105, T = 0.5, S0 = 100, = 0.3
and r = 0.05. Monte Carlo with M = 10,000 simulations.
Method Delta Std. dev. Conf. interval
Without recycling 0.662 18.96 [0.290, 1.033]
With recycling 0.499 0.600 [0.487, 0.511]
Width of confidence interval reduces by 30⇥ with recycling.
Algorithm is almost 1000⇥ faster.
More on variate recycling
Can apply variate recycling when:
I simulation of the stock price requires solution of an SDE
I want other derivatives, e.g. Gamma = @
2V
@s2
I Approximation:
Gamma ⇡ V (s +s) 2V (s) + V (s s)
(s)2
MATH5350M: Computations in Finance
Section 5.5: Computing delta using Malliavin calculus
Part a: Overview
Jonathan Ward, University of Leeds, 2023
When variate recyling fails
Variate recycling fails to deliver such excellent results when the
payo↵ is discontinuous, for example, for a binary call option
h(s) =
(
1, if s K ;
0, if s < K .
Recall variate recycling for Delta:
(St) ⇡ e
r(Tt)
2s
E
⇣
h((St +s)Xt) h((St s)Xt)
⌘
where Xt = exp
(r 122)(T t) + WTt
.
The variance of this estimator is large for a binary call because
payo↵s are either identical or their di↵erence is equal to 1, and s
is small. If h is continuous, then payo↵s are close to each other.
Delta via Malliavin calculus
In the Black–Scholes model, delta of payo↵ h(ST ) at T is given by
(St) =
er(Tt)
St
p
T tE
h
Zh
St exp
(r122)(Tt)+
p
T t Zi.
where Z ⇠ N (0, 1).
Proof. Integration by parts . . .
ir
aOlst》 : 录 er
< r "
E [ h ( Sπ]
⼆录 e
- r* E
[ h ( S ←exp} r -zo (T - t )
t σπ-ε E } ) ]
录 EIh ( sixt 1 ) = 录 Jihst 1π e- ⇌^dz 毙 ( h (s ) ]= %oh ( ae β z) 餐
r
—区
⼀ z
2
: J % 录录 hestxt ) e da ⼀~(z1el V( Z 1
- 2
= 98 h
'
Eax+a xtre dz IBP : SU
'
1o, V 1x) do = [ uv] - Ju ( x 1 U
'
cxidx
a = h ( ae
β t ) t v=te
- z
SiXt = Steap } ( r ±σ^ )(Ft ) -5FtE }
U
'
(E 1 = h
(
ae β
t )
ae β
aV
'
(
E) = ( - E说e -
E
'
= 2 e
β z
atlh π : { 红hixeitj 品je断
2 = Step } (r -t
0
) < T - t}
= E [ zhc α e∞ 了
β= 0Ft
0 =←
e
…
F ☆E [h]
: 七箭的正 { ( x@ P {γ=⑤} H +β0H} )
Performance of Malliavin method
Delta of binary call with K = 105, T = 0.5, S0 = 100, = 0.3
and r = 0.05. Monte Carlo with M = 100,000 simulations.
Compare central di↵erencing with s = 105 and variate recycling
with formula on previous slide.
Method Delta Std. dev. Conf. interval
Recycling 0.0293 3.7773 [0.0058, 0.0527]
Malliavin 0.0179 0.0270 [0.0177, 0.0181]
Confidence interval is more than 100⇥ smaller for Malliavin!
Computational cost is about the same.
Performance for continuous payo↵s
Delta of vanilla call with K = 105, T = 0.5, S0 = 100, = 0.3
and r = 0.05. Monte Carlo with M = 100,000 simulations.
Compare central di↵erencing with s = 105 and variate recycling
with formula on previous slide.
Method Delta Std. dev. Conf. interval
Recycling 0.4974 0.6009 [0.4937, 0.5011]
Malliavin 0.4992 1.2922 [0.4912, 0.5072]
Confidence interval is 2⇥ bigger for Malliavin method!
Malliavin method not necessary for continuous payo↵.
MATH5350M: Computations in Finance
Section 5.5: Computing delta using Malliavin calculus
Part b: Theory
Jonathan Ward, University of Leeds, 2023
MATH5350M: Computations in Finance
Section 5.6: Other Greeks
Jonathan Ward, University of Leeds, 2023
Other Greeks
I (gamma) = @
2V
@s2
I (theta) ⇥ = @V
@t
I (vega) V = @V
@
I (rho) ⇢ = @V
@r
Assume that V (s, t,, r) is the value of our portfolio under these
parameters. Taylor’s formula yields
V (s + s, t + t, + , r + r)
⇡ V (s, t,, r) + · s + 12 · s2 +⇥ · t + V · + ⇢ · r
Portfolio immunisation
Portfolio immunisation is the process of complementing the
portfolio with additional financial instruments to zero given Greeks.
This is a dynamic process: adjustments have to be made often.
Example:
I main ingredient: one call option with a strike price K and
maturity T ,
I additional instruments: shares, bonds.
This portfolio cannot be made immune to the changes of all
parameters. It can be made immune to the changes of t and S
(see the replicating strategy in the Black–Scholes model).