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MATH39032 Assignment

Creation date：2024-04-11 14:38:29

Belonging category：数学mathematics

Tag MATH39032

Assignment

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MATH39032

1.
A forward contract with delivery price F such that F < SerT (where S is the current value of an asset
S that does not pay out a dividend, r is the risk-free rate and T is the time to expiry) is trading on the
market with zero cost to enter a short or long position. Show that there exists an arbitrage opportunity.
2. Consider a financial contract depending on two assets, S1 and S2, that has the following payoff at
maturity
V (S1, S2, t = T ) = max(S1 −X1, S2 −X2).
Here X2 > X1 are constants, S1 ∈ [0,∞), S2 ∈ [0,∞), and t ∈ [0, T ].
(a) Provide a sketch of the payoff V against S2 when S1 = X2.
(b) Assuming that asset prices follow a geometric Brownian motion, interest rates are constant, and that
neither asset pays a dividend, state appropriate boundary conditions for the value of the financial
contract V (S1, S1, t) in the following two cases:
(i) S1 → 0 and S2 → 0;
(ii) S1 → 0 and S2 →∞.
3. Suppose that the interest rate process r is governed by the general stochastic differential equation
dr = κ(θ − r)dt+ σrdX,
where X is a Brownian motion. V1(r, t) is the value of a zero-coupon bond with maturity T1, and V2(r, t)
is the value of a zero-coupon bond with maturity T2.
Consider a portfolio comprising long one V1 bond and short ∆ of V2 bonds. Using Itoˆ’s Lemma (which
may be used without proof), show that the choice
∆ =
∂V1
∂r
/
∂V2
∂r
eliminates the random component of the portfolio. Hence show that the value of both V1 and V2 must
satisfy the equation
∂V
∂t
+
1
2
σ2r2
∂2V
∂r2
+ κ(θ − r − λσr)∂V
∂r
− rV = 0,
where λ(r, t) is the market price of risk.
Page 2 of 4 P.T.O.
MATH39032
4.
(a) Let F (S, t) = KeλtSα, where K, λ and α are constants. Determine λ (in terms of α, r and σ) so
that F satisfies the Black-Scholes PDE,
∂F
∂t
+
1
2
σ2S2
∂2F
∂S2
+ rS
∂F
∂S
− rF = 0,
where the volatility σ and interest rate r are constants.
(b) Now consider a particular financial instrument with price f(S, t) that satisfies the above PDE and
provides a single payoff at time t = T > 0 amounting to Sα; determine f(S, t < T ) using your
answer to part (a).
(c) Describe the monotonic behaviour of f(S, t) with respect to time for α < 1, α = 1 and α > 1.
5.
Consider the Black-Scholes equation for a put option, P , on a stock S which pays no dividends, namely
∂P
∂t
+
1
2
σ2S2
∂2P
∂S2
+ rS
∂P
∂S
− rP = 0
in the usual notation, where r and σ are both constants.
(a) By neglecting the time derivative in the above equation, seek solutions of the form
P (S, t) = P (S) = A1S
α1 + A2S
α2
where A1 and A2 are constants, and α1 > α2. Determine the values of α1 and α2.
(b) Consider a perpetual American put option P (S), i.e. an option with no expiry date but where it
is possible at any point in time to exercise, which has a (non-standard) exercise value of Ke−S,
where K is a constant. This can be valued using the time independent solutions in (i) above. The
exercise boundary is denoted by Sf . State and justify the two conditions at S = Sf and the other
appropriate boundary condition.
(c) Determine Sf and the values of A1 and A2.
(d) Describe the behaviour of P (S) and Sf as r → 0.
Page 3 of 4 P.T.O.
MATH39032
6. You may assume that the Black-Scholes equation (in the usual notation) for a put option on a non-
dividend paying asset is
∂P
∂t
+
1
2
σ2S2
∂2P
∂S2
+ rS
∂P
∂S
− rP = 0
where the interest rate r and the volatility σ are both constant.
(a) A binary put option Pb(S, t) has the payoff
Pb(S, T ) =
{
0 if S > X
K if S < X
.
Show that its value is given by
Pb(S, t) = K
∂P
∂X
where P (S, t;X) is the value of a vanilla put option with strike X.
(b) Given that the value of a European put option is
P (S, t;X) = Xe−r(T−t)N(−d2)− SN(−d1),
where N(x) is the cumulative distribution function of the standard normal distribution, namely
N(x) =
1√
2pi
∫ x
−∞
e−
y2
2 dy
and
d1 =
ln(S/X) + (r + σ2/2)(T − t)
σ
√
T − t , d2 = d1 − σ
√
T − t,
show that
Pb(S, t;X) = Ke
−r(T−t)N(−d2).
(c) Consider a stock whose price (6 months from the expiration of a binary put option) today is £10.50.
The option pays £0.5 if the stock value at expiry is less than £11, and nothing if the stock value
at expiry is greater than 11. The risk-free interest rate is 4% per annum (fixed) and the volatility
(constant) is 15% per (annum)
1
2 . Using the formula above for Pb(S, t;X), determine its value
today. The value of N(x) may be determined by a numerical (online) calculator.
END OF EXAMINATION PAPER

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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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