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Reflection principle
Creation date：2024-06-08 14:31:03
Belonging category：数学mathematics
Reflection principle

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Question 1

a) Assume an underlying Black–Scholes model with σ > 0, r > 0, S0 > 0 and without dividend
payments. Formulate a partial differential equation that is satisfied by the price V of the
European down-and-in barrier call option with barrier B and strike K, where K > B, and
where V (t, S) is a function of current time t and stock-value S. Assume S0 > B. Indicate
clearly the solution domain and the boundary conditions V (t, B) and V (T, S).
b) Assume an underlying Black–Scholes model with σ > 0, r > 0, S0 > 0 and without dividend
payments. Determine the linear complementarity problem for the American up-and-out put
option with Barrier B and strike K, K < B, S0 < B. Indicate clearly the solution domain
and the boundary condition at time T .
c) Find the probability density function of Mt := sup0≤s≤tWs where (Wt)t≥0 is a standard
Wiener process.
Hint: Reflection principle.
d) Let cfix(S,M, t,K) and pfl(S,M, t) denote the price function of a (European) fixed strike
lookback call option and a (European) floating strike lookback put option respectively, where
S is the time-t price of the underlying stock, M is the running maximum at time t and K is
the strike price. Suppose M > K. Compute cfix(S,M, t,K)− pfl(S,M, t).
Hint: Consider a portfolio long in the call option and short in the put option.
e) Compare the time-0 price of an up-and-out-call option with the price of a regular call option
with same strike and same maturity.
Hint: In-out-parity.
1
Question 2 (25 marks)
Assume a Black–Scholes model with drift µ ∈ R, risk-free rate r > 0, initial value S0 > 0 and
volatility σ > 0 and assume the stock does not pay any dividends. Suppose the arbitrage-free price
at time t with 0 ≤ t ≤ T of an up-and-out barrier call option with barrier B > S0 and strike K > 0
is given by V (t, St), where St denotes the value of the underlying asset at time t.
(a) Formulate (without proof) a partial differential equation that is satisfied by V and explicitly
state boundary conditions and the solution domain.
(b) Express the arbitrage-free price at time 0 of an up-and-in barrier put option (with the same
maturity T , same strike K and same barrier B as the up-and-out barrier call option) as a
function of the following four quantities:
• V (0, S0),
• p: the price of a regular put option with strike K and maturity T ,
• buo: the price of an up-and-out option with payoff 1, barrier B and maturity T ,
• suo: the price of an up-and-out option with payoff ST , barrier B and maturity T .
Question 3 (25 marks)
Suppose we have two traded risky assets S
(1)
t and S
(2)
t , and one risk-free asset with Bt = e
rt for
r > 0. The dynamics of the risky assets under the real-world probability measure are given by
dS
(1)
t = S
(1)
t µ1dt+ S
(1)
t σ1dW1,t
dS
(2)
t = S
(2)
t µ2dt+ S
(2)
t σ2dW2,t
dW1,tdW2,t = ρdt
where W1,t and W2,t are two correlated Wiener processes, µ1, µ2 ∈ R denote the drifts of the
processes, σ1, σ2 > 0 denote the volatilities of the first and second asset and where ρ ∈ [−1, 1].
Find the arbitrage-free price at time 0 of a financial derivative with payoff φ(S
(1)
T , S
(2)
T ) at maturity
T , where
φ(S
(1)
T , S
(2)
T ) := S
(1)
T 1
{
S
(2)
T ≤KS
(1)
T
}
for K > 0.
Question 4 (25 marks)
A down-and-in cash call is a claim that pays 1 at maturity date T > 0 provided the barrier B is
hit and that ST ≥ K. Suppose B ≤ K. Assume an underlying Black–Scholes model with σ > 0,
r > 0, S0 > 0 and without dividend payments.
Compute the price at time 0 of a down-and-in cash call.
Hint: Apply the reflection principle to the log-price process, similar to the approach from Lecture
10. You may use the results from the lecture without proof.
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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