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Semester 1 Examinations 2023

MA3071 FINANCIAL MATHEMATICS

FORMULA SHEET

(I) European Options

(1) Call Option Payoff: (ST - K)+;

(2) Call Option Proﬁt: P = (ST - K)+ - D ;

(3) Put Option Payoff: (K - ST)+;

(4) Put Option Proﬁt: P = (K - ST)+ - D.

(II) Binomial Tree Models

(1) Hedging Portfolio for single period tree:

V0 = 0S0 +ψ﹐ 0 = ﹐ ψ =

(III) Standard Brownian Motion

(1) 匝[f (t ﹐ Bt)|Fs] = 匝[f (t ﹐ Bs+(Bt - Bs))|Fs];

(2) 匝[f (s﹐ Bs)|Fs] = f (s﹐ Bs);

(3) 匝[(Bt - Bs)2m+1|Fs] = 0 ﹐ m = 0 ﹐ 1 ﹐．．．;

(4) 匝[(Bt - Bs)2m |Fs] = (t - s)m (2m - 1)!!﹐ m = 0 ﹐ 1 ﹐．．．﹐ ;

(5) If f (t ﹐ Bt) = g(t)h(Bt)﹐ 匝[g(t)h(Bt)|Fs] = g(t)匝[h(Bt)|Fs];

(6) 匝[g(t)|Fs] = g(t) and Var[g(t)] = 0; for all t > s ≥ 0.

(IV) Stochastic Integrals

(1) 0(t) YudBu ~ N ′0 ﹐ 匝、0(t) Yu(2)du┌╱;

(2) Var 、0(t)Audu┌ = 0 ;

(3) If Xt = X0 + 0(t)Audu + 0(t) YudBu, then Xt ~ N ′X0 + 0(t) 匝[Au]du﹐ 匝、0(t) Yu(2)du┌╱ .

(V) Ito’s Lemma

(1) df (t ﹐ Bt) = ft\dt + ft dBt + fBtdt;

(2) df (t ﹐ Xt) = ft\dt + ft dXt + fXt (dXt)2 ;

(3) If dXt = Atdt +YtdBt, then df (t ﹐ Xt) = (ft\ +At ft + Yt2fXt )dt +Yt ft dBt .

(VI) Geometric Brownian Motion

(1) log │S(S)0(t) 、~ N !│µ - 、t ﹐ σ2t!;

(2) 匝[St] = S0eµt ;

(3) Var[St] = S0(2)e2µt │eσ2t - 1、;

(4) 匝[f (ST)|Ft] = 匝 ┌f │Stea(T -t)+σ(BT -Bt )、┐Ft ┐ ;

(5) 匝[f (St)|Ft] = f (St);

(6) 匝[ST |Ft] = Steµ(T -t);

(7) 匝 ┌ │ea(T -t)+σ(BT -Bt)、k ┐Ft ┐ = e │ka+、(T -t) , k is a constant;

(8) St/ea(T -t)+σ(BT -Bt); for all T > t ≥ 0.

(VII) Black-Scholes Model

(1) Let g(t ﹐ St) be the option price at time t and St be a GBM, under the no arbitrage condition, the Black-Scholes equation is: gt(~) + pStgS(~)t + σ2St2gS(~~)tSt = pg;

(2) If no dividend payment:

• European call option price at time t: g(t ﹐ St) = Sto(d1 ) - Ke-p(T -t)o(d2 );

• European put option price at time t: g(t ﹐ St) = Ke-p(T -t)o(-d2 ) - Sto(-d1 ).

where o(x) is the cumulative distribution function of a standard normal random vari- able, and

• d1 = ,

• d2 = d1 - σ ^T - t ,

(3) For European options with no dividend, the exact expressions of the Greeks are

		European Call Option	European Put Option
A (Delta)	àg àSt	o(d1 )	-o(-d1 )
r (Gamma)	à2g	0(d1 ) Stσ^T -t	0(d1 ) Stσ^T -t
Ⅴ (Vega)	àg àσ	St0(d1 )^T - t	St0(d1 )^T - t
p (Rho)	àg àp	K(T - t)e-p(T -t)o(d2 )	-K(T - t)e-p(T -t)o(-d2 )
o (Theta)	àg àt	- - pKe-p(T -t)o(d2 )	- + pKe-p(T -t)o(-d2 )