The Implied Views of Bond Traders on the Spot Equity Market

This study delves into the temporal dynamics within the equity market through the lens of bond traders. Recognizing that the riskless interest rate fluctuates over time, we leverage the Black-Derman-Toy model to trace its temporal evolution. To gain insights from a bond trader's perspective, we focus on a specific type of bond: the zero-coupon bond. This paper introduces a pricing algorithm for this bond and presents a formula that can be used to ascertain its real value. By crafting an equation that juxtaposes the theoretical value of a zero-coupon bond with its actual value, we can deduce the risk-neutral probability. It is noteworthy that the risk-neutral probability correlates with variables like the instantaneous mean return, instantaneous volatility, and inherent upturn probability in the equity market. Examining these relationships enables us to discern the temporal shifts in these parameters. Our findings suggest that the mean starts at a negative value, eventually plateauing at a consistent level. The volatility, on the other hand, initially has a minimal positive value, peaks swiftly, and then stabilizes. Lastly, the upturn probability is initially significantly high, plunges rapidly, and ultimately reaches equilibrium.


Introduction
Our study begins with an exploration of the market, as characterized by the SPDR S&P 500 ETF Trust (SPY).We develop a binomial tree for this market, in line with the conventions of previous work on binomial option pricing (Hu et al., 2022), ensuring the preservation of the market's mean and natural upturn probability.Following this foundational understanding, we transition into the evaluation of the term structure of interest rates (TSIR).
A common approach within the TSIR involves drawing conclusions based on treasury data and subsequent hypotheses.It is noteworthy that standard TSIR models frequently assign riskneutral probabilities as follows: (p, q) = ( 12 , 1 2 ).However, this assumption is challenged by real-world data from the financial market, as highlighted by (Hu et al., 2022) and (Shreve, 2004).
A possible avenue then emerges: to decipher the risk-neutral dynamics, even if only through a largely theoretical model, that align with bond prices.By leveraging such probabilities, models such as (Hu et al., 2022) can offer a valuation of derivatives on the TSIR.
In the context of this paper, our meticulous preservation of the equity market's mean, natural upturn probability, and volatility in our bond valuation opens up an avenue to discern bond traders' perspective on these parameters.This perspective is in contrast to the implied probabilities perceived by option traders, a nuance previously overlooked by traditional TSIR models.
The crux of this paper, therefore, is to explore the nuances of the equity market through the lens of bond traders.This research is organized into three pivotal sections.
First, we elucidate the Black-Derman-Toy (BDT) model, tracing its historical significance in mathematical finance.Our emphasis remains primarily on the discrete incarnation of the BDT model, which offers insights into the market's temporal riskless interest rate transformations.
For the computation of the associated BDT model coefficients, we harness the prior 252 days of U.S. 10-year treasury bond interest rate data, which are processed using MATLAB.
Next, our study pivots to the vantage point of bond traders, introducing the concept of the zero-coupon bond.Through a binomial model, we endeavor to derive a systematic pricing algorithm for this bond; this is followed by a method that can be used to ascertain its real value.By establishing a relationship between the theoretical and real values of the zero-coupon bond, we identify the risk-neutral probability.Given that a simplistic risk-neutral probability assumption of (p, q) = (1 2 , 1 2 ) might yield unrealistic outcomes 1 , we expand this concept to derive parameters such as the instantaneous mean return, volatility, and natural upturn probability.This is further bolstered with simulated data, which are used to interpret these parameters over time.
In our conclusion, we discuss insights into the evolutionary trajectory of the equity market over the span of our study.

The History of the BDT Model
The BDT model, named after its creators, Black, Derman, and Toy, originated within Goldman Sachs during the 1980s.Initially conceived for the firm's internal utilization, its significance garnered broader recognition following its publication in the Financial Analysts Journal in 1990 (Black et al., 1990).Derman offers a first-hand account of the model's evolution in his 2004 memoir Derman (2004).
Within the realm of mathematical finance, the BDT model is highly regarded as a short-rate model that is instrumental in pricing bond options, swaptions, and various other interest rate derivatives.It is different from other finance models in that it is a one-factor model, meaning that the short rate serves as the sole stochastic determinant for the future trajectory of all interest rates.The BDT model pioneered the amalgamation of the mean-reverting behavior of the short rate with the log-normal distribution, a paradigm shift underscored by (Buetow et al., 2001).Its applicability and relevance persist today, as noted by (Fabozzi, 2007).

Theoretical Support
In (Shreve, 2004), one can find a presentation of the BDT model in the discrete case.In the present paper, we will also consider this form of the BDT model.Because the interest rate in the BDT model changes over time, we can consider the interest rate as a risky asset R and suppose that at every point in time, the interest rate will strictly go up or not.Additionally, we suppose that the interval between two adjacent points in time is ∆ = 1/2522 .Now, we define the interest rate return3 in the period [n∆, (n + 1)∆]: For every n in the range 0, • • • , N − 14 , the interest rate return can be described as follows: where the sequence of returns, denoted by r n∆5 for n = 1, • • • , N , comprises identically distributed binary random variables.
Furthermore, let the expected return of r (n+1)∆ be represented as E[r (n+1)∆ ] = µ rate (n+1)∆ = µ rate ∆, where µ rate stands for the instantaneous mean return of the interest rate.Therefore, the relationship becomes which implies that Now, let the variance of r (n+1)∆ be represented as follows: where (σ rate ) 2 is the instantaneous variance of the interest rate.By introducing Given that, By combining ( 4) and ( 5), we can derive By combining the expressions for µ rate and ν rate ∆ in (3) and ( 9), we obtain the following system of equations: ( If we treat µ rate and ν rate ∆ as constants and u and d as unknown variables, the solution of (10) should be Now, by combining (1), (2), and (11), we obtain for Next, considering the format of the BDT model on page 172 in (Shreve, 2004), • #H(ω 1 • • • ω n ) counts the occurrences of H in the given n periods, representing the number of times the interest rate increases.
• a n and b n are coefficients used to calibrate the model.
Following the form of a n and b n in (Shreve, 2004), we assume that a n = R 0 /c n 1 and b n = c 2 , where R 0 is the interest rate at time 0, while c 1 and c 2 are constants.By combining this assumption with (13), we deduce the following: From relations ( 14) and ( 15), we can write The coefficients in ( 16) and ( 17) represent our revised BDT model for this study.

Zero-Coupon Bond Pricing Algorithm
A zero-coupon bond can be viewed as a specific type of European contingent claim8 (ECC).
This characterization is due to its unique features: it has a predetermined maturity date, cannot be traded prior to its maturity, and guarantees a payoff of $1 upon maturity regardless of external conditions.
Let us denote the price of the zero-coupon bond at time t, with the bond reaching maturity at time T , as B(t, T ).It is a given that B(T, T ) ≡ 1.To price this bond, we adopt a traditional binary model, as described by (Shreve, 2004).Using the notation established earlier, • ∆ represents the time interval, which is set to 1/252.
• B(n∆, T ) indicates the bond price at time n∆, with a maturity T = N ∆.
• B[(n + 1)∆, T ] (u) and B[(n + 1)∆, T ] (d) are potential bond prices at time (n + 1)∆, with the former being greater than B(n∆, T ) and the latter being less than or equal to B(n∆, T ).
• R n∆ is the interest rate at n∆, derived using the BDT model.
• p is the risk-neutral upturn probability in the equity market.
Given the bond's ECC nature, it should be backed by an underlying risky asset S. To ascertain the risk-neutral probabilities, one could design a risk-free portfolio: 3.2 Linking Zero-Coupon Bonds to the Equity Market Our objective in this subsection is to deduce the implied value of the natural upturn probability, instantaneous mean return, and instantaneous volatility for the equity market by analyzing them through the lens of a zero-coupon bond.
The expectation is that the zero-coupon bond's price, as calculated using the algorithm detailed in the previous subsection, aligns with the bond market's price9 .To ascertain the price of the zero-coupon bond within the bond market, we refer to (Hull, 2022) and utilize the following formula: where • t represents the present time.
• T denotes the maturity date of the specific zero-coupon bond.(For robustness in our findings, we will be considering bonds with maturities spanning from two months to thirty years.) • Y (t, T ) is the yield to maturity.
The algorithm we developed in the preceding subsection provides us with a means to calculate the theoretical price of the zero-coupon bond.This can be represented as follows: B(t, T, µ e.m. , p, σ e.m. ; theoretical value).
Our next step entails equating equations ( 27) and ( 28).Doing this will reveal the implied values for the natural upturn probability, instantaneous mean return, and instantaneous volatility, all of which are vital metrics for understanding the equity market when it is viewed through the prism of a zero-coupon bond.

Estimation of Parameters for the Equity Market
To deduce the values of the parameters µ e.m. , p, and σ e.m. , we will utilize the data associated with the SPY from June 15, 2022 to June 16, 2023.This duration encompasses precisely 252 business days.Our data source will be Yahoo Finance.
The rationale behind selecting these data is multifaceted.Bond prices often mirror the broader economic landscape.Given that the SPY can act as a representative index for the economy, it is plausible to infer that SPY data offer insight not only into the economy's state but also into the nuances of the equity market.Our preliminary estimates for µ e.m. , p, and σ e.m.
As mentioned in the preceding subsection, our task is to equate ( 27) and ( 28).By doing this, we can ascertain the value of the risk-neutral upturn probability p.By leveraging equation ( 26) and concentrating on the parameters p, µ e.m. , and σ e.m. , we find that any two of the parameters can be utilized to express the third.Specifically, From these graphs, we can observe the following trends: • In Figure 2, the implied return rises sharply from approximately -0.11.It then starts to climb at a decelerated pace and eventually stabilizes around 0.025.
• Figure 3 shows that the volatility initially surges, reaching its zenith at 0.04 by the 10th year.Following this peak, there is a slight decline, and it settles near 0.035.
• As for Figure 4, the implied upturn probability experiences a sharp dip from its starting point of roughly 0.83.This decline slows over time, with the probability ultimately converging to approximately 0.48.

Conclusion
In this study, we delved into the temporal evolution of the SPY index within the equity market through the lens of bond traders.Our focus has been on deducing the implied values of the instantaneous mean return, instantaneous volatility, and natural upturn probability over time.Our primary conclusions can be summarized as follows: • The implied value of the instantaneous mean return is initially negative.However, with the passage of time, this value experiences an increase, eventually reaching a stable point.
• As for the implied value of the instantaneous volatility, it initially has a marginal positive value.Shortly thereafter, it peaks before possibly descending to a steady state.
• The implied upturn probability starts at a notably high value but experiences a pronounced decline as time progresses.It too attains stability over time.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :
Figure 1: Comparison of market and BDT model interest rates