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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Mathematical FinanceOption Pricing under the Risk-Neutral Measure
Cory Barnes
Department of MathematicsUniversity of Washington
June 11, 2013
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Outline
1 Probability Background
2 Black Scholes for European Call/Put Options
3 Risk-Neutral Measure
4 American Options and Duality
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
We work in (Ω,F ,P), with a 1-d Brownian Motion Wt , on timeinterval [0,T ].
W0 = 0 a.s.For partition 0 = t0 < t1, . . . , < tk = T , incrementsWtj −Wtj−1 independent.For s < t , increment Wt −Ws ∼ N(0, t − s).For ω ∈ Ω, the mapping t 7→Wt (ω) is continuous a.s.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
DefinitionA filtration of F is a an increasing sequence of sub σ-algebrasof F , i.e. (Ft ) where Fs ⊂ Ft for s < t .
We take (Ft ) to be the filtration generated by Wt .
All the information available at time t is the data that Wsattained for 0 ≤ s ≤ t .
Intuitively, we can make choices on the past and present, butwe don’t know the future.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
DefinitionA filtration of F is a an increasing sequence of sub σ-algebrasof F , i.e. (Ft ) where Fs ⊂ Ft for s < t .
We take (Ft ) to be the filtration generated by Wt .
All the information available at time t is the data that Wsattained for 0 ≤ s ≤ t .
Intuitively, we can make choices on the past and present, butwe don’t know the future.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
DefinitionA stochastic process is a family (Xt ) of real valued randomvariables indexed by time (we take t ∈ [0,T ]). It is continuous ift 7→ Xt (ω) is continuous almost surely.
We will construct various stochastic processes from (Wt )
DefinitionA stochastic process (Xt ) is adapted to the filtration (Ft ) if forevery s ∈ [0,T ], the random variable Xs is Fs measurable.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
DefinitionA stochastic process is a family (Xt ) of real valued randomvariables indexed by time (we take t ∈ [0,T ]). It is continuous ift 7→ Xt (ω) is continuous almost surely.
We will construct various stochastic processes from (Wt )
DefinitionA stochastic process (Xt ) is adapted to the filtration (Ft ) if forevery s ∈ [0,T ], the random variable Xs is Fs measurable.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
DefinitionAn Ft -adapted stochastic process (Xt ) is a martingale withrespect to Ft if for all 0 ≤ s ≤ t ,
E [Xt |Fs] = Xs.
As an example, take (Wt ) on [0,T ], w.r.t. Ft . Verify
E [Wt |Fs] = E [Wt −Ws + Ws|Fs]
= E [Wt −Ws|Fs] + E [Ws|Fs]
= Ws.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
We wish to make sense of the following object:
dSt = µtSt dt + σtSt dWt .
(St ) stock price as an adapted continuous stochasticprocessDeterministic growth µtSt , proportional to current stockpriceRandom Brownian noise, volatility σtSt , proportional tocurrent stock priceBoth (µt ) and (σt ) adapted continuous processes(constants in simple case)
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
We wish to make sense of the following object:
dSt = µtSt dt + σtSt dWt .
(St ) stock price as an adapted continuous stochasticprocessDeterministic growth µtSt , proportional to current stockpriceRandom Brownian noise, volatility σtSt , proportional tocurrent stock priceBoth (µt ) and (σt ) adapted continuous processes(constants in simple case)
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
How does dWt behave? Look at quadratic variation. Forpartition s0 < s1 < . . . < sn of interval sn − s0 = t ,
Qn =n−1∑j=0
(Wsj+1 −Wsj
)2 → t .
The convergence here happens in mean-square, sinceE(Qn)→ t and Var(Qn)→ 0. Thus we write the following’formal’ multiplication rule:
dW · dW = dt .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Similarly, we can show that as n→∞,
n−1∑j=0
(Wsj+1 −Wsj
) (sj+1 − sj
)→ 0,
n−1∑j=0
(sj+1 − sj
)2 → 0.
Thus we add the following to our multiplication rules:
dW · dt = 0, dt · dt = 0.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
We make sense of the stochastic differential equation
dXt = at dt + bt dWt ,
where at and bt are adapted to the Brownian filtration, by
Xt = X0 +
∫ t
0as ds +
∫ t
0bs dWs.
But what does this last term mean? Turns out it will be amartingale.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
As long as E[∫ T
0 b2t dt
]<∞, we can approximate the integral
with elementary functions. Take
φt =n−1∑j=0
ej · 1[tj ,tj+1)(t),
where ej is Ftj measurable and φ has finite squaredexpectation. Define∫ t
0φs dWs =
n−1∑j=0
ej · (Wtj+1 −Wtj ).
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Our SDE is a little bit more complicated:
dSt = µtSt dt + σtSt · dWt .
Need some sort of chain rule....
Theorem (Ito’s Formula)
Let f (t , x) have continuous second partial derivatives. Then
f (t ,Wt ) = f (0,W0) +
∫ t
0
∂f∂t
(s,Ws) ds +
∫ t
0
∂f∂x
(s,Ws) dWs+
+12
∫ t
0
∂2f∂x2 (s,Ws) ds.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The SDE dSt = µtSt dt + σtSt dWt has solution
St = S0 · exp(X (t)), Xt =
∫ t
0σs dWs +
∫ t
0(µs −
12σ2
s ) ds.
To verify this we differentiate f (Xt ) where f (x) = S0ex usingIto’s formula.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
dSt = df (Xt ) = f ′(Xt ) dXt +12
f ′′(Xt ) dXt · dXt
= St dXt +12
St dXt · dXt
= St
(σt dWt + (µt −
12σ2
t ) dt)
+
+12
St
(σt dWt + (µt −
12σ2
t ) dt)2
= µtSt dt + σtSt dWt .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Black Scholes for European Call/Put Options
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
European style call option is the right to purchase S for K at TStrike Price KTime to maturity TUnderlying stock price at t = 0, denoted S0
Payoff (ST − K )+ = maxST − K ,0
Put option is the right to sell, payoff (K − ST )+
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Market assumptions/characterizations:Single underlying stock with geometric brownian motionStock has expected growth µt and volatility σt
No taxes, transaction costs or bid-ask spreadCan borrow/lend at risk free rate rf
No arbitrage
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The goal is to construct f (t ,St ) that measures the value of theoption at time t , based on the stock price path. We do this witha hedge.Purchase some amount, ∆, of the underlying stock and sellone option. The value of this portfolio at time t is then
Πt = ∆ · St − f (t ,St ).
Over an infinitesimal time window dt , the portfolio’s valuechanges
dΠt = ∆ · dSt − df (t ,St ).
Use Ito’s formula to track this change.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The goal is to construct f (t ,St ) that measures the value of theoption at time t , based on the stock price path. We do this witha hedge.Purchase some amount, ∆, of the underlying stock and sellone option. The value of this portfolio at time t is then
Πt = ∆ · St − f (t ,St ).
Over an infinitesimal time window dt , the portfolio’s valuechanges
dΠt = ∆ · dSt − df (t ,St ).
Use Ito’s formula to track this change.
![Page 23: Mathematical Finance - Option Pricing under the Risk-Neutral Measuremorrow/papers/cory... · 2013-06-21 · Probability BackgroundBlack Scholes for European Call/Put OptionsRisk-Neutral](https://reader033.vdocuments.us/reader033/viewer/2022050313/5f7597d4f12ca06daa6e9a7a/html5/thumbnails/23.jpg)
Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The goal is to construct f (t ,St ) that measures the value of theoption at time t , based on the stock price path. We do this witha hedge.Purchase some amount, ∆, of the underlying stock and sellone option. The value of this portfolio at time t is then
Πt = ∆ · St − f (t ,St ).
Over an infinitesimal time window dt , the portfolio’s valuechanges
dΠt = ∆ · dSt − df (t ,St ).
Use Ito’s formula to track this change.
![Page 24: Mathematical Finance - Option Pricing under the Risk-Neutral Measuremorrow/papers/cory... · 2013-06-21 · Probability BackgroundBlack Scholes for European Call/Put OptionsRisk-Neutral](https://reader033.vdocuments.us/reader033/viewer/2022050313/5f7597d4f12ca06daa6e9a7a/html5/thumbnails/24.jpg)
Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
dΠt = ∆ · dSt −(∂f∂t
dt +∂f∂St
dSt +12∂2f∂S2
tdS2
t
)= ∆ · dSt −
(∂f∂t
dt +∂f∂St
dSt +12∂2f∂S2
tσ2S2
t dt)
=
(∆− ∂f
∂St
)dSt −
(∂f∂t
+12∂2f∂S2
tσ2S2
t
)dt .
The only Brownian randomness comes from dSt terms. Take
∆ =∂f∂St
. ’Continuously update’ number of stocks ∆ to form a
’Delta Hedge’.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
With ∆ =∂f∂St
stocks in the portfolio, Πt grows deterministically.
Because there is no arbitrage, Πt must grow at the risk freerate:
dΠt = rΠt dt .
Substitute previous expressions and obtain the Black-ScholesPDE:
r f =∂f∂t
+ r∂f∂St
St +12∂2f∂S2
tσ2S2
t .
Notice the lack of µ in this PDE!
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
With ∆ =∂f∂St
stocks in the portfolio, Πt grows deterministically.
Because there is no arbitrage, Πt must grow at the risk freerate:
dΠt = rΠt dt .
Substitute previous expressions and obtain the Black-ScholesPDE:
r f =∂f∂t
+ r∂f∂St
St +12∂2f∂S2
tσ2S2
t .
Notice the lack of µ in this PDE!
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
With ∆ =∂f∂St
stocks in the portfolio, Πt grows deterministically.
Because there is no arbitrage, Πt must grow at the risk freerate:
dΠt = rΠt dt .
Substitute previous expressions and obtain the Black-ScholesPDE:
r f =∂f∂t
+ r∂f∂St
St +12∂2f∂S2
tσ2S2
t .
Notice the lack of µ in this PDE!
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
With ∆ =∂f∂St
stocks in the portfolio, Πt grows deterministically.
Because there is no arbitrage, Πt must grow at the risk freerate:
dΠt = rΠt dt .
Substitute previous expressions and obtain the Black-ScholesPDE:
r f =∂f∂t
+ r∂f∂St
St +12∂2f∂S2
tσ2S2
t .
Notice the lack of µ in this PDE!
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Boundary conditions (for call option):f (t ,0) = 0f (T ,ST ) = (ST − K )+
f (t ,St )→∞ as St →∞How can we solve this?
rf =∂f∂t
+ r∂f∂St
St +12∂2f∂S2
tσ2S2
t .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Change variables....
S = Kex , t = T − 2τσ2 , f (T ,S) = K · v(x , τ).
Change variables again and set c = 2r/σ2 ...
v(x , τ) = exp(−1
2(c − 1)x − 1
4(c + 1)2τ
)· u(x , τ).
Miraculously this reduces to
∂u∂τ
=∂2u∂x2 .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Change variables....
S = Kex , t = T − 2τσ2 , f (T ,S) = K · v(x , τ).
Change variables again and set c = 2r/σ2 ...
v(x , τ) = exp(−1
2(c − 1)x − 1
4(c + 1)2τ
)· u(x , τ).
Miraculously this reduces to
∂u∂τ
=∂2u∂x2 .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Change variables....
S = Kex , t = T − 2τσ2 , f (T ,S) = K · v(x , τ).
Change variables again and set c = 2r/σ2 ...
v(x , τ) = exp(−1
2(c − 1)x − 1
4(c + 1)2τ
)· u(x , τ).
Miraculously this reduces to
∂u∂τ
=∂2u∂x2 .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Can use the Fourier Transform to solve this!
f (t ,St ) = SΦ(d+)− Ke−r(T−t)Φ(d−) (call),
f (t ,St ) = Ke−r(T−t)Φ(−d−)− SΦ(−d+) (put),
where Φ(x) is the normal CDF,
Φ(x) =
∫ x
−∞
1√2π
exp(−x2
2
)dx ,
d± =log(S/K ) + (r ± σ2/2)(T − t)
σ√
T − t.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Risk-Neutral Measure
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Suppose Z is a RV with E(Z ) = 1 and Z > 0 a.s. on (Ω,F ,P).Can define
P(A) =
∫A
Z (ω)dP(ω), for A ∈ F .
We call Z the Radon-Nikodym derivative of P w.r.t P.
Definition
If Z is the R-N derivative of P w.r.t P, then
Zt = E [Z |Ft ], 0 ≤ t ≤ T
is the R-N derivative process w.r.t. Ft .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
LemmaThe R-N derivative process is a martingale w.r.t Ft .
Proof.
E [Zt |Fs] = E [E [Z |Ft ]|Fs] = E [Z |Fs] = Zs.
LemmaLet Y be an Ft measurable RV. Then for 0 ≤ s ≤ t ,
E [Y |Fs] =1Zs
E [YZt |Fs].
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
LemmaThe R-N derivative process is a martingale w.r.t Ft .
Proof.
E [Zt |Fs] = E [E [Z |Ft ]|Fs] = E [Z |Fs] = Zs.
LemmaLet Y be an Ft measurable RV. Then for 0 ≤ s ≤ t ,
E [Y |Fs] =1Zs
E [YZt |Fs].
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Theorem (Girsanov)
On (Ω,F ,P) with Wt and Ft , let Θt be an adapted process for0 ≤ t ≤ T . Define
Wt = Wt +
∫ t
0Θs ds, i.e. dWt = dWt + Θt dt .
Then there exists an explicit Zt that defines P for whichE(ZT ) = 1 and under P the process Wt is a BM.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Recall our stock process
dSt = µtSt dt + σtSt dWt .
The risk free rate can be given as a stochastic process rt , sowe have discount factor
Dt = exp(−∫ t
0rs ds
).
Discount process has zero quadratic variation, so
DtSt = S0 exp(∫ t
0σs dWs +
∫ t
0µs − rs −
12σ2
s ds).
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
In differential form, the discounted stock process is
d(DtSt ) = σtDtSt (Θt dt + dWt ) , Θt =µt − rt
σt.
Use Girsanov with Θt to change to P, then
d(DtSt ) = σtDtSt dWt .
Discounted stock process is a martingale! From this we getdiscounted payoff DtVt also a martingale.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The solution to this SDE is
St = S0 exp(∫ t
0σs dWs +
∫ t
0rs −
12σ2
s ds).
Evaluate call option price from martingale statement:
f (0,S0) = E[e−rT (ST − K )+|F0
].
Straightforward integral to recover Black-Scholes.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Theorem (Martingale Representation)
On (Ω,F ,P) with Wt and Ft for 0 ≤ t ≤ T , let Mt be amartingale w.r.t. Ft . Then there exists a unique Ft adaptedprocess Γt such that
Mt = M0 +
∫ t
0Γs dWs, 0 ≤ t ≤ T .
We really want the existence of Ft adapted Γ for martingale Munder P, such that
Mt = M0 +
∫ t
0Γs dWs.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
General hedging problem for European style option:
Let VT be FT measurable - derivative payoffStart with initial capital X0 and form portfolio process Xt
Want XT = VT
Call V0 = X0 the price of the option at time zero
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Change to risk-neutral measure, get DtVt to be a martingale:
DtVt = V0 +
∫ t
0ΓsdWs, 0 ≤ t ≤ T .
On the other hand, we want the portfolio process Xt to be’self-financing’:
dXt = ∆tdSt + rt (Xt −∆tSt ) dt ,
where we hold ∆t shares of the stock, and invest/borrow at rt tofinance with X0 initial capital.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Write this under P and get
DtXt = X0 +
∫ t
0∆sσsDsSs dWs, 0 ≤ t ≤ T .
Equating this with our previous expression
DtVt = V0 +
∫ t
0ΓsdWs, 0 ≤ t ≤ T ,
we should pick
∆t =Γt
σtDtSt, 0 ≤ t ≤ T .
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Theorem (Fundamental Theorem of Asset Pricing)
A market has a risk-neutral measure if and only if it does notadmit arbitrage.
In an arbitrage free market, every derivative can be hedged ifand only if the risk neutral measure is unique.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
American Options and Duality
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
An American style option can be exercised at any time up tomaturity T .
Closed form solutions rarely availableApproximate with discrete exercise timesG = 0,1,2, . . . ,TDiscounted payoff ht (St ) if exercised at time tDiscounted value Vt (St ), inherent value/price at time t
Notice that
Vt (St ) ≥ ht (St ), V0 = supτ
E(hτ (Sτ )).
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
An American style option can be exercised at any time up tomaturity T .
Closed form solutions rarely availableApproximate with discrete exercise timesG = 0,1,2, . . . ,TDiscounted payoff ht (St ) if exercised at time tDiscounted value Vt (St ), inherent value/price at time t
Notice that
Vt (St ) ≥ ht (St ), V0 = supτ
E(hτ (Sτ )).
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The value of an American option followsVT (ST ) = hT (ST )
Vi(Si) = maxhi(Si), E [Vi+1(Si+1)|Fi ]
Use some sort of dynamic programming algorithm to find V0.
How do you compute this continuation value?LatticeLeast squaresWhatever works...
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
The value of an American option followsVT (ST ) = hT (ST )
Vi(Si) = maxhi(Si), E [Vi+1(Si+1)|Fi ]
Use some sort of dynamic programming algorithm to find V0.
How do you compute this continuation value?LatticeLeast squaresWhatever works...
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Consider martingale Mt with M0 = 0 and τ attaining values inG. Then by Optional Sampling,
E(hτ (Sτ )) = E(hτ (Sτ )−Mτ ) ≤ E(maxk∈G
(hk (Sk )−Mk )).
Take inf over M and sup over τ to arrive at
V0 = supτ
E(hτ (Sτ )) ≤ infM
E(maxk∈G
(hk (Sk )−Mk )).
Claim: This is actually an equality!
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
TheoremThere exists a martingale Mt with M0 = 0 for which
V0 = E(maxk∈G
(hk (Sk )−Mk )).
Proof.
For i = 1, . . . ,T define Ni = Vi(Si)− E [Vi(Si)|Fi−1]. Then take
M0 = 0, Mi =i∑
j=1
Nj , i = 1, . . . ,T .
Notice that E [Ni |Fi−1] = 0. Thus Mt is a martingale.
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
continued...With induction: Vi(Si) =maxhi(Si),hi+1(Si+1)− Ni+1, . . . ,hT (ST )− NT − · · · − Ni+1.True at T since VT (ST ) = hT (ST ). Assuming true at i ,
Vi−1(Si−1) = maxhi−1(Si−1), E [Vi(Si)|Fi−1]
= maxhi−1(Si−1),Vi(Si)− Ni
Therefore
V0 ≥ E [V1(S1)|F0] = V1(S1)− N1 = maxk=1,...,T
(hk (Sk )−Mk ).
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Probability Background Black Scholes for European Call/Put Options Risk-Neutral Measure American Options and Duality
Duality gives a way to form lower/upper bounds:
supτ
E(hτ (Sτ )) = V0 = infM
E(maxk∈G
(hk (Sk )−Mk )).
An approximate stopping time gives a lower boundAn approximate martingale gives an upper boundWhat constitutes a good approximation?