Theory of Uncertain Finance

Xiaowei Chen
Nankai University
Theory of Uncertain Finance
Xiaowei Chen
School of Finance, Nankai University
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chenx@nankai.edu.cn
http://orsc.edu.cn/∼xwchen
Xiaowei Chen
Nankai University
How to describe the future price?
Log-Return: r1 = ln
X1
X0
Xk
= r1 + r2 + · · · + rk
X0
Continuous Form: dXt = µXt dt + σXt dBt
Multi-period Log-Return: ln
µ is called the drift and σ is called the diffusion.
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Figure: Further Asset Price
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Xiaowei Chen
Nankai University
Stochastic Finance Theory – Stock Price
dXt
= rXt
dt
⇓
dXt
= rXt + σXt · “noise”
dt
⇓
“noise” =
dWt
dt
⇓
dXt
dWt
= rXt + σXt
dt
dt
⇓
dXt = rXt dt + σXt dWt
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Xiaowei Chen
Nankai University
Ito’s stochastic Differential Equation
dXt = eXt dt + σXt dWt
Xt is the stock price, and Wt is a Wiener process
⇓
Wt =
ln Xt − ln X0 − (e − σ 2 /2)t
σ
⇓
∆Wt =
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ln Xt+∆t − ln Xt − (e − σ 2 /2)∆t
σ
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Xiaowei Chen
Nankai University
Paradox
ln Xt+∆t − ln Xt − (e − σ 2 /2)∆t
.
σ
During a fixed period, the stock price jumps 100 times. Divide the period
into 10000 equal intervals and obtain 10000 samples of increment ∆Wt :
∆Wt =
A, A, · · · , A, B, C , · · · , Z
|
{z
} |
{z
}
9900
100
Do you believe they follow a normal probability distribution?
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Xiaowei Chen
Nankai University
Who could believe a normal probability distribution (curve) is able to
approximate to the frequency (histogram) of increment of stock price?
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Xiaowei Chen
Nankai University
Fact: Some people think that the stock price does behave like a Wiener
process in macroscopy although they recognize the paradox in microscopy.
Question: However, as the very core of stochastic finance theory, Ito’s
calculus is just built on the microscopic structure of Wiener process rather
than macroscopic structure.
Conclusion: Ito’s calculus cannot play the essential tool of finance theory
because Ito’s stochastic differential equation is impossible to model stock
price.
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Xiaowei Chen
Nankai University
Liu Process
Definition
An uncertain process Ct is said to be a canonical Liu process if
(i) C0 = 0 and almost all sample paths are Lipschitz continuous,
(ii) Ct has stationary and independent increments,
(iii) every increment Cs+t − Cs is a normal uncertain variable with
expected value 0 and variance t 2 .
−1
πx
Φt (x) = 1 + exp − √
3t
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Xiaowei Chen
Nankai University
Wiener Process vs Liu Process
Wiener Process (Wiener, 1923)
A stochastic process Wt is said to be a standard Wiener process if
(i) W0 = 0 and almost all sample paths are continuous,
(ii) Wt has stationary and independent increments,
(iii) every increment Ws+t − Ws is a normal random variable with
expected value 0 and variance t.
Liu Process (Liu, 2009)
An uncertain process Ct is said to be a canonical Liu process if
(i) C0 = 0 and almost all sample paths are Lipschitz continuous,
(ii) Ct has stationary and independent increments,
(iii) every increment Cs+t − Cs is a normal uncertain variable with
expected value 0 and variance t 2 .
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Xiaowei Chen
Nankai University
Liu Integral (Liu1 , 2009)
Definition
Let Xt be an uncertain process and let Ct be a canonical Liu process.
Then Liu integral of Xt with respect to Ct is
Z
b
Xt dCt = lim
a
∆→0
k
X
Xti · (Cti+1 − Cti )
i=1
provided that the limit exists almost surely and is finite.
1
Liu B, Some research problems in uncertainty theory, Journal of Uncertain
Systems, Vol.3, No.1, 3-10, 2009.
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Xiaowei Chen
Nankai University
Fundamental Theorem of Uncertain Calculus (Liu2 , 2009)
Theorem
Let Ct be a canonical Liu process, and let h(t, c) be a continuously
differentiable function. Then Zt = h(t, Ct ) has an uncertain differential
dZt =
∂h
∂h
(t, Ct )dt +
(t, Ct )dCt .
∂t
∂c
Note that ∆t and ∆Ct are infinitesimals with the same order. The infinitesimal
increment of Zt has a first-order approximation
∂h
∂h
(t, Ct )∆t +
(t, Ct )∆Ct .
∂t
∂c
Hence we obtain the uncertain differential because it makes
Z s
Z s
∂h
∂h
Zs = Z0 +
(t, Ct )dt +
(t, Ct )dCt .
∂t
0
0 ∂c
∆Zt =
2
Liu B, Some research problems in uncertainty theory, Journal of Uncertain
Systems, Vol.3, No.1, 3-10, 2009.
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Xiaowei Chen
Nankai University
Uncertain Differential Equation (Liu3 , 2008)
Definition
Suppose Ct is a canonical Liu process, and f and g are two functions.
Then
dXt = f (t, Xt )dt + g (t, Xt )dCt
is called an uncertain differential equation.
3
Liu B, Fuzzy process, hybrid process and uncertain process, Journal of
Uncertain Systems, Vol.2, No.1, 3-16, 2008.
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Xiaowei Chen
Nankai University
(Yao-Chen Formula, IJFS, 2013)
Let Xt and Xtα be the solution and α-path of the uncertain differential
equation
dXt = f (t, Xt )dt + g (t, Xt )dCt ,
(1)
respectively. Then
M{Xt ≤ Xtα , ∀t} = α;
M{Xt >
Xtα , ∀t}
= 1 − α.
(2)
(3)
M{Xt ≤ Xtα } = Φt (Xtα ) = α
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Xiaowei Chen
Nankai University
(Yao and Chen, IJFS, 2013)
Let Xt and Xtα be the solution and α-path of the uncertain differential
equation
dXt = f (t, Xt )dt + g (t, Xt )dCt ,
(4)
respectively. Then for any monotone (increasing or decreasing) function J,
we have
Z 1
E [J(Xt )] =
J(Xtα )dα.
(5)
0
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Xiaowei Chen
Nankai University
Table: Stochastic Assets Pricing vs Uncertain Assets Pricing
Stochastic Calculus
Uncertain Calculus
Wiener Process (Wt )
Liu process (Ct )
Ito
R s Integral
0 Xt dWt
Liu
R s Integral
0 Xt dCt
Ito Formula
1 ∂2f
df (t, Wt ) = ( ∂f
∂t + 2 ∂x 2 )dt +
∂f
∂x dWt
Stochastic Differential Equation
dXt = eXt dt + σXt dWt
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Liu Formula
df (t, Ct ) = ∂f
∂t dt +
∂f
∂x dCt
Uncertain Differential Equation
dXt = eXt dt + σXt dCt utlogo
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Xiaowei Chen
Nankai University
Uncertain Finance
1. Uncertain Stock Models
2. Uncertain Foreign Exchange Models
3. Uncertain Term Structure Models
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Xiaowei Chen
Nankai University
1. Liu’s Stock Models
Two assets in the market, Xt is the price of riskless asset and Yt is the
price of risk asset. Liu’s Stock Model(Liu, JUS, 2009))
dXt = rXt dt
dYt = eYt dt + σYt dCt
r : riskless interest rate
e: the drift
σ: the diffusion.
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Xiaowei Chen
Nankai University
European Call Option Pricing
A European option is a contract that gives the holder the right to buy a
stock only at an expiration time s for a strike price K .
European call option price is
fc = exp(−rs)E [(Ys − K )+ ]
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Xiaowei Chen
Nankai University
(Yao and Chen, IJFS, 2013)
Let Xt and Xtα be the solution and α-path of the uncertain differential
equation
dXt = f (t, Xt )dt + g (t, Xt )dCt ,
(6)
respectively. Then for any monotone (increasing or decreasing) function J,
we have
Z 1
E [J(Xt )] =
J(Xtα )dα.
(7)
0
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Xiaowei Chen
Nankai University
European Call Option Pricing
A European option is a contract that gives the holder the right to buy a
stock only at an expiration time s for a strike price K .
European call option price is
fc = exp(−rs)E [(Ys − K )+ ](Yao-Chen Formula)
Z 1
+
= exp(−rs)
Φ−1
dα.
s (α) − K
0
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Xiaowei Chen
Nankai University
European Put Option Pricing
A European option is a contract that gives the holder the right to sell a
stock only at an expiration time s for a strike price K .
European put option price is
fc = exp(−rs)E [(K − Ys )+ ](Yao-Chen Formula)
Z 1
+
= exp(−rs)
K − Φ−1
dα.
s (α)
0
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Xiaowei Chen
Nankai University
American Call Option Pricing(Chen, International Journal
of Operations Research, 2010)
An American option is a contract that gives the holder the right to buy a
stock at any time prior to an expiration time s for a strike price K .
American call option price is
+
fc = E max exp(−rs)(Ys − K )
0≤t≤s
Z
= exp(−rs)
1
max Φ−1
t (α) − K
0 0≤t≤s
+
dα.
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Xiaowei Chen
Nankai University
American Put Option Pricing (Chen, International Journal
of Operations Research, 2010)
An American option is a contract that gives the holder the right to sell a
stock at any time prior to an expiration time s for a strike price K .
American put option price is
+
fp = E max exp(−rt)(K − Yt )
0≤t≤s
Z
= exp(−rs)
1
+
dα.
max K − Φ−1
t (α)
0 0≤t≤s
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Xiaowei Chen
Nankai University
Asian Option Pricing(Sun and Chen, JUAA, 2015)
An
R s Asian option is a contract whose payoff at the expiration time s is
( 0 Yt /sdt − K ) where K is a strike price.
Asian call option:
«+ –
»„ Z s
1
Yt dt − K
fc = exp(−rs)E
s 0
«+
Z 1„ Z s
1
Φ−1
(α)dt
−
K
= exp(−rs)
dα.
t
s 0
0
Asian put option:
»„
K−
fc = exp(−rs)E
Z
= exp(−rs)
0
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1
1
s
Z
«+ –
s
Yt dt
0
„
«+
Z
1 s −1
K−
Φt (α)dt
dα.
s 0
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Xiaowei Chen
Nankai University
Other Stock Models
Peng-Yao’s Stock Model(Peng and Yao, IJOR, 2011))
dXt = rXt dt
dYt = (a − mYt )dt + σdCt
Periodic Dividends Model(Chen, Liu and Ralescu, FODM, 2013)
Xt = X0 exp(rt)
Yt = Y0 (1 − δ)n[t] exp(at + σCt )
Uncertain Stock Model with Jumps(Yu, IJUFKS, 2012)
Xt = X0 exp(rt)
dYt = eYt dt + σYt dCt + δYt dNt
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Xiaowei Chen
Nankai University
Other Papers
1. Ji XY, and Zhou J, Option pricing for an uncertain stock model with
jumps, Soft Computing, to be published.
2. Yao K, A no-arbitrage theorem for uncertain stock model, Fuzzy
Optimization and Decision Making, Vol.14, No.2, 227-242, 2015.
3. Yao K, Uncertain contour process and its application in stock model
with floating interest rate, Fuzzy Optimization and Decision Making, to be
published.
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Xiaowei Chen
Nankai University
2. Uncertain Currency Model
Liu, Chen and Ralescu’s (IJIS, 2013)

dXt = uXt dt (Domestic Currency)





dYt = vYt dt (Foreign Currency)





dZt = eZt dt + σZt dCt (Exchange rate)
where u domestic interest rate, v foreign interest rate, e is the drift and σ
is the diffusion.
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Xiaowei Chen
Nankai University
Foreign Currency Option
European Currency Option
1
1
exp(−us)E [(Zs − K )+] + exp(−vs)Z0 E [(1 − K /Zs )+].
2
2
f =
American Currency Option
1
f = E
2
1
sup exp(−ut)(Zt − K )+ + Z0 E sup exp(−vt)(1 − K /Zt )+ .
2
0≤t≤s
0≤t≤s
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Xiaowei Chen
Nankai University
Other models
Shen and Yao (2013)

dXt = uXt dt (Domestic Currency)





dYt = vYt dt (Foreign Currency)




√

dZt = (a − eZt )dt + σ Zt dCt (Exchange rate)
Wang X, and Ning YF, The pricing of put options on uncertain currency
exchange, Technical Report, 2015.
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Xiaowei Chen
Nankai University
3. Uncertain Term Stucture
Chen and Gao (Soft Computing, 2013)
rt is the short interest rate, m is the mean reverting level, a is the rate of
mean reverting and σ is the diffusion.
drt = (m − art )dt + σdCt
Jiao and Yao (Soft Computing, 2014)
√
drt = (m − art )dt + σ rt dCt
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Xiaowei Chen
Nankai University
Uncertain Term Structure
Zero-coupon bond price (Chen and Gao, soft computing, 2013)
Z s
Z 1
α
P(s) =
exp −
rt dt dα
0
0
Interest rate floor (Zhang, Liu and Sheng, FODM, 2015) is
Z
T
[L − rt ]
f = E exp
+
dt − 1
0
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Xiaowei Chen
Nankai University
The Multi-Factor Model
Two risk factors.
C1t is the risk process of the short interest rate.
C2t is the risk process of the drift process.
(
drt = (θ(t) + µt − αrt )dt + σdC1t
(8)
dµt = −bµt dt + σ2 dC2t
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Xiaowei Chen
Nankai University
Theorem
Assume that f (t, r , µ) is a continuous function monotone increasing with
respect to µ and g (t, r ) is a continuous function monotone. Then the
α-path of the uncertain differential equation
drt = f (t, rt , µt )dt + g1 (t, rt )dCt
is rtα subject to
drtα = f (t, rtα , µαt )dt + |g (t, rtα )|Φ−1 (α)dt.
Then M{rs ≤ rsα } = α, i.e., rs has an inverse uncertainty distribution
Ψs−1 (α) = rsα ,
Then we get E [J(rt )] =
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R1
0
J(rtα )dα.
0 < α < 1.
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Xiaowei Chen
Nankai University
The Multi-Factor Model
Two risk factors model.
(
drt = (θ(t) + µt − αrt )dt + σdC1t
(9)
dµt = −bµt dt + σ2 dC2t
Zero-coupon bond price (Chen and Gao, soft computing, 2013)
Z s
Z 1
α
P(s) =
exp −
rt dt dα
0
0
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Xiaowei Chen
Nankai University
Stochastic Finance
Uncertain Finance
⇑
⇑
Stochastic Calculus
Uncertain Calculus
⇑
⇑
Wiener Process
Liu Process
⇑
⇑
Probability Theory
Uncertainty Theory
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Xiaowei Chen
Nankai University
Thank you very much!
Xiaowei Chen, Theory of Uncertain Finance,
http://orsc.edu.cn/chen/tuf.pdf
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