Explanation Given: The given equations are, ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 = 0 with 0 < r < 1 , 0 < θ < π , u ( r , 0 ) = 0 , 0 ≤ r ≤ 1 , u ( r , π ) = 0 , 0 ≤ r ≤ 1 , u ( 1 , θ ) = sin 3 θ , 0 ≤ θ ≤ π , u ( 0 , θ ) bounded. Calculation: Consider the given equation, ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 = 0 … ( 1 ) Consider the relation, u ( r , θ ) = R ( r ) T ( θ ) … ( 2 ) Differentiate Equation ( 2 ) with respect to r . ∂ u ∂ r = R ′ ( r ) T ( θ ) … ( 3 ) Differentiate Equation ( 3 ) with respect to r . ∂ 2 u ∂ r 2 = R ″ ( r ) T ( θ ) Differentiate Equation ( 2 ) with respect to θ . ∂ u ∂ θ = R ( r ) T ′ ( θ ) … ( 4 ) Differentiate Equation ( 4 ) with respect to r . ∂ 2 u ∂ θ 2 = R ( r ) T ″ ( θ ) Substitute R ( r ) T ″ ( θ ) for ∂ 2 u ∂ θ 2 , R ″ ( r ) T ( θ ) for ∂ 2 u ∂ r 2 and R ′ ( r ) T ( θ ) for in ∂ u ∂ r Equation ( 1 ) . R ″ ( r ) T ( θ ) + 1 r R ′ ( r ) T ( θ ) + 1 r 2 R ( r ) T ″ ( θ ) = 0 r 2 R ″ ( r ) + r R ′ ( r ) R ( r ) = − T ″ ( θ ) T ( θ ) … (5) Equate the ratios of equation ( 5 ) to λ . r 2 R ″ ( r ) + r R ′ ( r ) − λ R ( r ) = 0 … ( 6 ) T ″ ( θ ) + λ T ( θ ) = 0 … ( 7 ) The auxiliary equation is, r 2 + λ = 0 r = ± i λ The solution for equation ( 7 ) is given as,

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In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)

Problem 4. Margrabe formula and the Greeks (20 pts) In the homework, we determined the Margrabe formula for the price of an option allowing you to swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility σ and correlation p, the formula was given by Fo=yo (d+)-x0Þ(d_), where In (±² Ꭲ d+ õ√T and σ = σ√√√2(1 - p). дго (a) We want to determine a "Greek" for ỡ on the option: find a formula for θα (b) Is дго θα positive or negative? (c) We consider a situation in which the correlation p between the two stocks increases: what can you say about the price Fo? (d) Assume that yo< xo and p = 1. What is the price of the option?

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Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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