FININCIAL MATHEMATICS
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solution_from_prof._mgk.docxMath 381– Financial Mathematics I – Fall Term 2014-15
Practice Problems: 13 December 2014
1. Assume that the stock price St follows a log-normal distribution, meaning ln St is normal random variable. If the variance of ln St is σ2 , then use the no arbitrage condition on St to determine the mean of ln St .
t
solution
St ̴ lognormal distribution; then
Therefore E(St) = exp ( μ + )
Var(St) = exp (2μ+)[exp()-1]
Since ln St has a normal distribution with mean μ and a variance of is , then the mean of St will be
E(St)=ln( )
2. For any α > 0, use the no arbitrage condition to determine the fair market value of the European style option whose payoff function is g(St ) = Sα .
t
Solution
The European call option will be exercised only when St<k; where St is the exercise price
While K is the strike price.
Therefore the present value of the option will be
K=S0
Where S0= g(St )
3. Starting from the Black-Scholes-Merton option price for the European call option, determine the asymptotic value as t → 0.
Solution
Price of a European Call Option follows the stochastic differential equation
{dS£t = r · S£t dt + (σ0(t) · S£t + £σ1(t)) dWt
{S£0= S0.
the underlying stock price of the European call option is given by;
S£t = eYt ·{S£0− £ ·(s)ds + (s)dws
Yt = rt + (s)ds (s)dWs
We define random variables f0 and f1 by
f0(t) := eYt · S£0
f0(t)= = eYt ·{S£0− £ ·(s)ds + (s)dws
which implies;
= f0(t) + £ · f1(t).
Which means as t -> 0;
The price of the European call option becomes;
S£0= S0
6. Let P call (K, S, r, t, σ) denote the Black-Scholes price for a European call option.
show that ntfor any α > 0, one has the relation
Pcall (αK, αS, r, t, σ) = αPcall (K, S, r, t, σ).
What is the economic meaning of α
Solution;
K is the strike price of the European call option
S is the spot price of the European call option
r is the risk free rate
σ is the volatility of the European call option
therefore the price of the option will be;
f(st,t)=st-k
which implies;
f(st,t)= αSt-αk
which is equal to after factoring α;
f(st,t)= α(St-k)
which proves that;
Pcall (αK, αS, r, t, σ) = αPcall (K, S, r, t, σ)
The economic implication of α is that the price of the European option will vary depending on the changes in the changes of the economy of a given country or a an economic factor directly affecting the European call option.
7. Let gcall and gput denote the payoff functions of a European call and European put
option, respectively, with the input parameters. Show that
gcall − gput = St − Ke−rt ,
and, from this relation, deduce the put-call parity without assuming that the stock
price is log-normal, but rather only assuming the no arbitrage condition.
The payoff of the put option will be given by (K-St)
And K, the strike price will be given by
K=St
Which implies
St= Ke−rt
Therefore
gcall − gput = St − Ke−rt
1. Use the integral representation of the Black-Scholes price for the European style call option (rather than the form involving the cumulative normal distribution function) to show that as K increases, the call option price decreases
Solution
C(St,K,T) = [(ST-K)+|Ft]
=
=
= EQ[ST|ST>K]
=
=exp(lnst+(r-)r+
=SterTØ(dt)
So as k increases, the call option price decreases
9. Consider a European capped call option whose payoff function is given by
g(S, K, M ) = min{max{S − K, 0}, 0}
where K is the strike price of the European style call option and M is the capped
price. Show that the present value of such an option is equal to
Pcall (K, S, r, t, σ) − Pcall (K + M, S, r, t, σ).
Solution
Pcall (K, S, r, t, σ) will be the strike price
Pcall (K + M, S, r, t, σ) will be exercise price
The European call option will only be exercised only when t=t or when K<St
This implies that pay offs will be given by (K-St) max
Which implies that
K=S0
K- S0
Which is the same as
Pcall (K, S, r, t, σ) − Pcall (K + M, S, r, t, σ).
