Introduction:  All students are expected to produce a Portfolio for Part B of 4AMF Advanced Mathematical Finance. The Computer Programming for this Component of your Portfolio is expected to be programmed in MatLab.  Students will be given a number of MatLab files to assist them with their Portfolio.

The majority of this Assessment concerns European Call Options.  Let C(S,t) be the value of a European Call Option where S is the value of the underlying asset and t is time. The risk-free interest rate is r and the volatility of the underlying asset is σ . The option has an Exercise price of E and Expiry date of t = T. The Black Scholes solution for the value of a European Vanilla Call Option is given by

C(S,t) = SN(d1 ) − Ee−r(T−t)N(d2 )                                    (1)

where N is the cumulative distribution function for a standardized normal random variable

given by

N(x) =  \ e −y2 /2dy .

Here

log(S/E) + (r + σ2 /2)(T − t)

σ ^T − t

log(S/E) + (r − σ 2 /2)(T − t)

σ ^T − t                  .

The value of a European Call Option can be calculated using the MatLab function ‘Euro- pean Call.m’. This MatLab function calculates the exact Black Scholes value of a European Call Option and it will be very helpful throughout this Portfolio.

Please investigate all of the following and include MatLab Code, graphs/tables as well as a written discussion in your Portfolio.  Several other examples of MatLab code will be indicated below to assist you.

1. Let C = C(S,t) be the Black Scholes value of a European Vanilla Call. Use equation

(1) and the MatLab function ‘European Call.m’to plot C versus S for 0 ≤ S ≤ 200. The Excercise price is E = 100 and the Expiry date is T = 1.

(a) Assume that r = 0.05 and σ = 0.3. Plot the value of C(S,t) at different values of t between t = 0 and t = T. Explain the limit of C(S,t) as t → T.

(b) Now keep r = 0.05 fixed and investigate the impact of varying σ . Comment on your results.

(c) Now keep σ = 0.3 fixed and investigate the impact of varying r . Comment on your results.

2. As we know the value of ∆ for a Delta hedged Call Option C(S,t) is

∂C

∂S

which clearly depends on the current value of S and t.   The Central Differencing Scheme can be used to approximate ∆(S,t) as

C(S + dS,t) − C(S − dS,t)

This approximation becomes increasingly accurate as dS → 0. Using the same nota- tion as in equation (1) the exact value is

∂C

∂S

which can be proved by several pages of Algebra which you do not need to provide. The exact solution is available in the file ‘Delta European Call.m’.

Consider the case of E  =  100 with r  = 0.05 and σ  = 0.3.   The current time is t = 0 and the expiry date is T = 1. Plot the numerical approximation of ∆(S,t) for

dS  < S  < 200 − dS for differing values of dS .  Plot the error in the absolute error in your approximation according to Error =  |∆numerical  − ∆exact |.   Comment on the  convergence of the numerical method as dS → 0.

3. Consider the function y = f(x) = x2  − 2.  Observe that f(0) = −2 and f(2) = 2. We denote the root of f(x) as the value of x = x∗ such f(x∗ ) = 0.  Implement the bisection root-finding method to approximate the value of ^2. Use the internet if you are not familiar with the bisection method.

4. Let Cexact  be the exact Black Scholes value of a European Call option as given by equtation (1) and let Cmarket be the current market value of the option with an Exercise price E .  The current value of time is t = 0 and the current value of the underlying  is S(0) = 3950.  The Call option expires at time t = T  = 0.14 and the risk free interest rate is r = 0.05.  The file ‘Implied Volatility Data.csv’contains market data of Call options. Column 1 represents the Exercise Price E and Column 2 corresponds  to the current market value of the Call Option Cmarket .  Plot the Volatility Smile for  this Call option for σimplied  as a function of E .  You can read the data in the file ‘Implied Volatility Data.csv’ into a matrix M in MatLab using the command:  M = csvread(’Implied Volatility Data.csv’).

5. This question involves implementing the standard Binomial Method to determine the value of a European Call Option.  The current value of the underlying asset at time t = 0 is S = S(0) = 100.  The risk-free interest rate is r = 0.05 and the volatility is σ = 0.3. The underlying asset S does not pay dividends. The European Call Option has Exercise price E = 100 and Expiry date T = 1. Divide the time interval 0 W t W T into M discrete time steps of size δt with T = Mδt. Examine the convergence of the standard Binomial Method as M t 8 for both the cases of p = 1/2 and ud = 1 using the formulas for (u,d,p) from your Mathematical Finance lectures.

6. The Monte Carlo Method of option valuation is based on a risk-neutral lognormal

random walk of the form

dS

= rdt + σdX

S

which has solution

S(t + δt) = S(t)exp((r − σ 2 /2)δt + σ <δtϕ)

where S is the value of the underlying asset, t is time, r is the risk-free interest rate, σ is the volatility and ϕ is a random variable from the standard normal distribution with mean 0 and variance 1.

The Monte Carlo Method involves performing a large number of simulations of the paths taken by the underlying asset S .  If the number of paths is large enough the Monte Carlo Method will converge to the Black Scholes value of the option V .  For simplicity we will only consider European style options which begin at time t = 0 and expire at time t = T. The payoff of the option V at time t = T is denoted as Λ . The Monte Carlo estimate of the value of the option is obtained by performing nPaths random walks and calculating

1     nPaths

nPaths   i=1

where Λi represents the payoff for path i and the e −rT  factor represents the correction factor for the time value of money.  The method will converge to the Black Scholes value of the option in the limit as nPaths t 8.  The time interval 0 W t W T can be discretised as tj   = jδtMC  for j = 0, 1, 2, ...,nSteps where nSteps represents the number of discrete timesteps with δtMC  = T/nSteps.  For path independent options the Monte Carlo Method will converge to the Black Scholes value of the option with nSteps = 1 in the limit as nPaths t 8. For path dependent options the Monte Carlo Method also requires nSteps t 8 where the necessary value of infinity for nSteps will depend on the particular path dependent option which is being valued.