Financial Mathematics 1 - Final Review

Instructor: Dr. Le Nhat Tan

1 Portfolio Analysis

Suppose we invest $\(X\) in \(n\) assets of price \(S_1,...,S_n\) with corresponding amount \(X_1,...,X_n\) in the sense that \[X=\sum_{i=1}^nX_i.\]

  1. The weight vector of this portfolio is \[\omega=(\omega_1,...,\omega_n)=\left(\frac{X_1}{X},...,\frac{X_n}{X}\right).\]

  2. Let \(r_i\) be the (rate of) return on asset \(i,\) then the portfolio return is \[r_P=\sum_{i=1}^nr_i\omega_i.\]

  3. If assets have return volatilites \(\sigma_i\) and correlations \(\rho_{ij},1\leq i<j\leq n,\) then the variance of portfolio return is \[\sigma_P^2=\sum_{i=1}^n\omega_i^2\sigma_i^2+\sum_{1\leq i<j\leq n}2\omega_i\omega_j\sigma_i\sigma_j\rho_{ij}.\]

  4. The minimum variance portfolio is the solution to the nonlinear program \[\begin{matrix} \textrm{minimize} & \sigma_P^2\\ \textrm{subject to} & \sum\omega_i=1\\ & \omega_i\in\mathbb{R} \end{matrix}\] where the efficient portfolio of return \(\alpha>0\) is the solution to the nonlinear program \[\begin{matrix} \textrm{minimize} & \sigma_P^2\\ \textrm{subject to} & r_P=\alpha\\ & \sum\omega_i=1\\ & \omega_i\in\mathbb{R} \end{matrix}\] Note that if short sell is not permitted, the programs above become \[\begin{matrix} \textrm{minimize} & \sigma_P^2\\ \textrm{subject to} & \sum\omega_i=1\\ & \omega_i\geq0\\ & \omega_i\in\mathbb{R} \end{matrix}\] and \[\begin{matrix} \textrm{minimize} & \sigma_P^2\\ \textrm{subject to} & r_P=\alpha\\ & \sum\omega_i=1\\ & \omega_i\geq0\\ & \omega_i\in\mathbb{R} \end{matrix}\]

  5. Karush\(-\)Kuhn\(-\)Tucker Theorem: consider the program \[\begin{matrix} \textrm{minimize} & f(x)\\ \textrm{subject to} & h_1(x)=...=h_m(x)=0\\ & g_1(x),...,g_p(x)\leq0 \end{matrix}\] and \(x^*\) is a minimum point of this program. Its Lagragian is \[\mathcal{L}(x,\lambda,\mu)=f(x)+\sum_{i=1}^m\lambda_ih_i(x)+\sum_{j=1}^p\mu_jg_j(x)\] and there exist numbers \(\lambda_i,\mu_j\) such that \(\mu_j\geq0\) and \[\left\{\begin{matrix} \mathcal{L}_{x_k}=0,\forall k\\ h_i(x)=0,\forall i\\ \mu_jg_j(x)=0,\forall j \end{matrix}\right.\]

2 Forward Contract

2.1 No Dividend

Consider a \(T-\)year forward contract on an asset paying no dividend with current price \(S_0,\) interest rate \(r\) p.a. compounding annually. We set up a portfolio \(V\) as follows:

  1. Today \((t=0):\)
  • Borrow \(S_0\) to buy \(1\) asset.
  • Enter the contract to sell \(1\) asset (at maturity) for \(K.\)
  • Additional fees: \(0.\)
  • Contract value: \(V_0=S_0-S_0+0=0.\)
  1. At maturity \((t=T):\)
  • Repay the loan with \(S_0\cdot(1+r)^T.\)
  • Hand over \(1\) asset and receive the delivery price \(K.\)
  • Additional fees: \(0.\)
  • Contract value: \(V_T=-S_0\cdot(1+r)^T+K+0=K-S_0\cdot(1+r)^T.\)
  1. Under the assumption of no arbitrage, \(V_T=0\Rightarrow K=S_0\cdot(1+r)^T.\)
  2. The delivery price is, therefore, \(S_0\cdot(1+r)^T.\)

Similarly, if \(r\) compounds \(n\) times p.a., then \(K=S_0\cdot(1+r/n)^{Tn}.\) It implies that as \(n\rightarrow\infty,\) i.e. \(r\) compounds continuously, \[K=\lim_{n\rightarrow\infty}S_0\cdot\left(1+\frac{r}{n}\right)^{Tn}=S_0\cdot e^{rT}.\]

2.2 Discrete Dividend

Consider a \(T-\)year forward contract on an asset with current price \(S_0,\) interest rate \(r\) p.a. compounding annually. This asset pays dividend discretely at time \(0<t_1<...<t_m<T\) with amount \(c_1,...,c_m.\) We set up a portfolio \(V\) as follows:

  1. Today \((t=0):\)
  • Borrow \(c_i\cdot(1+r)^{-t_i}\) for \(t_i\) years, \(i=\overline{1,m}.\)
  • Borrow \(S_0-\sum c_i\cdot(1+r)^{-t_i}\) for \(T\) years.
  • Use the total money borrowed, \(S_0,\) to buy \(1\) asset.
  • Enter the contract to sell \(1\) asset (at maturity) for \(K.\)
  • Additional fees: \(0.\)
  • Contract value: \(V_0=S_0-S_0+0=0.\)
  1. At each time \(t=t_i,i=\overline{1,m}:\)
  • The loan \(c_i\cdot(1+r)^{-t_i}\) becomes \(c_i.\)
  • Repay this loan with the dividend \(c_i\) paid.
  1. At maturity \((t=T):\)
  • Repay the remaining loan with \(\left(S_0-\sum c_i\cdot(1+r)^{-t_i}\right)\cdot(1+r)^T.\)
  • Hand over \(1\) asset and receive the delivery price \(K.\)
  • Additional fees: \(0.\)
  • Contract value: \[\begin{align*}V_T &= -\left(S_0-\sum c_i\cdot(1+r)^{-t_i}\right)\cdot(1+r)^T+K+0\\&= K-\left(S_0-\sum c_i\cdot(1+r)^{-t_i}\right)\cdot(1+r)^T.\end{align*}.\]
  1. Under the assumption of no arbitrage, \(V_T=0\Rightarrow K=\left(S_0-\sum c_i\cdot(1+r)^{-t_i}\right)\cdot(1+r)^T.\)
  2. The delivery price is, therefore, \(\left(S_0-\sum c_i\cdot(1+r)^{-t_i}\right)\cdot(1+r)^T.\)

Similarly, if \(r\) compounds \(n\) times p.a., then \(K=\left(S_0-\sum c_i\cdot(1+r/n)^{-t_in}\right)\cdot(1+r/n)^{Tn}.\) It implies that as \(n\rightarrow\infty,\) i.e. \(r\) compounds continuously, \[K=\lim_{n\rightarrow\infty}\left(S_0-\sum_{i=1}^m \frac{c_i}{(1+r/n)^{t_in}}\right)\cdot\left(1+\frac{r}{n}\right)^{Tn}=\left(S_0-\sum_{i=1}^m \frac{c_i}{e^{rt_i}}\right)\cdot e^{rT}.\]

2.3 Continuous Dividend

Consider a \(T-\)year forward contract on an asset with current price \(S_0,\) interest rate \(r\) p.a. compounding annually. This asset pays dividend continuously with rate \(\delta\) p.a., meaning that the dividend paid will be reinvested. We set up a portfolio \(V\) as follows:

  1. Today \((t=0):\)
  • Borrow \(S_0\cdot e^{-\delta T}\) to buy \(e^{-\delta T}\) asset.
  • Enter the contract to sell \(1\) asset (at maturity) for \(K.\)
  • Additional fees: \(0.\)
  • Contract value: \(V_0=S_0\cdot e^{-\delta T}-S_0\cdot e^{-\delta T}+0=0.\)
  1. At maturity \((t=T):\)
  • Repay the loan with \(S_0\cdot e^{-\delta T}\cdot(1+r)^T.\)
  • Hand over \(1\) asset and receive the delivery price \(K.\)
  • Additional fees: \(0.\)
  • Contract value: \(V_T=-S_0\cdot e^{-\delta T}\cdot(1+r)^T+K+0=K-S_0\cdot e^{-\delta T}\cdot(1+r)^T.\)
  1. Under the assumption of no arbitrage, \(V_T=0\Rightarrow K=S_0\cdot e^{-\delta T}\cdot(1+r)^T.\)
  2. The delivery price is, therefore, \(S_0\cdot e^{-\delta T}\cdot(1+r)^T.\)

Similarly, if \(r\) compounds \(n\) times p.a., then \(K=S_0\cdot e^{-\delta T}\cdot(1+r/n)^{Tn}.\) It implies that as \(n\rightarrow\infty,\) i.e. \(r\) compounds continuously, \[K=\lim_{n\rightarrow\infty}S_0\cdot e^{-\delta T}\cdot\left(1+\frac{r}{n}\right)^{Tn}=S_0\cdot e^{(r-\delta)T}.\]

3 Future Contract

Assume an investor enters \(C\) future contracts, each of size \(S\) with the initial price \(P_0,\) initial margin \(I,\) maintenance margin \(M.\) Then the margin ratio is \(I/(P_0\cdot S)\) and the maintenance ratio is \(M/I.\) The contract lasts for \(n\) days.

3.1 Long Future Contract

Day Future Price Contract Value Daily Gain Cumulative Gain Margin Account Balance Margin Call
\(0\) \(P_0\) \(CV_0\) \(I\)
… … … … … … …
\(i\) \(P_i\) \(CV_i\) \(DG_i\) \(CG_i\) \(MB_i\) \(MC_i\)
… … … … … … …
Variable Formula
\(CV_i\) \(P_i\cdot S\)
\(DG_i\) \(CV_i-CV_{i-1}\)
\(CG_i\) \(DG_1+...+DG_i=CG_{i-1}+DG_i\)
\(MB_i\) \((MB_{i-1}+DG_i)\cdot1_{MB_{i-1}+DG_i\geq M}+I\cdot1_{MB_{i-1}+DG_i< M}\)
\(MC_i\) \((I-(MB_{i-1}+DG_i))\cdot1_{MB_{i-1}+DG_i< M}\)

The overall gain is \(CG_n\cdot C,\) and the rate of return is \(CG_n/(I+\sum MC_i).\)

3.2 Short Future Contract

Everything stays the same, except \(DG_i=CV_{i-1}-CV_i.\)

4 Options

Consider call and put options on a same asset with current price \(S_t,\) strike price \(K,\) maturity \(T\) years, interest rate \(r\) compounds continuously. Their intrinsic values and payoffs are \[\left\{\begin{matrix}I_C(t)=S_t-K\\I_P(t)=K-S_t\end{matrix}\right.\textrm{ and }\left\{\begin{matrix}C_T=\max\left\{I_C(T),0\right\}\\P_T=\max\left\{I_P(T),0\right\}\end{matrix}\right.\] An option is out\(-\)of\(-\)money/at\(-\)the\(-\)money/in\(-\)the\(-\)money at time \(t\) if the intrinsic value is negative/\(0\)/positive.

4.1 European Put-Call Parity

Assume the option premiums are \(C_t,P_t.\)

  1. If the asset pays no dividend, the put\(-\)call parity is \[C_t+K\cdot e^{-r(T-t)}=P_t+S_t.\] Hint: if \(C_t+K\cdot e^{-r(T-t)}<P_t+S_t\) we short sell \(1\) share, write \(1\) put and buy \(1\) call. If \(C_t+K\cdot e^{-r(T-t)}>P_t+S_t\) we short sell \(1\) zero\(-\)coupon bond with face value \(K,\) write \(1\) put, buy \(1\) call and \(1\) share.
  2. If the asset pays dividends \(D_i\) at time \(t_i,i=\overline{1,n},\) the put\(-\)call parity is \[C_t+K\cdot e^{-r(T-t)}=P_t+S_t-\sum_{i=1}^nD_i\cdot e^{-r(t_i-t)}\]
  3. If the asset pays dividend continuously at rate \(\delta\) p.a., meaning that the dividend paid will be reinvested, the put\(-\)call parity is \[C_t+K\cdot e^{-r(T-t)}=P_t+S_t\cdot e^{-\delta(T-t)}\]

4.2 American Put-Call Symmetry

Assume the option premiums are \(C_A(S_t,t),P_A(S_t,t)\) and the asset pays no dividend. The put\(-\)call symmetry is \[S_t-K\leq C_A(S_t,t)-P_A(S_t,t)\leq S_t-K\cdot e^{-r(T-t)}\]

  1. If \(S_t-K>C_A(S_t,t)-P_A(S_t,t),\) we short sell \(1\) share, write \(1\) put and buy \(1\) call.
  2. If \(C_A(S_t,t)-P_A(S_t,t)>S_t-K\cdot e^{-r(T-t)},\) we short sell \(1\) zero\(-\)coupon bond with face value \(K,\) write \(1\) call, buy \(1\) put and \(1\) share.

5 Binomial Pricing Method

Consider an option on a stock with price \(S_t,\) strike price \(K,\) maturity \(T,\) interest rate \(r\) p.a. compound continuously.

5.1 Binomial Tree Model

Assume the up factor \(u,\) down factor \(d\) and the \(n-\)step binomial tree model.

  1. Step size: \(\Delta t=T/n.\)
  2. Up\(-\)move probability: \(p=(e^{r\Delta t}-d)/(u-d).\)
  3. On the stock price tree, at each step the stock price \(S_t\) goes up to \(uS_t\) or goes down to \(dS_t.\)
  4. On the option price tree, at step \(n\) the option price is its payoff. For alternative \(i\) at step \(j<n,\) \[V_{(i-1)j}=e^{-r\Delta t}(p\cdot V_{(i-1)(j+1)}+(1-p)V_{i(j+1)})\]

5.2 CRR Model

Assume the stock volatility is \(\sigma\) p.a. Then the up and down factors are \[u=e^{\sigma\sqrt{\Delta t}}\textrm{ and }d=e^{-\sigma\sqrt{\Delta t}}\] then everything follows the binomial tree model.

5.3 American Option

Note that for American options, the price at step \(n\) equals the corresponding European option but for step \(j<n,\) \[V_{(i-1)j}=\max\left\{e^{-r\Delta t}(p\cdot V_{(i-1)(j+1)}+(1-p)V_{i(j+1)}),O(S_{(i-1)j})\right\}\] where \(O(S_t)\) is the payoff if we exercise the option. At all nodes, the price of an American option should be at least equal to the corresponding European option since it gives the buyer more rights.