# Finance Question

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Question 3

[5 + 4 + 2 + 4 = 15 marks]

Word gets out that you are doing such a great job in your new role and so it is not long before

you are asked back to UTS to give a guest lecture. The subject coordinator has asked you to

help with some of the trickier aspects of lectures 7, 8, and 9 of Derivative Securities (25620).

Specifically, you have been asked to go through the calculation of various option prices using

both binomial trees and the Black-Scholes model.

The example to be used is as follows: The S&P/TSX 60 (which covers the 60 largest stocks

traded on the Toronto Stock Exchange in Canada) is currently trading at 1,232 (index points).

The dividend yield of the index is 5.2% per annum and the risk-free interest rate in Canada is

3.5% per annum (both with continuous compounding for all maturities). The volatility of the

index over the next six months is also estimated to be 28% per annum. Given your expertise

in option valuation you decide to use a six-step binomial tree to calculate the following

derivative values (in units of index points, to two decimal places):

(a) A six-month European call option on the index with a strike of 1,200. Calculate also the

value of the option by using the Black-Scholes formula. Compare the values and comment.

(b) A six-month American call option on the index with a strike of 1,200. Is the answer different

to the answer from (a)? Is so, explain why, making explicit reference to the dividend yield

and risk-free rate.

(c) A long position in a forward contract on the index for delivery in six months at a price of

1,200. Calculate also the theoretical value of this forward position. Compare these values

(the theoretical and binomial tree values) and comment. Hint: the fair forward price that

would be agreed for new contracts today is different to 1,200 and hence the forward

contract in question has a non-zero value.

(d) An American down-and-out barrier call option with a strike of 1,200 and knockout barrier

of 1,100 maturing in six months. An American down-and-out call option gives the holder

the right to buy the underlying asset at the strike price at any time on or prior to the

expiration date so long as the price of that asset did not go below a pre-determined barrier

during the options lifetime. When the price of the underlying asset falls below the barrier,

the option is “knocked-out” and no longer carries any value. Comment on the value of this

option relative to the option in (b) and explain any differences. Would the value of this

option change if the knockout barrier was decreased from 1,100 to 1,000? Is so, comment

on the reason why.

Question 4

[5 + 5 + 5 = 15 marks]

Satoshi Nakamoto, the mysterious inventor of Bitcoin and developer of the worlds first

blockchain,4 has approached your financial institution (confidentially of course) for advice on

hedging his existing crypto portfolio. The portfolio is currently worth USD1.2 billion and also

has an income yield (from yield farming) of around 2.5% per annum with simple

compounding. Satoshi (or Ian as he insists on being called) has seen the value of his portfolio

significantly decline over the last two years, due in large part to the latest crypto winter.5

While there has been a resurgence in bitcoins price in 2023, Satoshi/Ian expects a bear

market for the remainder of the year. As such, he is seeking advice on how best to protect

the value of his existing crypto portfolio over the next 4 months. He also does not want to

miss out on any potential upside if the bear market doesnt eventuate.

You know just the derivative for the job and suggest that a position in Bitcoin (BTC) options

will provide the desired protection. Since Satoshis portfolio is highly correlated with Bitcoin

prices, options on Bitcoin can be used as a portfolio hedge (in the same way that stock index

options can be used to hedge an equity portfolio). To this end, you estimate from historical

price data that the Bitcoin beta6 of Satoshis portfolio is 0.7. You also note that Bitcoin does

not pay any income (and hence has a zero income yield) and that the risk-free interest rate in

the US is currently 4.2% per annum with simple compounding for all maturities.

The current spot price of Bitcoin is USD25,830 and the following table provides the market

prices (in USD) for various European call and put options written on Bitcoin with different

strikes and maturing in four months time:

Call price | Strike (USD) | Put price |

12,135 | 14,000 | 108 |

10,244 | 16,000 | 191 |

8,395 | 18,000 | 310 |

6,631 | 20,000 | 517 |

5,008 | 22,000 | 865 |

3,603 | 24,000 | 1,434 |

2,495 | 26,000 | 2,291 |

1,715 | 28,000 | 3,482 |

120 | 30,000 | 4,934 |

865 | 32,000 | 6,571 |

646 | 34,000 | 8,299 |

493 | 36,000 | 10,112 |

387 | 38,000 | 11,983 |

305 | 40,000 | 13,873 |

(Source: https://www.deribit.com)

4 https://en.wikipedia.org/wiki/Satoshi_Nakamoto

5 https://www.forbes.com/advisor/investing/cryptocur….

6 The Bitcoin beta tells you the sensitivity (elasticity) of a portfolios returns to the returns on Bitcoin (BTC).

This is analogous to the market beta which measures the sensitivity of an equity portfolios returns to the

return on a market index.

(a) Describe the options portfolio insurance strategy that would insure against Satoshis

cryptocurrency portfolio falling below USD1 billion over the next four months.7 Explain

why this strategy fulfils his request and why hedging with Bitcoin futures does not suffice.

Also, highlight some of the potential downsides of the strategy if implemented in practice.

(b) Calculate the gains/losses on the strategy if the price of Bitcoin in four months is either (i)

$15,000, or (ii) $35,000, and prepare a short summary for Satoshi to discuss the outcome

of the insurance strategy in these two scenarios. Note that here you should include the

cost of the options purchase (inferred from the table above) in your calculation of the P&L

in each scenario. Ignoring the time value of money, what value of Bitcoin in four months

would the portfolio, including the option costs, breakeven (i.e., be worth USD1.2 billion)?

Satoshi is somewhat interested in the proposed strategy, but he is surprised by how expensive

the options are. You state that this is due to the very high volatility in BTC and the wider crypto

market. To explain things further you decide to investigate the implied volatility of BTC using

the option prices given above.

(c) Use Excels GoalSeek (or otherwise) to estimate the implied volatility of the spot price of

BTC, based on the market prices of the BTC options above. Specifically, complete the

following table with the estimated implied volatilities (to 3 significant figures):

Implied volatility from call |
Call price (USD) |
Strike (USD) |
Put price (USD) |
Implied volatility from put |

12,135 | 14,000 | 108 | ||

10,244 | 16,000 | 191 | ||

8,395 | 18,000 | 310 | ||

6,631 | 20,000 | 517 | ||

5,008 | 22,000 | 865 | ||

3,603 | 24,000 | 1,434 | ||

2,495 | 26,000 | 2,291 | ||

1,715 | 28,000 | 3,482 | ||

120 | 30,000 | 4,934 | ||

865 | 32,000 | 6,571 | ||

646 | 34,000 | 8,299 | ||

493 | 36,000 | 10,112 | ||

387 | 38,000 | 11,983 | ||

305 | 40,000 | 13,873 |

You should also perform the following tasks and answer the following questions:

i. Plot the implied volatility from each option type (put/call) as a function of the strike.

ii. Is the implied volatility of one option class higher than the other? If so, explain why.

iii. Does the implied volatility depend on the moneyness of the option? Is so, explain.

iv. What do your results tell you about the observed market prices and their consistency

with the assumptions underlying the Black-Scholes model?

7 Note that the USD1 billion value should not include the cost of the options used for insurance and each

option contract is written on exactly one unit of BTC.

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