Simple Sports Betting

  • Odds: X:1 odds on an event happening means that, for every one dollar you would lose if the event happens, you gain X dollars if it doesn’t happen
    • In this class, we used fractional odds: a-to-b odds, written as a/b
    • For every b dollars bet, you get a dollars in profit
  • Hedging: Managing investments/bets to eliminate risk; guarantee profit
Assume you have N dollars and that there are two sets of odds: OA:1 on A and OB:1 on B.Finding guaranteed profit:πA=πB=p∗OB(n−x∗)−x∗=OA(x∗)−(n−x∗)x∗=n(1+OB)2+OA+OBp∗=n(OAOB−1)2+OA+OBRange of x values such that profit is always positive:Profit if A happens: −x+OB(n−x)>0x<OBn1+OBProfit if B happens: −(n−x)+OAx>0x>n1+OAn1+OA<x<OBn1+OB\text{Assume you have N dollars and that there are two sets of odds: } O_A : 1 \text{ on A and } O_B : 1 \text{ on B.} \\ \text{Finding guaranteed profit:} \\ \pi _A = \pi _B = p^* \\ O_B (n - x^*) - x^* = O_A (x^*) - (n - x^*) \\ x^* = \frac{n(1 + O_B)}{2 + O_A + O_B} \\ p^* = \frac{n(O_A O_B - 1)}{2 + O_A + O_B} \\ \text{Range of x values such that profit is always positive:} \\ \text{Profit if A happens: } -x + O_B (n - x) > 0 \\ x < \frac{O_B n}{1 + O_B} \\ \text{Profit if B happens: } -(n - x) + O_A x > 0 \\ x > \frac{n}{1 + O_A} \\ \frac{n}{1 + O_A} < x < \frac{O_B n}{1 + O_B}
  • A fair bet is when the expected value of winning is 0
    • The sum of implied probabilities in a fair bet equals 100%
    • Bookmakers set the sum of implied probabilities to over 100% in order to ensure a profit
    • If you are offered a set of bets with a sum of implied probabilities being less than 100%, then you have an arbitrage opportunity
Odds of A happening if OA:1 odds are given is OA1+OA\text{Odds of A happening if } O_A : 1 \text{ odds are given is } \frac{O_A}{1 + O_A}

Long and Short Option Pricing

Option Definitions

  • Option: A contract that allows you to purchase some product at a specified value at the expiration date
  • Strike price: The cost of purchasing an option at the current moment
  • Owning the option allows you to take an action; should always cost something
    • Owning an option is known as being long, and the cost of an option is calld a premium
  • Selling an option means you sell the rights to a counterparty and must allow them to exercise their action
    • Selling an option is known as being short
  • Options are derivative securities, as they derive their values from the prices of other securities like stocks, gold, or oil
  • Call Option $C_E (S)$
    • A long call option is the right to buy an asset S at an agreed value E (aka the strike or exercise price) at a future date
  • Put Option $P_E (S)$
    • A long put option is the right to sell an asset S at an agreed value E on a future date
  • Four types of options: long put, short put, long call, short call
    • Linear combinations of puts and calls can be created for custom payoff structures
  • Options are defined by the strike/exercise price E, expiration date, and exercise style
    • American style: can be exercised any time before expiration
    • European style: can be exercised only at expiration
  • Options can be in/at/out of the money depending on S compared to E

Values at Expiration

  • For a long call, its value at expiration is $max(0, S-E)$
    • If the asset price is less than the strike price, then the option is worthless
    • Any difference in strike and asset price is profit, adds value to the option

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  • For a short call, you only get the profit of the strike price and incur a loss depending on the asset value

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  • For a long put, you are given the option of selling a stock S at the strike price E; you want the stock S to go down in price in order to make profit

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  • For a short put, you want the stock price to go up; risk is bounded at -E

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  • You can put together puts and calls to make portfolios and minimize risk

Put-Call Parity

  • Basic compounding equation
    • $P_{t + n} = (1 + r)^n P_t$
  • Compounding equation where interest compounds M times per year
    • $P_{final} = (1 + r/M)^M P_0$
    • $\lim_{M \rightarrow \infty} (1 + r/M)^M = e^r$
      • Derived from the fact that $\lim_{n \rightarrow \infty} (1 + 1/n)^n = e$
    • Continuous compounding is better for modeling compared to discrete modeling
    • Final compounding model: $P_{final} = e^{rt} P_0$, where r is the APR and t is the number of years
    • Present value of money: $P_0 = P_{final} e^{-rt}$
  • Value of money over time
    • $\Delta M(t) = r M(t) \Delta t \rightarrow \sum \frac{\Delta M}{M} = r \sum \Delta t$
    • As the change in t goes to 0, the equation becomes $\int \frac{1}{M} dM = r \int dt$
    • If $E = M(T = t)$, then $\int_{M(T)}^E \frac{1}{M} dM = r\int_{t}^T dt \rightarrow M(t) = Ee^{-r(T - t)}$
  • Put-Call Parity Equation
Consider two portfolios. Having a stock and the right to sell it: S+PE(S,t)Having the right to buy a stock and enough money to purchase it: CE(S,t)+Ee−r(T−t)The payoff of these portfolios will have identical values at expiration in the cases E>S,E≤STherefore, S+PE(S,t)=CE(S,t)+Ee−r(T−t); this is the Put-Call Parity Equation.\text{Consider two portfolios. } \\ \text{Having a stock and the right to sell it: } S + P_E (S,t)\\ \text{Having the right to buy a stock and enough money to purchase it: } C_E(S, t) + Ee^{-r(T-t)} \\ \text{The payoff of these portfolios will have identical values at expiration in the cases } E > S, E \leq S \\ \text{Therefore, } S + P_E (S,t) = C_E(S, t) + Ee^{-r(T-t)} \text{; this is the Put-Call Parity Equation.}
  • $\text{If } S + P_E (S,t) > C_E(S, t) + Ee^{-r(T-t)} \text{, then there is an arbitrage opportunity}$
    • Sell the lefthand side and buy the righthand side, depositing $Ee^{-r(T-t)}$ in the bank
    • Arbitrage profit: $S + P_E (S,t) - C_E(S, t) - Ee^{-r(T-t)} > 0$
    • This portfolio ($-S - P_E + C_E + E$) will costlessly liquidate
    • At expiration, if $S \geq E$:
      • The put option is worthless and won’t be exercised
      • Use the call option to buy the stock at a lower price E using your bank balance
      • Use the stock you purchased to cover the short sale of S
    • If $S < E$:
      • The call option is worthless and you won’t use it
      • Use the bank balance E to buy the stock at the strike price E when the put option you sold is exercised
      • Use the stock you purchased to cover the short sale of S
  • $S + P_E (S,t) < C_E(S, t) + Ee^{-r(T-t)} \text{ case}$
    • Buy the lefthand side and sell the righthand side, borrowing $Ee^{-r(T-t)}$ from the bank
    • At expiration, if $S \geq E$:
      • The call option will be exercised and you must sell the stock you own at price E
      • Use the money you got from the sale to payoff the bank loan
      • The put option is worthless and wont’t be used

Cheat Sheet

Finding range of profitable values:

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Taylor series formula:

One variable: ∑n=0∞f(n)(a)n!(x−a)nTwo variables: ∑j=0∞∑k=0∞1j!k!(∂j+kf(a1,a2)∂xj∂yk)(x−a1)j(y−a2)k\text{One variable: } \sum^\infty_{n=0} \frac{f^{(n)} (a)}{n!} (x-a)^n \\ \text{Two variables: } \sum^\infty_{j=0} \sum^\infty_{k=0} \frac{1}{j!k!} (\frac{\partial ^{j+k} f(a_1, a_2)}{\partial x^j \partial y^k}) (x - a_1)^j(y - a_2)^k

Calculating PDF from known distribution:

Suppose Y=f(X) and the PDF/CDF of X is known. The PDF of Y can be derived as follows:Fy(y)=∫g(y)h(y)fx(x)dxfy(y)=ddy∫g(y)h(y)fx(x)dx=fx(h(y))h′(y)−fx(g(y))g′(y)\text{Suppose } Y = f(X) \text{ and the PDF/CDF of X is known. The PDF of Y can be derived as follows:} \\ F_y(y) = \int_{g(y)}^{h(y)} f_x(x)dx \\ f_y(y) = \frac{d}{dy}\int_{g(y)}^{h(y)} f_x(x)dx = f_x(h(y))h'(y) - f_x(g(y))g'(y)

Shannon’s Demon

  • One can gain expected returns if they rebalance their portfolio
  • Consider a portfolio that invests in a stock that either doubles or halves after each day; if we always keep half of the portfolio in cash, then we are expected to make money
  • Tells us that we can harvest volatility and make money even if the drift of a stock is 0

Black-Scholes

  • Aims to price a European option before expiration
    • Two aspects: intrinsic value (what is it worth if it expired right now) and time value (speculative aspect)

Modeling Stock Prices

  • Price includes a predictable component and a random (stochastic) component
  • Random walk model: $\frac{dS}{S} = \mu dt + \sigma dw \rightarrow dS = \mu Sdt + \sigma Sdw$
    • Predictable component: $\mu dt$, where $\mu$ is the average annual growth rate
    • Random component: $\sigma dw$ where $\sigma$ is a measure of the standard deviation of annual returns and dw is a random number generated from $N(0, dt)$
      • This means that $W_t \sim N(0, t)$
  • If $S(0) = S_0$, then $S_t = S_0 \exp\left(\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma W_t\right)$
    • By taking the natural log of both sides, we find that $\ln(S_t)$ is normally distributed, so $S_t$ is log-normally distributed
    • $\ln(S_t) \sim N(\ln(S_0) + (\mu - \frac{1}{2}\sigma^2)t, \sigma^2 t)$
    • $E[\ln(S_t)] = \ln(S_0) + (\mu - \frac{1}{2}\sigma^2)t$
    • $Var[\ln(S_t)] = \sigma^2 t$
    • Mode is located at $e^{\mu - \sigma^2}$

Ito’s Lemma

Uses the fact that $(\Delta t)^2 = 0$ to eliminate terms in the second order Taylor expansion

ΔS=μSΔt+σSΔXΔf(S,t)≈σS∂f∂SΔX+[μS∂f∂S+12σ2S2∂2f∂S2+∂f∂t]Δt\Delta S = \mu S \Delta t + \sigma S \Delta X\\ \Delta f(S,t) \approx \sigma S \frac{\partial f}{\partial S} \Delta X + \left[\mu S \frac{\partial f}{\partial S} + \frac{1}{2}\sigma^2S^2 \frac{\partial^2 f}{\partial S^2} + \frac{\partial f}{\partial t}\right]\Delta t

Example: $f(S,t) = \ln S$

ΔS=μSΔt+σSΔXΔln⁡S≈σS∂ln⁡S∂SΔX+[μS∂ln⁡S∂S+12σ2S2∂2ln⁡S∂S2+∂ln⁡S∂t]Δt∂ln⁡S∂S=1S,∂2ln⁡S∂S2=−1S2,∂ln⁡S∂t=0Δln⁡S≈σΔX+(μ−12σ2)Δt\begin{gather*} \Delta S = \mu S\Delta t + \sigma S\Delta X\\ \Delta \ln S \approx \sigma S\frac{\partial \ln S}{\partial S}\Delta X + \left[\mu S \frac{\partial \ln S}{\partial S} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 \ln S}{\partial S^2} + \frac{\partial \ln S}{\partial t}\right]\Delta t \\ \frac{\partial \ln S}{\partial S} = \frac{1}{S}, \frac{\partial^2 \ln S}{\partial S^2} = -\frac{1}{S^2}, \frac{\partial \ln S}{\partial t} = 0\\ \Delta \ln S \approx \sigma \Delta X + \left(\mu - \frac{1}{2}\sigma^2\right)\Delta t \end{gather*}

Base Formula

  • Assume that our portfolio is given by $\Pi = V - \delta S$; we buy one option and sell $\delta$ units of stock
  • $d\Pi = dV - \delta dS$; plug in the Ito’s Lemma value for $dV$ and the fact that $dS = \mu Sdt + \sigma SdX$
  • Set $\delta = \frac{\partial V}{\partial S}$ to eliminate the random portion $dX$
  • Market formula: $d\Pi_{market} = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)$
  • Arbitrage free world; this derivative should equal to risk-free interest rate from banks: $d\Pi_{bank} = r\Pi dt$
  • Setting these equal to each other, we find that $\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$
  • Plug different formulas into $V$ to get the value of European options

Option Formulas

  • Call option: $C_E(S, t) = SN(d_1) - Ee^{-r(T-t)}N(d_2)$
  • Put option: $P_E(S, t) = Ee^{-r(T-t)}N(-d_2) - SN(-d_1)$
  • $d_1 = \frac{\ln(S/E) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}$
  • $d_2 = \frac{\ln(S/E) + (r - \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} = d_1 - \sigma\sqrt{T-t}$
  • Normal distribution: $N = \Phi$, $N’ = \phi$

Greeks Formulas

  • Delta ($\delta$)
    • $\frac{\partial C_E(S,t)}{\partial S} = N(d_1) \geq 0$
    • $\frac{\partial P_E(S,t)}{\partial S} = -N(-d_1) = N(d_1) - 1 \leq 0$
  • Gamma ($\Gamma$)
    • $\frac{\partial^2 C_E(S,t)}{\partial S^2} = \frac{\partial^2 P_E(S,t)}{\partial S^2} = \frac{\phi(d_1)}{S\sigma\sqrt{T-t}}$
  • Vega ($\nu$)
    • $\frac{\partial C_E(S,t)}{\partial \sigma} = \frac{\partial P_E(S,t)}{\partial \sigma} = S\sqrt{T-t}\phi(d_1) \geq 0$
  • Rho ($\rho$)
    • $\frac{\partial C_E(S,t)}{\partial r} = E (T-t)e^{-r(T-t)}\Phi(d_2) \geq 0$
    • $\frac{\partial P_E(S,t)}{\partial r} = - E(T-t)e^{-r(T-t)}\Phi(-d_2) \leq 0$
  • Theta ($\theta$)
    • $\frac{\partial C_E(S,t)}{\partial t} = - \frac{S\sigma\phi(d_1)}{2\sqrt{T-t}} - Ere^{-r(T-t)}\Phi(d_2) \leq 0$
    • $\frac{\partial P_E(S,t)}{\partial r} = -\frac{S\sigma\phi(d_1)}{2\sqrt{T-t}} + Ere^{-r(T-t)}\Phi(-d_2)$; can be positive or negative depending on the values of E and S

American Options

  • The value of an American call option should be above $S - E$; if $S - E$ is greater than the value of the option, then there is risk-free arbitrage
    • This means that the value of an American call option should equal the price of a European call option
  • The value of an American put option should generally be below $E - S$, but as S gets lower, it might be above $E - S$ due to how little profit is made
    • American puts have different values than European puts because being able to exercise it early can give value, especially when $S$ is close to 0