Mathematical Models in Finance and Industry 832G1, spring term 16/17 Project 2: Black-Scholes model for option pricing
Submission deadline: Wed 10 May 2017, 16:00. School office.
1. The Black-Scholes formula for a European Call option is given by C(S,t) = SΦ(d(S,t))−Ee−r(T−t)Φ(d(S,t)−σ√T −t), (1)
where d(S,t) =
log(S/E)+(r+1 2σ2)(T−t) σ√T−t and E,T,σ, and r denote the exercise price, exercise time, volatility, and riskless interest rate, respectively.
(a) Compute (give all details) the following derivatives of the option price, the so-called Greeks,
∆ =
∂C ∂S
, Γ =
∂2C ∂S2
, κ =
∂C ∂σ
, ρ =
∂C ∂r
, θ =
∂C ∂t
.
(b) Verify that C is a solution of the equation
∂C ∂t
+
1 2
σ2S2∂2C ∂S2
+ rS
∂C ∂S −rC = 0, S > 0, 0 < t < T.
2. A European asset-or-nothing call option CAoN is similar to a European call option, it also solves the Black-Scholes PDE but has the payoff function (final condition) CAoN(S,T) = 0 S < E, S/2 S = E, S S > E. (a) Derive the Black-Scholes formula for the European asset-or-nothing call option. Proceed along the lines of the corresponding calculations for the European call option presented in the lectures, starting out from the Black-Scholes PDE with the payoff given above as final condition. (b) Use put-call-parity to derive the Black-Scholes formula for the European asset-ornothing put PAoN which is similar to a European put option but has the payoff function (final condition) PAoN(S,T) = S S < E, S/2 S = E, 0 S > E. 1
3. Consider the UBS Discount Certificate on the Allianz share, with details given in the attached product information.
(a) Construct a portfolio consisting of options and underlying Allianz shares which represents (i.e. has equivalent cash flows at maturity) this Discount Certificate. What are the exercise price, initial call option price, and maturity of the option(s)? (b) Plot the payoff diagram and the profit diagram of the Discount Certificate. (c) On 20/03/14, the Discount Certificate and the Allianz share were traded for EUR 116.40 and EUR 121.30, respectively. How would you advise an investor (regarding opportunities and risks of the two investments) who is considering to either buy the Discount Certificate or invest in the Allianz share directly on this day? (d) Find the maximum annual yield (annual return) they can achieve by investing in the discount certificate on 20/03/14.
4. Write a Matlab program to evaluate the Black-Scholes formula (1) for a European Call option using appropriate special functions of Matlab.
(a) Graph the option price for E = 70, σ = 0.25, r = 3% (per annum), and four months to maturity. (b) Repeat the calculations for the parameter choices • E = 90 and E = 50, keeping the others fixed, • σ = 0.05 and σ = 0.5, keeping the others fixed, • r = 1% and r = 10%, keeping the others fixed. Describe how the option price changes in each of these cases. Try to give a financial interpretation and a mathematical explanation for the changes.
5. Write a Matlab program to implement the numerical method presented in the lecture notes for valuing an American Put option.
(a) Graph the option price for E = 60, σ = 0.25, r = 4% (per annum), and four months to maturity. (b) Graph the free boundary t 7→ S∗(t) for T − 2 months < t < T using the above parameters.
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