Josh O'Byrne

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					Explain and illustrate with the aid of pay-off diagrams the outcome of buying a
call option on the FTSE100 index while short-selling the FTSE100 index itself.

       Synthetic long put.
       Payoff maximised when S=0.
       Useful for turning a bullish position (long call) into a bearish position (long put).

What strategies would an investor adopt if she wants to create (i) a long
straddle and (ii) a short straddle? Explain the reasons for choosing these
strategies in different market conditions. Show the relevant pay-off diagrams.

Long straddle can be created by buying a call option and a put option at the same strike
price. An investor choosing this option is predicting high volatility in the underlying asset,
and receives maximum payoff when the underlying is much lower or much higher in
value than the strike price.

Short straddle can be created by selling a call option and a put option at the same strike
price. An investor choosing this option is predicting that the share price will remain
relatively unchanged throughout the life of the options. Maximum payoff is received if
the terminal stock price is equal to the strike price. LOW VOLATILITY

Differentiate between a futures contract and a forward contract.

   •   Futures are traded on exchanges whereas forwards are traded over the counter.
   •   Futures contracts are standardised – i.e. specific delivery dates & volume. In a
       forward contract the terms can be decided by the trading parties.
   •   Futures contracts are settled daily whereas a forward contract is usually settled at
   •   Forward contracts hold some credit risk, whereas futures contracts do not.
   •   Future contracts are usually closed out before expiry, whereas forward contracts
       are usually completed (i.e. delivery).
   •   Price difference depends on correlation of underlying with interest rate

“The futures price of a contract might be the same as, or higher than, or less
than, the forward price of a contract with the same time to maturity”. Discuss
this statement with reference to the empirical literature.

       Difference arises because of the daily settlement of futures contracts.
       If underlying is positively correlated with the interest rate rises in the interest
       rate will increase the value of the futures contract when compared to the forward.
       If the interest rate falls the futures contract will be worth less than the forward
       If underlying is negatively correlated the opposite is true.
       This is because when for example the interest rate rises the underlying will rise
       and the investor with a long position is in a better position. Not only because (S-K)
       is greater, but the discount rate is higher, so in real terms the final price is
       When the interest rate falls the underlying falls and the investor with a long
       position will have to make margin payments, and purchase of the underlying
       would seem more attractive.

                                                                                  Josh O’Byrne 
Demonstrate how the fair price of a futures contract (with no uncertainty) can
be derived. Discuss the assumptions underlying the derivation of this fair

Assume no arbitrage.
Assume rate of interest r.
K=price of future contract

Consider two portfolios A: long one futures contract B: borrowed Kexp(-rT) and buy
underlying S(0) with cash. On expiry the value of the future contract =S(T)-K and the
value of portfolio B = S(T) – K. By the principle of no arbitrage these portfolios always
hold an equal value. Therefore at time 0 K=S(0)exp(rT).

“If there is no basis risk, the minimum variance hedge ratio is always 1.0”. Is
this statement true or false? Explain your answer in detail.

If there is no basis risk this suggests a perfect hedge.
In which case the optimal minimum variance hedge ratio is 1.0.
The statement is therefore true.

On the 30th January 2005, a UK equity pension fund is valued at £53 million.
The fund has a beta (β) of 1.25; the FTSE100 index is currently at 5150. The
March FTSE100 futures contract (valued at £10.00 per index point) is settled
on March 25th. The funding rate is 4.4%, the historic dividend yield on the
FTSE100 is 4.0% and you expect 15% of the total dividend payment to be made
by the FTSE100 stocks between now and March 25th. (Work to the nearest
   (i)    Price the March FTSE100 futures contract.

F= 10 * 5150 * exp {[4.4% - (15% * 4%)]*(54/365)} = 5179*10= £51790

   (ii)    If the fund manager wants to hedge the portfolio until March using
           FTSE100 stock index futures, does she have to buy or sell the futures
           contracts? How many would she need?

Short index futures.
Using formula Number of contracts=Beta*Portfolio/Futures.
N=1.25*53000000/51790=1279.20 contracts.

   (iii)   Calculate the final value of the portfolio if on the 25th March, the
           FTSE100 index rises to 5200.

   Using the security market line, R0= 4.4% * 54/365 = 0.651%.
   R=0.651%+1.25*[{(5200/5150)-1} – 0.651%] = 1.051%. 53*1.01051 = £53.557
   Loss from futures contract = Deliver index at 5179*10*1279.20 – 5200*10*1279.20
   Portfolio = $53.288 million

                                                                             Josh O’Byrne 
(iv) Calculate the final value of the portfolio if on the 25th March, the FTSE100
index falls to 5100.

From SML as before R=-1.376% Portfolio value= 53*(1+R)=53*0.9862=52.271 million.
Gain from short futures contacts = (5179-5100)*10*1279.20= £1,010,568
Portfolio= £58.282 million

Explain carefully the difference between a calendar spread and a butterfly
spread. Illustrate with the aid of pay-off diagrams. Under what market
conditions will an investor choose each of the two strategies?

A calendar spread involves two call options or two put options with the same strike price
but different maturity dates. Depending on the investors outlook on the market they can
choose a strike price to reflect their views – higher strike if they’re bullish.

Butterfly spread is created using call or put options. Using one long call at K1 and one
long call at K3 where K1<K3, and two short calls with strike K2 where K1<K2<K3. Same
with puts. An investor who chooses this spread expects the option to expire at a price
close to K2, however unlike a shorting a straddle the downside risk is limited.

State the assumptions underlying the Black-Scholes model.

   •   Log-normal property of stock prices
   •   Unlimited short selling possible
   •   Borrowing and lending at the risk free rate of interest
   •   No transaction costs
   •   The stock is does not pay a dividend
   •   Perfectly divisible securities
   •   No arbitrage opportunities
   •   European options

Six factors affect the value of call options on stocks. Identify those factors and
explain how and why changes in each of these factors affect the value of call

   •   Volatility – An increase in volatility will increase the value of European and
       American puts and calls. The increase in price reflects the premium required for a
       certain price at expiry (European). In terms of American increased volatility gives
       the underlying a greater chance of moving into the money.
   •   Time to expiry – For American options an increase in the time to maturity will
       increase the value. This is because the owner of the option will have more
       opportunities to exercise the option.
   •   Underlying price – An increase in the price of the underlying will increase the
       value of call options which give the holder of the option the payoff max(S-K,0).
       For a put option an increase in the value of the underlying reflects a fall in the
       value of the put option as the payoff is max(K-S,0) therefore lower S means
       smaller payoff.
   •   Strike price – Increasing the strike price decreases the value of calls and
       increases the value of puts.
   •   Risk free rate – Increasing the risk free rate increases the value of calls and
       decreases the value of puts. This is because the holder of a call option is now
       discounting the payment for the option at expiry at a higher rate of interest and
       they can invest the money in the bank at a higher rate of interest. The opposite
       holds for puts. It should also be consider though that the stock market is
       negatively correlated with interest rates, as a rise in interest rates will erode the

                                                                                Josh O’Byrne 
       value of companies who may be valued using DCF for which r plays some role in
       the discount rate.
   •   Dividend yield – Increasing the dividend yield will reduce the value of calls - the
       underlying is now more attractive. Put value increases.

Describe three strategies that utilize financial derivatives and could be
implemented to take advantage of the company’s market expectations.
Expecting rise in the market

Long index calls. Short index puts. Long index futures.

                                                                              Josh O’Byrne 

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