Arbitrages and pricing of stock options Gonzalo Mateos Dept. of ECE - - PowerPoint PPT Presentation
Arbitrages and pricing of stock options Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November 30, 2018 Introduction to Random
Introduction to Random Processes Arbitrages and pricing of stock options 1
Introduction to Random Processes Arbitrages and pricing of stock options 2
◮ Bet on different events with each outcome paying a random return ◮ Arbitrage: possibility of devising a betting strategy that
◮ Arbitrages often involve operating in two (or more) different markets
Introduction to Random Processes Arbitrages and pricing of stock options 3
◮ Bet x on Yankees and y against Yankees. Guaranteed earnings?
◮ Bet x on Yankees and y against Yankees. Guaranteed earnings?
Introduction to Random Processes Arbitrages and pricing of stock options 4
◮ First condition on Booker 1 and second on Booker 2 are compatible ◮ Bet x on Yankees on Booker 1, y against Yankees on Booker 2 ◮ Guaranteed earnings possible. Make e.g., x = 2066, y = 1000
◮ Sport bookies coordinate their odds to avoid arbitrage opportunities
◮ If you plan on doing this, do it on, e.g., currency exchange markets
Introduction to Random Processes Arbitrages and pricing of stock options 5
◮ Let events on which bets are posted be k = 1, 2, . . . , K ◮ Let j = 1, 2, . . . , J index possible joint outcomes
◮ Joint realizations, also called “world realization”, or “world outcome”
◮ If world outcome is j, event k yields return rjk per unit invested (bet) ◮ Invest (bet) xk in event k ⇒ return for world j is xkr jk
◮ Total earnings ⇒ K
◮ Vectors of returns for outcome j ⇒ rj := [rj1, . . . , rjK]T (given) ◮ Vector of bets ⇒ x := [x1, . . . , xK]T (controlled by gambler) Introduction to Random Processes Arbitrages and pricing of stock options 6
◮ There are K = 2 events to bet on
◮ Naturally, there are J = 2 possible outcomes
◮ Q: What are the returns?
◮ Bet x on Yankees and y against Yankees, vector of bets x = [x, y]T
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◮ Arbitrage is possible if there exists investment strategy x such that
◮ Equivalently, arbitrage is possible if
x
j
◮ Earnings xTrj are the inner product of x and rj (i.e., ⊥ projection)
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◮ There is a line that leaves all rj
◮ There is not a line that leaves all rj
◮ Arbitrage possible ◮ Arbitrage not possible ◮ Prob. vector p = [p1, . . . , pJ]T
J
◮ There is prob. vector p = [p1, . . . , pJ]T
J
◮ Think of pj as scaling factors
Introduction to Random Processes Arbitrages and pricing of stock options 9
◮ Have demonstrated the following result, called arbitrage theorem
x
j
J
◮ Prob. vector p is NOT the prob. distribution of events j = 1, . . . , J
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◮ There are K = 2 events to bet on, J = 2 possible outcomes ◮ Q: What are the returns?
◮ Arbitrage impossible if there is 0 ≤ p ≤ 1 such that
Introduction to Random Processes Arbitrages and pricing of stock options 11
◮ Consider a stock price X(nh) that follows a geometric random walk
√ hYn ◮ Yn is a binary random variable with probability distribution
◮ Q: Are there arbitrage opportunities in trading this stock?
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◮ Consider the following investment strategy (stock flip):
◮ Cost of transaction is $1. Units of stock purchased are 1/X(0)
◮ There are two possible outcomes for the price of the stock at time h
√ h,
√ h ◮ Possible returns are therefore
√ h
√ h − 1,
√ h
√ h − 1
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◮ One dollar at time h is not the same as 1 dollar at time 0
◮ Interest rate of a risk-free investment is α continuously compounded
◮ Prices have to be compared at their present value ◮ The present value (at time 0) of X(h) is X(h)e−αh
◮ Present value of possible returns (whether Y0 = 1 or Y0 = −1) are
√ h
√ h − 1,
√ h
√ h − 1
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◮ Arbitrage not possible if and only if there exists 0 ≤ q ≤ 1 such that
◮ Substituting r1 and r2 for their respective values
√ h − 1
√ h − 1
◮ Can be easily solved for q. Expanding product and reordering terms
√ h + (1 − q)e−αhe−σ √ h = 1 ◮ Multiplying by eαh and grouping terms with a q factor
√ h − e−σ √ h
√ h
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◮ Solving for q finally yields ⇒ q = eαh − e−σ √ h
√ h − e−σ √ h ◮ For small h we have eαh ≈ 1 + αh and e±σ √ h ≈ 1 ± σ
◮ Thus, the value of q as h → 0 may be approximated as
◮ Also, suspiciously similar to probabilities of geometric random walk
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Introduction to Random Processes Arbitrages and pricing of stock options 17
◮ Stock prices X(nh) follow geometric random walk (drift µ, variance σ2)
◮ Arbitrage is not possible in stock flips if there is 0 ≤ q ≤ 1 such that
√ h
√ h − e−σ √ h ◮ Notice that q satisfies the equation (which we’ll use later on)
√ h + (1 − q)e−σ √ h = eαh ◮ Q: Can we have arbitrage using a more complex set of possible bets?
Introduction to Random Processes Arbitrages and pricing of stock options 18
◮ Consider the following general investment strategy:
◮ Possible outcomes are value at time mh and observed values
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◮ There are 2n possible bets:
◮ Bet 1 = n price increases in 1, . . . , n ◮ Bet 2 = price increases in 1, . . . , n − 1 and price decrease in n ◮ . . .
◮ For each bet we have 2m−n possible outcomes:
◮ m − n price increases in n + 1, . . . , m ◮ Price increases in n + 1, . . . , m − 1 and price decrease in m ◮ . . .
X(h) X(2h) X(3h) X(nh) bet 1 eσ
√ h
e2σ
√ h
e3σ
√ h
enσ
√ h
bet 2 eσ
√ h
e2σ
√ h
e3σ
√ h
e(n−2)σ
√ h
bet 2n e−σ
√ h e−2σ √ h e−3σ √ h
e−nσ
√ h
X((n+1)h) X((n+2)h) X(mh) X(nh)eσ
√ h
X(nh)e2σ
√ h
X(nh)emσ
√ h
X(nh)eσ
√ h
X(nh)e2σ
√ h
X(nh)e(m
− 2)σ √ h
X(nh)e
−σ √ h X(nh)e −2σ √ h
X(nh)e
−mσ √ h
◮ Table assumes X(0) = 1 for simplicity
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◮ Define the prob. distribution q over possible outcomes as follows ◮ Start with a sequence of i.i.d. binary RVs Yn, probabilities
√ h
√ h − e−σ √ h
◮ Joint prob. distribution q on X(h), X(2h), . . . , X
√ hYn
◮ Will show that expected value of earnings with respect to q is null
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◮ Consider a time 0 unit investment in given arbitrary outcome ◮ Stock units purchased depend on the price X(nh) at buying time
◮ Cash after selling stock given by price X(mh) at sell time m
◮ Return is then ⇒ r
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◮ Expected value of all possible returns with respect to q is
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◮ Need to find expectation of future value Eq(n+1:m)
√ hYm−1
√ hYm−1eσ √ hYm−2
√ hYm−1eσ √ hYm−2 . . . eσ √ hYn
◮ All the Yn are independent. Then, upon taking expectations
√ hYm−1
√ hYm−2
√ hYn
◮ Need to determine expectation of relative price change E
√ hYn
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◮ The expected value of the relative price change E
√ hYn
√ hYn
√ h Pr [Yn = 1] + e−σ √ h Pr [Yn = −1] ◮ According to definition of measure q, it holds
◮ Substituting in expression for E
√ hYn
√ hYn
√ h q + e−σ √ h (1 − q) = eαh
◮ Reweave the quilt:
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◮ Plug E
√ hYn
◮ Substitute result into expression for expected return
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◮ Suppose stock prices follow a geometric Brownian motion, i.e.,
◮ Q: What is the no arbitrage condition? ◮ Approximate geometric Brownian motion by geometric random walk
◮ No arbitrage measure q exists for geometric random walk
◮ This requires h sufficiently small ◮ Notice that prob. distribution q = q(h) is a function of h
◮ Existence of the prob. distribution q := lim h→0 q(h) proves that
Introduction to Random Processes Arbitrages and pricing of stock options 27
◮ Recall that as h → 0 ⇒ q ≈ 1
h→0 q(h) is a geometric Brownian motion
◮ Measure showing arbitrage impossible a geometric Brownian motion
◮ Furthermore, the variance is the same as that of stock prices
Introduction to Random Processes Arbitrages and pricing of stock options 28
◮ Compute expected return on an investment on stock X(t)
◮ Expected value of cash after sell given X(0) is
◮ Alternatively, invest X(0) risk free in the money market
◮ Invest in stock only if µ + σ2/2 > α
Introduction to Random Processes Arbitrages and pricing of stock options 29
◮ Stock prices follow a geometric Brownian motion X(t) = X(0)eY (t)
◮ Q: What is the expected return E
◮ Note first that E
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◮ Since Y (t) ∼ N(µt, σ2t)
−∞
2σ2t dy
◮ Completing the squares in the argument of the exponential we have
◮ The blue term does not depend on y, red integral equals 1
2
−∞
2 2σ2t
2
◮ Putting the pieces together, we obtain
Introduction to Random Processes Arbitrages and pricing of stock options 31
◮ Compute expected return as if q were the actual distribution
◮ Expected cash value is different because prob. distribution is different
◮ Measure q is called risk neutral measure
◮ Pricing of derivatives, e.g., options, is always based on expected returns
Introduction to Random Processes Arbitrages and pricing of stock options 32
◮ A continuous-time process X(t) is a martingale if for t, s ≥ 0
◮ Model of a fair, e.g., gambling game. Excludes winning strategies
◮ For risk-neutral measure q, time 0 prices e−αtX(t) form a martingale
◮ Key principle: stock price = expected discounted payoff
Introduction to Random Processes Arbitrages and pricing of stock options 33
◮ Recall measure q is a geometric Brownian motion X(t) = eY (t)
Introduction to Random Processes Arbitrages and pricing of stock options 34
Introduction to Random Processes Arbitrages and pricing of stock options 35
◮ An option is a contract to buy shares of a stock at a future time
◮ Strike time t = Convened time for stock purchase ◮ Strike price K = Price at which stock is purchased at strike time
◮ At time t, option holder may decide to
◮ May buy option at time 0 for price c ◮ Q: How do we determine the option’s worth, i.e., price c at time 0? ◮ A: Given by the Black-Scholes formula for option pricing
Introduction to Random Processes Arbitrages and pricing of stock options 36
◮ Let eαt be the compounding of a risk-free investment ◮ Let X(t) be the stock’s price at time t
◮ Risk neutral measure q is also a geometric Brownian motion
Introduction to Random Processes Arbitrages and pricing of stock options 37
◮ At time t, the option’s worth depends on the stock’s price X(t) ◮ If stock’s price smaller or equal than strike price ⇒ X(t) ≤ K
◮ Since had paid c for the option at time 0, lost c on this investment
◮ If stock’s price larger than strike price ⇒ X(t) > K
◮ To obtain return express as time 0 values and subtract c
◮ May combine both in single equation ⇒ r = e−αt
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◮ Select option price c to prevent arbitrage opportunities
◮ From above condition, the no-arbitrage price of the option is
Introduction to Random Processes Arbitrages and pricing of stock options 39
◮ Let us evaluate Eq
◮ Recall q is a geometric Brownian motion ⇒ X(t) = X0eY (t)
◮ Can rewrite no arbitrage condition as
−∞
Introduction to Random Processes Arbitrages and pricing of stock options 40
◮ Note that
◮ Because integrand is null for Y (t) ≤ log(K/X0) can write
log(K/X0)
◮ Change of variables z = (y − µt)/
◮ Option price then given by
a
√ σ2tz+µt − K
Introduction to Random Processes Arbitrages and pricing of stock options 41
◮ Separate in two integrals c = e−αt(I1 − I2) where
a
√ σ2tz+µte−z2/2 dz
a
◮ Gaussian Φ function (ccdf of standard normal RV)
x
◮ Integral I1 requires some more work
Introduction to Random Processes Arbitrages and pricing of stock options 42
◮ Reorder terms in integral I1
a
√ σ2tz+µte−z2/2 dz = X0eµt
a
√ σ2tz−z2/2 dz ◮ The exponent can be written as a square minus a “constant” (no z)
◮ Substituting the latter into I1 yields
a
−
√ σ2t 2/2+σ2t/2 dz
a
−
√ σ2t 2/2 dz
Introduction to Random Processes Arbitrages and pricing of stock options 43
◮ Change of variables u = z −
◮ Implementing change of variables in I1
b
◮ Putting together results for I1 and I2
◮ For non-arbitrage stock prices (measure q) ⇒ α = µ + σ2/2
Introduction to Random Processes Arbitrages and pricing of stock options 44
◮ Black-Scholes formula for option pricing. Option cost at time 0 is
◮ Note further that µ = α − σ2/2. Can then write a as
◮ Black-Scholes formula independent of stock’s mean tendency µ
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◮ Arbitrage ◮ Investment strategy ◮ Bets, events, outcomes ◮ Returns and earnings ◮ Arbitrage theorem ◮ Geometric Brownian motion ◮ Stock flip ◮ Time value of money ◮ Continuously-compounded interest ◮ Present value ◮ Risk-free investment ◮ Expected return ◮ Risk premium ◮ Risk neutral measure ◮ Pricing of derivatives ◮ Stock option ◮ Strike time and price ◮ Option price ◮ Stock volatility ◮ Black-Scholes formula
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