Definitions Flashcards
Risk Free assets
Over the time period considered there is no uncertainty in the future, hence no risk in the return, we model it as Aexp{-r(T-t)}
Explain what A, t, r and T mean in terms of a risk free asset
r is the continuous compound interest rate
A is the amount invested at time t
T is the expiry time
Risky assets
Stocks or shares usually, traded on the open market. We don’t know what their price will be in the future, so we treat it as random
What is arbitrage
An opportunity to make a risk free profit with no initial outlay
This returns a profit regardless of what the asset does in the future
No Arbitrage principle (1st version)
There can be no opportunities to make a riskfree profit with no initial outlay.
No Arbitrage principle (2nd version)
There can be no riskfree way to get a better rate of return than the bank interest rate r
What assumptions are made in the no Arbitrage principles (2 criteria)
i) Traders buy and sell at the same price, so we are ignoring taxes etc.
ii) the market’s adjustment to the pressures of supply and demand does not happen instantaneously
Short selling
Here we ‘borrow’ shares and trade with them, but have to then return the same number of shares to their owner
Could mean you have to buy the shares back at the market price later on, could be very expensive
Frictionless/Liquidity/Divisibility/Short-Selling Assumption:
An investor can hold any quantity (positive or negative, integer or fractional) of an asset, and at any time can buy and sell any quantity at the prevailing price (with no transaction charges).
What is a derivative
Derivative securities are assets which depend on some other (risky) asset, called the underlying asset
Give examples of derivatives
Forward contracts & options
What is a forward contract
A deal to sell a specified amount of some commodity (shares) at a specified future time T, for a fixed price F.
The contract is assumed to not have any value, though both sides have to buy or sell the agreed contracts at the expiry time, regardless of who wins or loses
Give the ‘fair’ price of a forward contract, according to the no arbitrage principle
S_0 exp{rT}, so initial price multiplied by exponential of the product of the interest rate and expiry date.
If a forward contract was priced unfairly how could you make an arbitrage opportunities (2 cases)
i) if F>S_0 exp{rT} a guaranteed profit can be made at time T by selling a forward contract and taking out a loan at time 0 to buy the commodity
ii ) If F<S_0 exp{rT} can guarantee profit by buying a forward contract and short selling the commodity at time 0, investing the proceeds risk free until time T
What is an option
They are derivative securities that give the holder the option but not the obligation to buy or sell an asset for a specified price at some time in the future
Difference between a put and call option
In the call option the person buying the asset gets to choose if they want to, in a put option its the person selling that gets to make the choice at time T
In both cases the person who gets to make the choice is known as the holder of the option, as they are the ones that has paid to have the choice
European call option
gives the holder the right to buy one unit of the asset for a price E (the exercise price) at some future time T (the expiry
time).
European Put option
gives the holder the right to sell one unit of the asset for the price E at time T
What makes American options different to European ones
the holder may choose to exercise them at any time up to T
What are the payoff equations for a call option
max(S-E, 0) for final asset price (on the open market) of S and the agreed exercise price E
Payoff formula of put option
max(E-S, 0)
Put -Call parity formula
P(t) = C(t) + Eexp{-r(T-t)} - S(t)
Explain what P, C E, r, T, t and S mean in the put call parity formula
P = value of put option (at t)
C = value of call option
r = compound interest rate
T = expiry time
t = variable time we are interested in
S = market price of the asset
What 2 things is it important to understand about the Put-call parity
(i) it does not tell us the actual value of either P(t) or C(t), only how these two are related
(ii) it will be true whatever assumptions we make about the possible future values of the asset price (although these assumptions will affect the actual values of P(t) and C(t)). In particular, it will be true whether we treat the share price as a discrete time stochastic process or a continuous time stochastic process
Whats the lower bound for the price of a call option
C(t) >= S(t) - E exp{-r(T-t)}
What are the assumptions of the One-Step Binomial Model
The share has initial price S(0) at t=0 and the share price is given at S(T) by:
S(T) = S+ with prob p or S- with prob 1-p
S(^-)exp{-rT} <S(0) <S(^+)exp{-rT}
Give an example of a payoff of a One-Step binomial model
To give a specific example, suppose that X is a call option with expiry time T and exercise price E, where S − < E < S+, then X+ = S + − E and X− = 0, since the payoff of the option is max(S(T) − E, 0). Similarly for a put option, we would have X+ = 0 and X− = E − S −. Note that we do not necessarily have X+ > X−
State the One-Step Binomial theorem giving the intial value of the option
X(0) = ((S^+X^- - S^-X^+)/(S^+-S^-))exp{-rt} + ((X^+-X^-)/(S^+-S^-))S(0)
What is the expected payoff of the one step binomial model
E[X(T)] = pX^(+) + (1-p)X^(-)
Whats the expected discounted payoff of the claim
E[exp{-rT}X(T)] = exp{-rT}(pX^(+) + (1-p)X^(-))
Why is the expected discounted payoff not the correct price
It depends on probability which can lead to an arbitrage oppourtunity
Write the risk neutral probability
p_(*) = (S(0)exp{rT) - S^(-)) / (S^(+) - S^(-))
Why is the risk neutral probability importatn
When you use it to calculate the discounted expected payoff of the claim you get the correct arbitrage value of the claim, something you don’t get when using the actual market free value of p
What is a stochastic process
A sequence of random variables, X_0, X-1,…. representing observations of some random system at successive times 0,1,…
The probability distribution of the general X_n can depend on n and its history up to n, they can also be finite of infinite
Whats the random walk
stochastic process with probability to move forward of backward at each time step, can be written interatively of explicitly
How do your represent a random walk with drift iteratively
S_0 = 0;
S_n = S_n-1 + mu + sigmaY_n for n>= 1
with Y_n iid random variables with equal probability of generating {1,-1}
Find E[Y_i], Var[Y_i], E[S_n], Var[S_n] for a symmetric random walk with drift
E[Y_i] = 1/2(1^2) + 1/2(-1^2) = 0
Var[Yi] = 1/2(+1)^2 +1/2(−1)^2 − 0^2 = 1.
E[Sn] = nµ + σSum{Y_i} = nmu
Var(S_n) = sigma^2 SumVar(Y_i) = nsigma^2
What does central limit theorem say about the distribution of S_n for large n
It will have normal distribution with mean and variance:
S_N ~ N(n mu, n sigma^2)
What is a Markov process
A stochastic process which is memoryless, future evolution only depends on the state it is currently in
What is the filtration of a stochastic process
F_m is the set of all events which are determined by time m, and the family (F_m)_(m>=0) is a filtration of the underlying probability space
What are the three rules of conditional expectation
i ) Time zero rule: E[Z|F_0] = E[Z]
ii ) Tower law: E[E[Z | Fn] | Fm] = E[Z | Fm]
iii) taking out a known factor: If Y is a random variable whose value is already determined by time m then;
E[Y Z | Fm] = Y E[Z | Fm]
What is a Martingale
A discreter time stochastic process (X_n)_(n>=0) such that
i ) E[Xn | Fm] = Xm for all n > m
ii ) E[ |Xn| ] < infinity for all n.
What does the risk neutral probability do if we consider the one step binomial method as a stochastic process
It makes it a martingale
