金融代写|MATH3975 Financial Derivatives

Chooser options are a type of exotic option that, at some pre-specified time $T_1$ in the future, can be converted into either a put or call option with expiry $T_2>T_1$ and strike $K$. We consider these options in the Black-Sholes setting. See the website http://demonstrations.wolfram.com/ChooserOptions/ and the interactive demonstrations there (you will need to download a free player). Then, solve the following problems.

• Explain the put-call parity argument (“It can be shown using general put-call parity considerations…”) and derive the price of a chooser option. You can check your solution for example here: http://www.haas.berkeley . edu/groups/finance/WP/ rpf220.pdf, pp. 56-57 (but you need to provide a more detailed explanation).
• Read the paper ${ }^1$
  Martinkute-Kauliene, Raimonda. “Exotic Options: a Chooser Option and its Pricing.” Business, Management and Education 2 (2012): 289-301.
  and, for selected values of parameters, make similar plots as those in Figure 2 (dependence of chooser option price on the time $T_1$ when it is converted into put or call; note that in general this dependence is not necessarily so close to linear, as claimed in the paper) and Figure 3 (dependence of chooser option price on the strike price $K$ ). Give a financial interpretation for their increasing/decreasing behaviour.

Sharpe ratio of a derivative (of an underlying stock $S$ ) is given by
$$
\frac{\text { expected return – risk free rate }}{\text { return volatility }}
$$
which is equal to $\frac{\tilde{\mu}-r}{\tilde{\sigma}}$, if $r$ is the risk free rate and the value $x$ of the derivative satisfies the stochastic differential equation $d x / x=\tilde{\mu}(S, t) d t+\tilde{\sigma}(S, t) d w$. Now, consider the Black-Scholes setting, the stock which does not pay dividends and the derivative with the payoff $\bar{V}(S)=S^2$

• Find the price $V(S, t)$ of the derivative.
  HinT: See lecture slides for a form of the solution.
• Use Itō lemma to compute $d V$ and use it to compute the Sharpe ratio of this derivative. Show that it is same as the Sharpe ratio for the underlying stock.

Sharpe ratio of a derivative (of an underlying stock $S$ ) is given by
expected return – risk free rate
return volatility
which is equal to $\frac{\tilde{\mu}-r}{\tilde{\sigma}}$, if $r$ is the risk free rate and the value $x$ of the derivative satisfies the stochastic differential equation $d x / x=\tilde{\mu}(S, t) d t+\tilde{\sigma}(S, t) d w$. Now, consider the Black-Scholes setting and some general European-style derivative.

Show that the Sharpe ratio of this derivative is same as the Sharpe ratio for the underlying stock.
HinT: Use Itō lemma to compute $d V$ and connect it to Black-Scholes PDE which is satisfied by the derivative price.

Source: beanactuary.com, acturial exams. Consider a 3-month American call option on a nondividend-paying stock with strike price 41.5. You are given:
The Black-Scholes framework holds.
The stock is currently selling for 40 .
The stock’s volatility is $30 \%$ (i.e. 0.3 )
The current call option delta is 0.5 .
Determine the current price of the option.
Hint: The underlying stock does not pay dividends, so the value of the American call option equals to the value of the European call option with the same parameters. Recall the formula for the delta of such an option and the graph of the CDF of $\mathcal{N}(0,1)$ distribution.

Let $A, K>0$ be given constants. Consider the Black-Scholes framework and a European option with payoff equal to $(\min (S-K, A))^{+}$, where $x^{+}=\max (x, 0)$ and $S$ is a stock price. Prove that delta of this option is positive.