Xiaowei Chen Nankai University Theory of Uncertain Finance Xiaowei Chen School of Finance, Nankai University utlogo [email protected] 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. utlogo Figure: Further Asset Price [email protected] http://orsc.edu.cn/∼xwchen 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 [email protected] utlogo http://orsc.edu.cn/∼xwchen 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 = [email protected] ln Xt+∆t − ln Xt − (e − σ 2 /2)∆t σ utlogo http://orsc.edu.cn/∼xwchen 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? [email protected] utlogo http://orsc.edu.cn/∼xwchen 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? [email protected] utlogo http://orsc.edu.cn/∼xwchen 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. utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 . utlogo [email protected] http://orsc.edu.cn/∼xwchen 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. [email protected] utlogo http://orsc.edu.cn/∼xwchen 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. [email protected] utlogo http://orsc.edu.cn/∼xwchen 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. [email protected] utlogo http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 [email protected] Liu Formula df (t, Ct ) = ∂f ∂t dt + ∂f ∂x dCt Uncertain Differential Equation dXt = eXt dt + σXt dCt utlogo http://orsc.edu.cn/∼xwchen Xiaowei Chen Nankai University Uncertain Finance 1. Uncertain Stock Models 2. Uncertain Foreign Exchange Models 3. Uncertain Term Structure Models utlogo [email protected] http://orsc.edu.cn/∼xwchen 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. utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 )+ ] utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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. utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 [email protected] 1 1 s Z «+ – s Yt dt 0 „ «+ Z 1 s −1 K− Φt (α)dt dα. s 0 utlogo http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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. utlogo [email protected] http://orsc.edu.cn/∼xwchen 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. utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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. utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen 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 )] = [email protected] R1 0 J(rtα )dα. 0 < α < 1. utlogo http://orsc.edu.cn/∼xwchen 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 utlogo [email protected] http://orsc.edu.cn/∼xwchen Xiaowei Chen Nankai University Stochastic Finance Uncertain Finance ⇑ ⇑ Stochastic Calculus Uncertain Calculus ⇑ ⇑ Wiener Process Liu Process ⇑ ⇑ Probability Theory Uncertainty Theory utlogo [email protected] http://orsc.edu.cn/∼xwchen Xiaowei Chen Nankai University Thank you very much! Xiaowei Chen, Theory of Uncertain Finance, http://orsc.edu.cn/chen/tuf.pdf [email protected] utlogo [email protected] http://orsc.edu.cn/∼xwchen
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